Growing 3 1 & 3 2
View more documents from Kathy Favazza.
Wednesday, March 31, 2010
Tuesday, March 30, 2010
Math Reflection
1) You can recognize an exponential decay pattern from a table of data if it is divided my the same number each time.
2) You can recognize an exponential decay pattern from a graph when the line slopes down.
3) You can tell that an equation represents exponential decay if the growth factor is less than 1.
4) Exponential growth relationships and exponential decay relationships are similar because they both use exponents, they have growth/decay rates, and they both have growth/decay factors. They are different because exponential growth relationships have a growth factor greater than 1, when an exponential decay's is less than 1.
5) Exponential decay relationships and decreasing linear relationships are similar because they both decrease. They are different because exponential decay relationships are divided by the same number each time, while decreasing linear relationships get the same number smaller each time.
2) You can recognize an exponential decay pattern from a graph when the line slopes down.
3) You can tell that an equation represents exponential decay if the growth factor is less than 1.
4) Exponential growth relationships and exponential decay relationships are similar because they both use exponents, they have growth/decay rates, and they both have growth/decay factors. They are different because exponential growth relationships have a growth factor greater than 1, when an exponential decay's is less than 1.
5) Exponential decay relationships and decreasing linear relationships are similar because they both decrease. They are different because exponential decay relationships are divided by the same number each time, while decreasing linear relationships get the same number smaller each time.
Monday, March 29, 2010
Math Reflection 3
1.
a. If you know the initial value for a population, and you know the yearly growth rate, you can make an equation. For example, starting population=500, yearly growth rate=80%. To change the growth rate to the growth factor, you change the percent to a decimal, and add 1. The equation for this situation would be 500(1.8^x). If you want to find out the population after, say, 5 years, just plug in 5 for x, 500(1.8^5). The answer is 9447.84.
b. The growth rate of an equation is the growth factor of an equation, times 100 (as a percent). If the growth factor was 1.6, the growth rate would be 160%.
2.
a. If you know the initial value of a population and the yearly growth rate, you can make an equation. If the beginning population was 200, and yearly growth factor was 1.3, the equation would be 200(1.3^x). If you wanted to know the population 7 years from now, you would just plug in 7 for the x. The answer is 200(1.3^7), or 1254.97034.
b. You can determine the yearly growth rate by multiplying the growth factor by 100.
3. To get the population when it doubles, just multiply the starting value by 2. Then, guess and check what the x value is(the year), when the population doubles. When it is about the starting value times 2, that is it.
a. If you know the initial value for a population, and you know the yearly growth rate, you can make an equation. For example, starting population=500, yearly growth rate=80%. To change the growth rate to the growth factor, you change the percent to a decimal, and add 1. The equation for this situation would be 500(1.8^x). If you want to find out the population after, say, 5 years, just plug in 5 for x, 500(1.8^5). The answer is 9447.84.
b. The growth rate of an equation is the growth factor of an equation, times 100 (as a percent). If the growth factor was 1.6, the growth rate would be 160%.
2.
a. If you know the initial value of a population and the yearly growth rate, you can make an equation. If the beginning population was 200, and yearly growth factor was 1.3, the equation would be 200(1.3^x). If you wanted to know the population 7 years from now, you would just plug in 7 for the x. The answer is 200(1.3^7), or 1254.97034.
b. You can determine the yearly growth rate by multiplying the growth factor by 100.
3. To get the population when it doubles, just multiply the starting value by 2. Then, guess and check what the x value is(the year), when the population doubles. When it is about the starting value times 2, that is it.
Math Reflection 3
1. A) If you know the initial value and the growth rate of the population you can make an equation using the format y=a(b)^n. Say the initial value was 1000 and the growth rate was 4%, you first need to make the growth factor. You do this by making the percent into a decimal, (.04), then you add 1, (1.04), your equation would be y=1000(1.04)^n. to figure out the population in a certain amount of years you just put the number in as x and solve.
1. B) The growth rate is related to the growth factor because the growth factor is the rate in decimal form plus 1.
2. A) You do the same as above. You take the initial value and growth factor and put them into an equation. Then you put the number of years s x then solve.
2. B) You can determine the growth rate by first subtracting 1. Then you multiply the decimal by 100 to get the percent.
3. To find the doubling point of the population you take the initial value and double it. So now the equation is y=2p(b)^n. From there you take an educated guess to see it when you plug x in (number of years) it doubles. If it doesn't you go up or down from there to see when the doubling point is.
Math Reflection
1. a) To determine the population many years from now you have to figure out the growth factor. To do this, you change the percent to a decimal and add one. For example, if the growth rate is 20%, the growth factor is 1.2. You would then take the growth factor and the initial value and put it into this equation: y=a(b)^x.
y=total, a=initial value, b=growth factor, x=number of years.
If the initial value is 200 and the growth factor was 1.2 the equation would be y=200(1.2)^x.
b) The growth rate is related to the growth factor because they are the same thing just written differently. To change the growth rate (which is a percent) to the growth factor (a number...usually a decimal) you divide by 100 and add 1. To change the growth factor to the growth rate you subtract 1 and multiply by 100. (It's doing the opposite of changing the growth rate to the growth factor).
2. a) If you know the initial value and the growth factor you can just plug it into this equation to find out the total for any number of years. The equation is: y=a(b)^x.
y=total, a=initial value, b=growth factor, x=number of years.
b) To determine the yearly growth rate you subtract 1 from the growth factor and multiply by 100.
3. If you know the equation that represents the exponential relationship between the population and the number of years you can determine the doubling time for the population by first figuring out 2p. After that you plug that value into the equation p=a(b)^x to solve for x. The x will give you how long it will take for the population to double.
Math Reflection from Page 47
#1 When you know the initial value and yearly growth rate, it is simple to determine a population from year to tear. First, convert the growth rate to a growth factor by putting the growth factor into decimal form and adding 1. For example if the growth rate is 7% then 7% in decimal form is 0.07 and 0.07+1=1.07. The growth factor is 1.07. Next, start the equation with the initial population. Then multiply that with the growth rate to the power of the year. For example, if the initial population is 1,000 and the growth factor is 1.07, then the population in 4 years corresponds to the equation 1000(1.07)^4. The population is about 1310 in 4 years. Growth factor and growth rate are closely related. Growth rate is really growth factor but in percent form without including the current sum. Growth factor is growth rate plus one and in the decimal form.
#2 If you know both the initial value and the growth factor, finding a population in a few years is even simpler. All you need to do is multiply the initial value times the growth rate to the power of the year. If the initial population is 10,000 and the growth rate is 1.1 the population is 6 years is found with the following equation: 10,000(1.1)^6.
#3 To determine the doubling population's sum, you use the following equation p= 2^n. This means population equals the growth factor (2) to the power of the year, n.
Kate :-)
#2 If you know both the initial value and the growth factor, finding a population in a few years is even simpler. All you need to do is multiply the initial value times the growth rate to the power of the year. If the initial population is 10,000 and the growth rate is 1.1 the population is 6 years is found with the following equation: 10,000(1.1)^6.
#3 To determine the doubling population's sum, you use the following equation p= 2^n. This means population equals the growth factor (2) to the power of the year, n.
Kate :-)
Math Reflection 3
1.a. You could make an equation by substituting the growth rate(as a decimal and added to 1) as b and the initial value as a.You then substitute the number of years as the x.
1.b.The growth rate is what percent of the original is added to itself. The growth factor is the original value added to what ever is added to the original value.
2.a.You could make an equation using a=initial value b=growth factor. You could then substitute the number of years for x.
2.b. You make the growth factor a percent value and subtract 100% from it.
3. You figure out the amount of years it takes for the growth factor to equal 2 using guess and check.
1.b.The growth rate is what percent of the original is added to itself. The growth factor is the original value added to what ever is added to the original value.
2.a.You could make an equation using a=initial value b=growth factor. You could then substitute the number of years for x.
2.b. You make the growth factor a percent value and subtract 100% from it.
3. You figure out the amount of years it takes for the growth factor to equal 2 using guess and check.
Math Reflection 3
1a. To determine the population several years from the start using the initial value and the yearly growth rate, you can form an equation. To do this, you use the form of y=a(b^x) (a=initial value, b=growth factor, y=dependent variable, x=independent variable). Also, it's important that you take the growth rate, and convert it into its growth factor by taking the percent and adding 1 to its decimal form.
1b. A growth rate is the percent form of a growth factor. For instance, if a growth rate were 75%, its growth factor would be 1.75 because you have to take in account 100% of the previous value, plus the 75% that is going to be added to form the next value of y.
2a. To determine the population several years from the start using initial value and the yearly growth factor, you will also need to form an equation, only this time, there is no need to turn the growth rate into a growth factor, because there is no growth rate at all. Yet again, you use the form of y=a(b^x).
2b. You can determine the yearly growth rate by subtracting 1 from the number that is the growth factor and turning it into a percent.
3. If you know the equation that represents the exponential relationship between the population size p and the number of years n, you can determine the doubling time for the population by finding what 2p is, then finding the value of n that is closest to that amount.
1b. A growth rate is the percent form of a growth factor. For instance, if a growth rate were 75%, its growth factor would be 1.75 because you have to take in account 100% of the previous value, plus the 75% that is going to be added to form the next value of y.
2a. To determine the population several years from the start using initial value and the yearly growth factor, you will also need to form an equation, only this time, there is no need to turn the growth rate into a growth factor, because there is no growth rate at all. Yet again, you use the form of y=a(b^x).
2b. You can determine the yearly growth rate by subtracting 1 from the number that is the growth factor and turning it into a percent.
3. If you know the equation that represents the exponential relationship between the population size p and the number of years n, you can determine the doubling time for the population by finding what 2p is, then finding the value of n that is closest to that amount.
Math Reflection 3
1.a. With the initial value and yearly growth rate, you can determine the x value several years from now by using the equation y=a(b^x) where a is the initial value, b is the yearly growth factor and x is the given value of x. But, before you write the equation yo must turn the growth rate into a growth factor by adding one to the growth rates decimal value.
b. To turn the growth rate into the growth factor, you add one to the growth rate's decimal value. The growth rate is the percent of growth and the growth factor is what the x value is multiplied each time it goes up.
2.a. If you know the initial value and the growth factor, you simply place the numbers in the equations. Substitute the initial value for a, and the growth factor for b. Then you subsitute the x value for x.
b. To find the growth rate out of the growth factor, you subtract one and turn the decimal into a percent.
3. To find the doubling time, you guess and check until it is just about correct.
mathematical reflection 3
1. a. If you know the initial value and the growth rate for a population you can find the population years from now. You can do this by making the growth rate into the growth factor by making it a decimal and adding 1. You then find the growth factor to the how many years it has been power. Then you multiply the initial value by the total of the exponent problem.
1. b. The growth factor is the percent of growth and the growth rate is how much the y total is multiplied by each time x goes up. You can find the growth factor from the growth rate by putting the growth rate into decimal form and adding 1 to the decimal.
2. a. If you know the growth factor and the initial value for a population equation you can find the population several years from now. You can do this by finding the growth to the exponent of how many years it has been and multipying the initial value by the total of the exponent problem.
2. b. You can find the yearly growth rate by using the growth factor. You can do this by subtracting 1 from the growth factor and making the number you have left a percent.
3. You can find the doubling of a population p for the number of years n by making the equation p=2^ n and multiply that by the intial value if there is an initial value.
1. b. The growth factor is the percent of growth and the growth rate is how much the y total is multiplied by each time x goes up. You can find the growth factor from the growth rate by putting the growth rate into decimal form and adding 1 to the decimal.
2. a. If you know the growth factor and the initial value for a population equation you can find the population several years from now. You can do this by finding the growth to the exponent of how many years it has been and multipying the initial value by the total of the exponent problem.
2. b. You can find the yearly growth rate by using the growth factor. You can do this by subtracting 1 from the growth factor and making the number you have left a percent.
3. You can find the doubling of a population p for the number of years n by making the equation p=2^ n and multiply that by the intial value if there is an initial value.
Mathematical Reflections 3
1. a. If your know the initial value for a population and the yearly growth rate, you can determine the population several years from now. First ou find the growth factor, because the growth factor and the initial value are the two things you need to form an equation. To get the growth factor, you put the growth rate into decimal form, and add one. Then you form an equation using the y=a(b^x) form. You plug in the initial value as A and the growth factor as B. Then you can plug the number of years into the equation to find your answer.
b. The growth rate is the percentage of growth, while the growth factor is the amount of increase. You can find the growth factor from the growth rate by putting the percent into decimal form and adding 1.
2. a. If you know the initial value and the yearly growth factor you can determine the population in several years. To do this, you need to form an equation. You can plug th numbers you have into the form y=a(b^x). A is the initial value, and B is the growth factor. Solve the equation, and you have your answer.
b. You can determine the growth rate by subtracting 1 from the growth factor and turning that number into a percent.
3. You can determine the doubling time by making a table and observing when a number doubles.
b. The growth rate is the percentage of growth, while the growth factor is the amount of increase. You can find the growth factor from the growth rate by putting the percent into decimal form and adding 1.
2. a. If you know the initial value and the yearly growth factor you can determine the population in several years. To do this, you need to form an equation. You can plug th numbers you have into the form y=a(b^x). A is the initial value, and B is the growth factor. Solve the equation, and you have your answer.
b. You can determine the growth rate by subtracting 1 from the growth factor and turning that number into a percent.
3. You can determine the doubling time by making a table and observing when a number doubles.
Mathematical Reflection 3
1a. If we know the initial value for a population and the yearly growth rate we can determine the population any number of years from now. We can find the growth factor from the growth rate by adding 1, then writing in decimal form. After that, plug the values of a(the initial value), b(the growth factor), and x(the number of years) into the equation y=a(b^x), and solve for y.
1b. A growth rate is the percentage growth, while the growth factor is the fractional pattern of increase. We find the growth rate by adding 1 to the growth rate and then dividing by 100 to get the decimal form, or growth factor.
2a. In the equation y=a(b^x), we already have the values of a(the initial value) and b(the yearly growth factor). If we substitute the number of years in for x, we automatically can solve for y.
2b. We can find the yearly growth rate from the yearly growth factor by subtracting 1and writing it in a percentage form.
3. To find the doubling time for the population, find the value of 2p. Then, plug the new value of p into the equation to solve for n.
1b. A growth rate is the percentage growth, while the growth factor is the fractional pattern of increase. We find the growth rate by adding 1 to the growth rate and then dividing by 100 to get the decimal form, or growth factor.
2a. In the equation y=a(b^x), we already have the values of a(the initial value) and b(the yearly growth factor). If we substitute the number of years in for x, we automatically can solve for y.
2b. We can find the yearly growth rate from the yearly growth factor by subtracting 1and writing it in a percentage form.
3. To find the doubling time for the population, find the value of 2p. Then, plug the new value of p into the equation to solve for n.
Math Reflection 3
1a. To determine the population several years after and you have the growth rate and initial population you turn the growth rate into a growth factor then make an equation Y= initial value(growth factor to the x).
1b. The growth rate is the percent that the number is going up by and the growth factor is the number that you put to the xth power.
2a. You can once again make the equation and solve. It will be one less step from 1 because you do not have to change the percent to a decimal and then solve.
2b. You turn the number to a percent then subtract 100.
3. To find the time it will take an initial value to double you must find what x is so that the growth factor can equal 2 so it can double. Another way you could do it is by guessing and checking, you geuss for a number to be x and put the growth factor if it is over 2 it can work and you can try a smaller number.
1b. The growth rate is the percent that the number is going up by and the growth factor is the number that you put to the xth power.
2a. You can once again make the equation and solve. It will be one less step from 1 because you do not have to change the percent to a decimal and then solve.
2b. You turn the number to a percent then subtract 100.
3. To find the time it will take an initial value to double you must find what x is so that the growth factor can equal 2 so it can double. Another way you could do it is by guessing and checking, you geuss for a number to be x and put the growth factor if it is over 2 it can work and you can try a smaller number.
Math Reflection
MATH REFLECTION
1a. Provided you were knew the initial population and growth rate, finding the population x years is really quite simple. All you need to do is turn the growth rate into a growth factor. You do this by turning the percentage given to you as a growth rate and change it to a decimal or integer. You will then add one to the number you get from the growth rate. This number is a vital part of this process, as it will be the base of the exponent in the equation. You would then right an equation for maximum clarity. EXAMPLE:
Initial Population: 100
Growth Rate: 5%
Growth Factor: 1.05
Equation: Y=100(1.05 to the power of x)
You now choose the number of years and sub that in for x. After solving you should know the population.
1b. Growth rate and growth factor are very closely related. The growth rate is a percent and the factor a number. The growth rate shows the rate at which each number changes in the equation. While the growth factor is the number that "physically" represents the rate in the equation and is the base of the exponent in the equation.
2a. If you had the initial population and the growth factor of a situation finding the population after a set number of years would be simple. All you would need would be to right an equation. This is easy because you are provided with all the needed elements of the equation. You would make the population (a variable) equal the initial value multiplied by the growth rate to the power of a variable representing the number of years that have passed.
EXAMPLE
Initial Value: 24
Growth Factor: 1.24
Equation: Y = 24(1.24 to the power of x)
2b. When given the growth factor finding the growth rate is not hard. You turn the growth factor into a percentage and then subtract 100%. You can also subtract one from the factor and then turn in into a percent.
3. In order to determine the doubling point of a population when given the equation of that situation you would need to find the exponent that will bring the growth factor to two. This will then double the initial population. Finding the exponent that doubles the factor is purely educated guess and check.
Nick S.
Sunday, March 28, 2010
Mathematical Reflections 3
1)
a. You can determine the population in several years from now by making a equation out of what you know. The formula is y=a(b^x). Since you know the initial value, plug it in to the "a". Then change the growth rate to a growth factor. You can do this by dividing one hundred and adding 1. Then, plug it into "b". Then, you can plug in the amount of years that you want to solve for in the "x", and solve the equation.
b. The growth rate is the percentage of growth between each year in this case. To find the growth rate from a growth factor is to subtract the 1 and multiply by 100. To find the growth factor from a growth rate is to add the 1 and divide by 100. The reason that the growth rate doesn't have the 1, and the growth factor does is because the growth factor is already the original value and the growth. The growth rate is only the growth and not the original value.
2)
a. As I said before, you could use the equation, y=a(b^x). Plug the initial value into "a". Then, as a step less from question 1a, you can just plug in the growth factor. Then, plug in the amount of years that you want to solve for in the "x" then solve.
b. You can determine the yearly growth rate by the growth factor. Subtract the 1 from the growth factor and multiply by 100.
3) You could do two things to determine the doubling time for the population. You could plug in twice the amount of the initial value in to "y", to make it equal that number. Then, you could solve for "x". Or you can do the guess and check method that I usually use. Plug in twice the initial value for "y", but guess a number for "x" and work your way around that to determine the correct answer.
a. You can determine the population in several years from now by making a equation out of what you know. The formula is y=a(b^x). Since you know the initial value, plug it in to the "a". Then change the growth rate to a growth factor. You can do this by dividing one hundred and adding 1. Then, plug it into "b". Then, you can plug in the amount of years that you want to solve for in the "x", and solve the equation.
b. The growth rate is the percentage of growth between each year in this case. To find the growth rate from a growth factor is to subtract the 1 and multiply by 100. To find the growth factor from a growth rate is to add the 1 and divide by 100. The reason that the growth rate doesn't have the 1, and the growth factor does is because the growth factor is already the original value and the growth. The growth rate is only the growth and not the original value.
2)
a. As I said before, you could use the equation, y=a(b^x). Plug the initial value into "a". Then, as a step less from question 1a, you can just plug in the growth factor. Then, plug in the amount of years that you want to solve for in the "x" then solve.
b. You can determine the yearly growth rate by the growth factor. Subtract the 1 from the growth factor and multiply by 100.
3) You could do two things to determine the doubling time for the population. You could plug in twice the amount of the initial value in to "y", to make it equal that number. Then, you could solve for "x". Or you can do the guess and check method that I usually use. Plug in twice the initial value for "y", but guess a number for "x" and work your way around that to determine the correct answer.
Mathematical Reflections 3
1.a. If the initial value and growth rate of a population are known, you can easily determine the population several years from now. First, you must change the growth rate into the growth factor. The growth factor is usually in decimal form, but most important is that for exponential growth it is at least 1 or greater. Then, you just multiply the initial value by the growth factor raised to the n power.
b. The growth rate represents by how much the initial value increases each year, and the growth factor is what it is multiplied by each year. For example, the growth factor would be 1.7 when the growth rate is 70%. (The growth factor must have a 1 during exponential growth in order to make sure that the initial value is kept along with the increase.)
2.a. To find the population from the initial value and growth factor, you would again use the y=a(b)^x form, except this time you don't need to convert anything.
b. To determine the yearly growth rate, you subtract 1 from the growth factor if it is less than 2.
3. To find out how long it will take for a population to double, all you have to do is make y twice the initial value. Another thing you could do is to keep multiplying by the growth factor until the outcome is double or more the initial value.
b. The growth rate represents by how much the initial value increases each year, and the growth factor is what it is multiplied by each year. For example, the growth factor would be 1.7 when the growth rate is 70%. (The growth factor must have a 1 during exponential growth in order to make sure that the initial value is kept along with the increase.)
2.a. To find the population from the initial value and growth factor, you would again use the y=a(b)^x form, except this time you don't need to convert anything.
b. To determine the yearly growth rate, you subtract 1 from the growth factor if it is less than 2.
3. To find out how long it will take for a population to double, all you have to do is make y twice the initial value. Another thing you could do is to keep multiplying by the growth factor until the outcome is double or more the initial value.
Saturday, March 27, 2010
Math Reflections 3
1a. You can determine the population several years from now by changing the growth rate to a growth factor by dividing by 100 and adding 1. Then you write an exponential equation using the initial population and the growth factor and then substitute the number of years for x.
b. A growth rate is related to the growth factor of a population because they both represent the same thing, except a growth rate is in percent form and the growth factor is in decimal form.
2a. You can determine the population several years from now by writing an exponential equation in y=a(b)^x form where y=the population, a=the initial population, b=the growth factor, and x=the number of years. Then, you solve the equation by sunstituting the number of years for x.
b. You can determine the yearly growth rate by subtracting 1 from the growth factor and changing it to a percent.
3. You can determine the doubling time for the population by substituting twice the initial value for y and solving the equation. You could also guess and check until you find the right number of years.
b. A growth rate is related to the growth factor of a population because they both represent the same thing, except a growth rate is in percent form and the growth factor is in decimal form.
2a. You can determine the population several years from now by writing an exponential equation in y=a(b)^x form where y=the population, a=the initial population, b=the growth factor, and x=the number of years. Then, you solve the equation by sunstituting the number of years for x.
b. You can determine the yearly growth rate by subtracting 1 from the growth factor and changing it to a percent.
3. You can determine the doubling time for the population by substituting twice the initial value for y and solving the equation. You could also guess and check until you find the right number of years.
Math Reflections 3
1.a. You can determine the population several years from now, if you know the initial value in yearly growth rate, by changing the growth rate from a percent to a decimal, and by adding 1 to that decimal. Then you can multiply the initial value by this growth factor as many years in the future as you want to determine the population.
1.b. A growth factor is just the decimal conversion of a growth rate plus 1.
2.a. You can find the population several years from now, if you know the initial value and yearly growth factor, by multiplying the initial value by the yearly growth factor as many times as the number of years from now you want to determine the population.
2.b. You can determine the yearly growth rate by subtracting 1 from the growth factor and converting the value you are left with to a percent.
3. You can find the doubling time for a population from an equation by trying different values for the time variable until the population doubles.
1.b. A growth factor is just the decimal conversion of a growth rate plus 1.
2.a. You can find the population several years from now, if you know the initial value and yearly growth factor, by multiplying the initial value by the yearly growth factor as many times as the number of years from now you want to determine the population.
2.b. You can determine the yearly growth rate by subtracting 1 from the growth factor and converting the value you are left with to a percent.
3. You can find the doubling time for a population from an equation by trying different values for the time variable until the population doubles.
Exponential Decay
Exponential decay is when an equation decreases exponentially. Exponential decay equations have a decay factor instead of a growth factor. For example, a chart with exponential decay might look like this:
X / 1 / 2 / 3 /4/5/6
Y/64/32/16/8/4/2
The equation for this chart would be Y= 64(1/2)^x. Whenever you find the numbers are getting smaller each time, you know that there is a decay factor. The decay factor in this equation is 1/2. The decay factor will always be a decimal/fraction. The initial value in this equation is 64.
X / 1 / 2 / 3 /4/5/6
Y/64/32/16/8/4/2
The equation for this chart would be Y= 64(1/2)^x. Whenever you find the numbers are getting smaller each time, you know that there is a decay factor. The decay factor in this equation is 1/2. The decay factor will always be a decimal/fraction. The initial value in this equation is 64.
Thursday, March 25, 2010
Wednesday, March 24,2010
Yesterday in Algebra 1, we worked on connecting grow factor with growth rate. The growth factor is the number in an exponential function which the previous result is multiplied by in order to get the result for the next interval of the function. For example, in the equation y=100(2.2)* 2.2 is the growth factor because it is the number that *(x) is multiplied by to get the next sum in the function. Once we learned about growth factor, we learned about growth rate. The growth rate is the percent of increase in an exponential function, and it is sometimes given in place of the growth factor. When the population is growing 3% a year, 3% is the growth factor. Growth factor and growth rate are easily interchangeable. When the growth factor is 1.23, the growth rate is 23%. This is because when changing growth rate to growth factor you simply put the percentage in decimal form and add one to it (because you add the current amount to the growth rate when you are solving a function). When the money within a bank account is 4,000, and grows 7% a year, all you have to do to find the amount in it after the first year is do 4000 x 1.07. 7% is really 0.07, and once you add one to include the 4000, you get 1.07 as the growth factor. Switching from the growth factor to the growth rate is just as easy as the oppisite. Subtracting 1 from the growth factor and putting the remaining decimal into a percent is all it consists of. When the growth factor is 1.80, you do 1.80-1=.80 .80=80% and you find that the growth rate is eighty percent when the grwoth factor is 1.80. Growth factor and growth rate are simply connected. By Kate M :-)
Wednesday, March 24, 2010
Math Reflection
1.
a) From a table you can find the y-intercept because when x equals zero it's the y-intercept. You can find the growth factor by dividing one number by the number before it on the graph. Then use the answer to see if it works as the growth factor for the rest of the numbers.
From a graph you can find the y-intercept just by looking where the line crosses the y-axis. You can find the growth factor by seeing how many spaces go up each time it goes up by one.
From and equation you can find the y-intercept by looking at the a in y=a(b)^n. You can find the growth factor by looking at the b.
b) Use the y-intercept as the a and use the growth factor as the b.
2.
a) A represents the y-intercept except when c changes it. B represents the growth factor.
Monday, March 22, 2010
Mathematical Reflections 2
1)
a. You can use a table to find the y-intercept by looking at the y when x is 0. You can find the growth factor from the table by dividing two x's and two y's right in above each other. Then, the division of y over x is the growth factor. On a graph, you basically do the same as a table. Find two points preferably right next to each other. Then the division of the y's over the x's is the growth factor. The y-intercept is found when the x coordinate is 0. The initial equation is y=a(b^x). The a is the y-intercept which is just plainly stated. The b is the growth factor which is also plainly stated.
b. As I said in letter 1a. The variable a is the y-intercept. If you know the y-intercept you can plug it in. For example, let's say 2 is the y-intercept. This is how much of the equation you would have. y=2(b^x). B is the growth factor, and same as before just plug it into b. Let's say 3 is the growth factor. The equation now would be y=2(3^x). Now the equation is complete.
2)
a. In the equation y=a(b^x), the a is the initial amount or y-intercept. The b is growth factor.
b. The y-intercept or a is represents how up or down the curved line is.
c. The growth factor or b is represented the steepness of the curved line.
Mathematical Reflection 2
1. a. You can use a table, a graph, and an equation to find the y-intercept and growth factor for an exponential relationship by looking at the table and graph then finding what y is when x is zero
(y-intercept), and what the number multiplies by each time (growth factor).
1. b. You can use the y-intercept and growth factor to write an equation for an exponential relationship by using the equation y=a(b^x). A= Y-Intercept B= Growth Factor
2. a. In the equation y=a(b^x) a equals the y-intercept and b equals the growth factor.
2. b. A is represented in a graph of y=a(b^x) byecause it is the y-intercept, so a is what y equals when x equals zero.
2. c. B is represented in a graph of y=a(b^x) by making the line steeper or less steep.
Math Reflections 2
1.a. You can use a table to find the y-intercept of an exponential relationship by finding the dependent variable's value when the independent variable's value is 0. When finding the y-intercept of an exponential relationship in a graph, you look to see where the line crosses the y-axis. To find the y-intercept of an exponential equation, you substitute 0 for the independent variable and solve for y. You could also find this by looking for a in the equation y=a(b^X).
You find the growth factor of an exponential relationship from a table by calculating what the dependent variable is multiplied by every time the independent variable increases by 1. To find the growth factor from a graph, find what the dependent variable is multiplied by every time the independent variable increases by 1. This is just like a table, only you use points on the line. The growth factor in the equation is b in the equation y=a(b^X).
1.b. The y-intercept is a in the equation y=a(b^X). The growth factor is b in this equation. To create an exponential relationship's equation, you simply substitute these numbers in for a and b in the equation.
2.a. In the equation y=a(b^X), a is the initial value of the exponential relationship, and b is the growth factor of the exponential relationship.
2.b. In a graph of y=a(b^X), a is the y-intercept.
2.c. In a graph of y=a(b^X), b is the amount y is increased by every x, or the steepness of the line.
You find the growth factor of an exponential relationship from a table by calculating what the dependent variable is multiplied by every time the independent variable increases by 1. To find the growth factor from a graph, find what the dependent variable is multiplied by every time the independent variable increases by 1. This is just like a table, only you use points on the line. The growth factor in the equation is b in the equation y=a(b^X).
1.b. The y-intercept is a in the equation y=a(b^X). The growth factor is b in this equation. To create an exponential relationship's equation, you simply substitute these numbers in for a and b in the equation.
2.a. In the equation y=a(b^X), a is the initial value of the exponential relationship, and b is the growth factor of the exponential relationship.
2.b. In a graph of y=a(b^X), a is the y-intercept.
2.c. In a graph of y=a(b^X), b is the amount y is increased by every x, or the steepness of the line.
Mathematical Reflection 2
1. a. You can use a table to find the y-intercept and growth factor from an exponential relationship. To find the y-intercept you look or extend the table to when x is 0. You can find the growth factor by finding the relationship between the y section in the table.
b. You use the y-intercept and growth factor to form an equation. You use the form, y=a(b^x). A is the y-intercept and b is the growth factor. You plug the numbers in and form an equation.
2. a. In an exponential relationship you use the equation y=a(b^x). The A is the y-intercept (initial value). B is the growth factor.
b. A is represented because it is the y-intercept .
c. B is represented because it is the growth factor, so it affects the steepness of the curve. It is sort-of like the slope of a line.
b. You use the y-intercept and growth factor to form an equation. You use the form, y=a(b^x). A is the y-intercept and b is the growth factor. You plug the numbers in and form an equation.
2. a. In an exponential relationship you use the equation y=a(b^x). The A is the y-intercept (initial value). B is the growth factor.
b. A is represented because it is the y-intercept .
c. B is represented because it is the growth factor, so it affects the steepness of the curve. It is sort-of like the slope of a line.
Monday 3/22/10: Fractional Growth Factors
Today in class, we went over the Exploring Exponential Functions worksheet we started on Friday. We determined the effect of a, b, and c of the graph of the equation y=a(b^x)+c:
We were introduced to fractional growth factors today. We saw that whether the growth factor was fractional or not did not effect the form of the equation. We did a problem in class about rabbits. We were given a table of the growth of a particular rabbit population. Using the information, we found that the growth rate was about 1.8. This did not influence the appication of y=a(b^x) to its graph.
- a is the starting point of the equation. It is the original y-intercept. When a is a negative number,the line of the equation points downward. On the other hand, when a is a positive number, the resulting line points up, giving it a shape similar to an U.
- b is the growth rate of the equation. It defines the steepness of the graph.
- c shifts the curved line up or down on the y-axis. When added to the a value of the equation, it results in the final y-intercept of the graph.
We were introduced to fractional growth factors today. We saw that whether the growth factor was fractional or not did not effect the form of the equation. We did a problem in class about rabbits. We were given a table of the growth of a particular rabbit population. Using the information, we found that the growth rate was about 1.8. This did not influence the appication of y=a(b^x) to its graph.
Mathematical Reflections 2
1.
a. The growth factor in an exponential line is how steep the curve is. The y-intercept is where the line starts on the y-axis. In a table, you divide the y of any x by the y of the x before it, and you get the growth factor. In a graph, you do the same, only you have to find the points, and divide the y values. In the equation y= a(b^x), b is the growth factor. To find the y-intercept in a table, you just look to where the x=o. On a graph, you just look at where the line crosses the y-axis. In the equation y=a(b^x), the y-intercept is a.
b. If you know what the y-intercept and growth factor are, you can plug them into the standard equation y= a(b^x).
2.
a. The a is the y-intercept, the b is the growth factor.
b. a is the y when x=o, and where the line crosses the y-axis
c. b is the steepness of the line.
a. The growth factor in an exponential line is how steep the curve is. The y-intercept is where the line starts on the y-axis. In a table, you divide the y of any x by the y of the x before it, and you get the growth factor. In a graph, you do the same, only you have to find the points, and divide the y values. In the equation y= a(b^x), b is the growth factor. To find the y-intercept in a table, you just look to where the x=o. On a graph, you just look at where the line crosses the y-axis. In the equation y=a(b^x), the y-intercept is a.
b. If you know what the y-intercept and growth factor are, you can plug them into the standard equation y= a(b^x).
2.
a. The a is the y-intercept, the b is the growth factor.
b. a is the y when x=o, and where the line crosses the y-axis
c. b is the steepness of the line.
Yesterday, our class worked on graphing Exponential Functions in order to learn how certain variables effect the function's graph. In the equation y=a(b)*+c the variables a, b and c all change the way the graph looks. The a is the y-intercept or initial value, unless there is a c in the equation. If there is a c, the two variables values are added to create the y-intercept. The b in the equation is the growth factor. It represents the rate that the line is increasing or decreasing much like the slope does in a linear equation. For example, if the b is two, then the y is doubling every interval. Next, the c is, again, the added or subtracted variable that is able to change the y-intercept and the a. The c sometimes is the y-intercept when you add one to it, because in many Exponential Functions the initial factor (a) is one. These three important variable can reveal a lot of information about an Exponential Function's graph.
Mathematical reflections 2
1. a. You can find the y-intercept in an exponential relationship by using a table, a graph, and an equation. You can find the y-intercept from a table by looking at the y when the x is zero on the table. Whatever the y is will be the y-intercept. you can find the y-intercept from a graph by looking at where the line passes the y axis and that will be the y-intercept. You can find the y-intrcept from an equation by looking at the number that is being multiplied by a number with an exponent and that will be the y-intercept.
You can also find the growth factor in an exponential relationship by using a table, a graph, and an equation. you can find the growth factor from a table by looking at ho much y is being multiplied by for each x and that will be the growth factor. You can find the growth factor from a graph by looking at how much y is multiplied on the graph for every x and that will be the growth factor. You can find the growth factor of an equation by seeing what number has an exponent and that will be the growth factor.
1. b. You can use the y-intercept and growth factor to write an equation for an exponential relationship. You can put the y-intercept first and have it multiplied by the growth factor with an exponent of whatever number x the problem is at.
2. a. In the equation y=a(b^x), a is the initial value of the exponetial relationship and b is the growth factor of the exponetial relationship.
2. b. In a graph of y=a(b^x), a is the y-intercept.
2. c. In a graph of y=a(b^x), b is the amount y goes up for every time x goes up one interval or the steepness of the line.
You can also find the growth factor in an exponential relationship by using a table, a graph, and an equation. you can find the growth factor from a table by looking at ho much y is being multiplied by for each x and that will be the growth factor. You can find the growth factor from a graph by looking at how much y is multiplied on the graph for every x and that will be the growth factor. You can find the growth factor of an equation by seeing what number has an exponent and that will be the growth factor.
1. b. You can use the y-intercept and growth factor to write an equation for an exponential relationship. You can put the y-intercept first and have it multiplied by the growth factor with an exponent of whatever number x the problem is at.
2. a. In the equation y=a(b^x), a is the initial value of the exponetial relationship and b is the growth factor of the exponetial relationship.
2. b. In a graph of y=a(b^x), a is the y-intercept.
2. c. In a graph of y=a(b^x), b is the amount y goes up for every time x goes up one interval or the steepness of the line.
Math Reflection 2
1a. The growth factor of an expenentionally grown line is the steepness. It shows how fast it goes up for example doubling or tripleing. To find it in in a table you divide y when x equals 2 by y when x equals 1. To get it from a graph you do the same thing from the table exept you have to find the points. From the equation: Y= a(b to the x) the growth factor is the b. To find the y intercept in a table or graph you take the number when x is zero. To find it from the equation you fill in 0for x or take the a value.
1b. You fill in the Y intercept as the a value and the growth factor for the b value.
2a. A is the y intercept b is the growth factor.
2b. A is the spot where x equals 0.
2c. B is how rapidly the line is going up.
1b. You fill in the Y intercept as the a value and the growth factor for the b value.
2a. A is the y intercept b is the growth factor.
2b. A is the spot where x equals 0.
2c. B is how rapidly the line is going up.
Math Reflection
1.a.The growth factor is how fast the slope grows and the steepness of the graph. You can get it by dividing y of any given x value by the y of the x value below that. For example, for a graph with the points (0,1) and (1,4) you would divide 4 by 1 to get a growth factor of 4. To find the y-intercept you take the a in and equation or whatever the y is when x=0 in a graph or table.
1.b.You plug the y-intercept in as a and the growth factor as b.
2.a.A represents the original value or the y-intercept. B represents the growth rate.
2.b.A is the y-intercept of the graph.
2.c.B is how fast the slope grows and how curved the line is.
1.b.You plug the y-intercept in as a and the growth factor as b.
2.a.A represents the original value or the y-intercept. B represents the growth rate.
2.b.A is the y-intercept of the graph.
2.c.B is how fast the slope grows and how curved the line is.
Exploring Exponential Functions
In class on Friday we did a graphing activity called exploring exponential functions. In the activity we graphed different sets of equations that all were the same except 1 or 2 factors were changed to figure out what the different elements did to the graph.
In set 1, a and b were the same but c varied. From this we figured out that c moves the graph up and down. To figure out the y-intercept you can use the equation (when y is the intercept) c+a=y.
In set 2, b and c stayed the same but b varied. The higher b was, the higher the y-intercept was. The graphs look similar except that the graph accelerates faster to start out when the a is higher.
In set 3, the only thing that changed was the growth factor which effected how fast the slope grew and moreover the steepness of the graph. The higher the growth factor the steeper the graph.
Finally, in set four we explored a fraction as a growth factor and a negative. When the growth factor is a fraction the slope starts out large and gets smaller and smaller as the x continues. When the growth factor is negative the graph goes down instead of up and mirrors whatever the graph would be if the growth factor were not a negative.
In set 1, a and b were the same but c varied. From this we figured out that c moves the graph up and down. To figure out the y-intercept you can use the equation (when y is the intercept) c+a=y.
In set 2, b and c stayed the same but b varied. The higher b was, the higher the y-intercept was. The graphs look similar except that the graph accelerates faster to start out when the a is higher.
In set 3, the only thing that changed was the growth factor which effected how fast the slope grew and moreover the steepness of the graph. The higher the growth factor the steeper the graph.
Finally, in set four we explored a fraction as a growth factor and a negative. When the growth factor is a fraction the slope starts out large and gets smaller and smaller as the x continues. When the growth factor is negative the graph goes down instead of up and mirrors whatever the graph would be if the growth factor were not a negative.
Mathematical Reflecations 2
1. a. To find the y-intercept for an exponential equation using the graph, you must extend the graph to the point where the line in question crosses the y-axis. To find the growth factor of an exponential relationship using a graph, you need to compare different values. The easiest way to do this is to divide a y-value by the y-value preceding it, and the quotient is the growth factor. If you are not sure if it is an exponential relationship, it is always good to try this a few times with different numbers.
The y-intercept is most easily found using a table. To do this, you simply find where the value of x is 0 and look at the corresponding value of y. The growth factor is found with a table the same way it is found with a graph. By dividing a value of y by the one before it you find the growth factor of that particular exponential equation.
The third way to find the y-intercept and growth factor of an exponential equation is to look at the equation itself.
The y-intercept is most easily found using a table. To do this, you simply find where the value of x is 0 and look at the corresponding value of y. The growth factor is found with a table the same way it is found with a graph. By dividing a value of y by the one before it you find the growth factor of that particular exponential equation.
The third way to find the y-intercept and growth factor of an exponential equation is to look at the equation itself.
The equation above is the standard form for exponential equations. The a is the initial valuue, also known as the y-intercept. The b is the growth factor.
b. Writing and exponential equation is very simple. Since a is the initial value, or y-intercept, you just have to substitute your y-intercept into the equation given in 1.a. The growth factor is represented by b, and so all that is left to do is substitute your second variable in and you have your equation.
2.a. In the standard form of the exponential equation shown in 1.a., the a represents the intial values, or y-intercept. The b, in turn, represents the growth factor of the relationship.
b. On a graph, the a is represented by where the line crosses the y-axis.
c. On a graph, the b is represented by the slope or "shape" of the line.
Math Reflection p. 32
1. a. To find the y-intercept and growth factor from a table you look at when x is 0 and that is the y-intercept. To find the growth factor from the table you have to figure out how it got from when x is 0 to when x is 1 and then try it out to see if the pattern continues. To fing the y-intercept on a graph you just find where y is when x is 0. To find the growth factor from a graph you have to look at some of the points and see how it is getting from one point to another like in the graph. To find the y-intercept from the equation a(b)^x, a is the y-intercept. The growth factor is b.
b. To write an equation from the y-intercept and the growth factor, you plug it into the equation a(b)^x. The y-intercept is a and the growth factor is b.
2. a. a=y-intercept and b=growth factor
b. a is the y-intercept on the graph. It is where y is when x=0.
c. b is the steepness of the graph.
b. To write an equation from the y-intercept and the growth factor, you plug it into the equation a(b)^x. The y-intercept is a and the growth factor is b.
2. a. a=y-intercept and b=growth factor
b. a is the y-intercept on the graph. It is where y is when x=0.
c. b is the steepness of the graph.
Sunday, March 21, 2010
Mathematical Reflections 2
1a. To find the y-intercept for a table, you have to extend the table to when x=0. To find the growth factor, you have to find how much one value of y has grown from the previous value of y.
Using a graph, the same things basically apply. When x=0, you have the y-intercept, and when you find the growth between two values of y, you have the growth factor. In an equation, if you are thinking of the form "y=a(b^x)", then a=y-intercept and b=growth factor
1b. Thinking of the form of an exponential equation, if you find the y-intercept, it is a. If you find the growth factor, it is b. For example, if the y-intercept were 10 and the growth factor was 2, then the equation would be y=10(2^x).
2a. The values of a and b represent the y-intercept and the growth factor (respectively) in the exponential relationship.
2b. a is represented in a graph as the point where a line intersects the x value of 0.
2c. b is represented in a graph as the growth between two y values.
Using a graph, the same things basically apply. When x=0, you have the y-intercept, and when you find the growth between two values of y, you have the growth factor. In an equation, if you are thinking of the form "y=a(b^x)", then a=y-intercept and b=growth factor
1b. Thinking of the form of an exponential equation, if you find the y-intercept, it is a. If you find the growth factor, it is b. For example, if the y-intercept were 10 and the growth factor was 2, then the equation would be y=10(2^x).
2a. The values of a and b represent the y-intercept and the growth factor (respectively) in the exponential relationship.
2b. a is represented in a graph as the point where a line intersects the x value of 0.
2c. b is represented in a graph as the growth between two y values.
Math Reflections 2
1a. You can use a table to find the y-intercept by extending it until x=0. You can use it to find the growth factor by figuring out the pattern in the x- and y-values. You can use a graph to find the y-intercept by finding the point where the line crosses the y-axis. You can use it to find the growth factor by finding the relationship between different points. In the equation y=a(b)^x, a is the y-intercept and b is the growth factor.
b. The form for an exponential equation is y=a(b)^x, where y=the dependent variable, x=the independent variable and the exponent, a=the y-intercept, and b=the growth factor.
2a. In the equation y=a(b)^x, a=the y-intercept and b=the growth factor.
b. In a graph of y=a(b)^x, a is the y-intercept.
c. In a graph of y=a(b)^x, b is the growth factor and it determines the steepness of the line.
b. The form for an exponential equation is y=a(b)^x, where y=the dependent variable, x=the independent variable and the exponent, a=the y-intercept, and b=the growth factor.
2a. In the equation y=a(b)^x, a=the y-intercept and b=the growth factor.
b. In a graph of y=a(b)^x, a is the y-intercept.
c. In a graph of y=a(b)^x, b is the growth factor and it determines the steepness of the line.
Saturday, March 20, 2010
Math Reflections 2
1a. You can use a table, graph, and an equation to find the y-intercept and growth factor for an exponential relationship. In a table, the y-intercept is when x=0. The growth factor is by what number the y is multiplied by per x. Using a graph, wherever the line crosses the y-axis, that is the y-intercept. The growth factor is the number that is multipled to get from y when x=1 to when x=2, and etc. Also, taking the equation y=a(b^x), a is the starting point, or the y-intercept, and b is the growth factor.
1b. If given that the y-intercept is 2, and the growth factor is 5, we can come up with the exponential equation y=2(5^x). To form the equation, we use the form y=a(b^x). We plug in the y-intercept for a, and the growth factor for b.
2a. In the equation y=a(b^x), a is the y-intercept, and b is the growth factor.
2b. In the graph of y=a(b^x), a is the y-intercept for when x=o.
2c. In the graph of y=a(b^x), b is the multiplier to get from y when x=1 to when x=2, and so on.
1b. If given that the y-intercept is 2, and the growth factor is 5, we can come up with the exponential equation y=2(5^x). To form the equation, we use the form y=a(b^x). We plug in the y-intercept for a, and the growth factor for b.
2a. In the equation y=a(b^x), a is the y-intercept, and b is the growth factor.
2b. In the graph of y=a(b^x), a is the y-intercept for when x=o.
2c. In the graph of y=a(b^x), b is the multiplier to get from y when x=1 to when x=2, and so on.
math reflection 2
1.a. To find the y-intercept in a table, you have to extend the table until x equals 0. Whatever y is when x is 0 is your y-intercept. To find the y-intercept on a graph, you have to find the point where the line passes throught the y-axis. Again, when x is 0, whatever the value is of why is your y-intercept. To find the y-intercept in an equation, you have to find the initial value, which is the a variable in the equation and also when x is 0.
b. To write an equation for an exponential relationship with the y-intercept and the growth factor is simple. In the formula y=a(braised to the x), a is the y-intercept and b is the growth factor and you raise it to the value of x.
2.a. In the equation y=a(b raised to the x) the variable a represents the initial value or value when the value of x is 0(y-intercept). The variable b represents the groeth factor which decides the steepness of the line. If the value of y doubles from one x value to the next, the growth factor is two, and if it triples the growth factor is three etc.
b. The variable a is the y-intercept.
c. The variable b is the growth factor.
b. To write an equation for an exponential relationship with the y-intercept and the growth factor is simple. In the formula y=a(braised to the x), a is the y-intercept and b is the growth factor and you raise it to the value of x.
2.a. In the equation y=a(b raised to the x) the variable a represents the initial value or value when the value of x is 0(y-intercept). The variable b represents the groeth factor which decides the steepness of the line. If the value of y doubles from one x value to the next, the growth factor is two, and if it triples the growth factor is three etc.
b. The variable a is the y-intercept.
c. The variable b is the growth factor.
Friday, March 19, 2010
Thursday, March 18, 2010
Growing, Growing, Growing Problem 2.2 & 2.3
Exponential equations & graphs
Growing, Growing ,Growing Problem 2 2 & 2 3
View more documents from Kathy Favazza.
Tuesday, March 16, 2010
Exponential Growth
Today in class we finished up the problem we started yesturday, The Kingdom of Monterak. We also started problem 2.1 in the Growing, Growing, Growing book. Both these problems have to do with exponential growth. Exponential growth is based on the growth factor.
The growth factor-
-the fixed number of increase (ex. doubling, tripling)
-the base in the equation (see below)
-the ratio of change from one y value to the next.
When you do these kind of problems, you should use an equation like this.
y=a(b)[to the x power]
The y is the dependant variable.
The x is the independent variable. (the exponent)
The a is the initial value. (the y-intercept)
The b is the growth factor.
:) :)
The growth factor-
-the fixed number of increase (ex. doubling, tripling)
-the base in the equation (see below)
-the ratio of change from one y value to the next.
When you do these kind of problems, you should use an equation like this.
y=a(b)[to the x power]
The y is the dependant variable.
The x is the independent variable. (the exponent)
The a is the initial value. (the y-intercept)
The b is the growth factor.
:) :)
Monday, March 15, 2010
March 15
Today in class we started a problem called The Kingdom Of Marteka. It had to do with equations including exponents. There were five different plans for a reward for a peasent. All of the plans had a different number of squares on the board number of rubas per first square and the rate they went up. For each plan they had a different equation and we had to find it. An example of one of the equations was 2 to x-1 power.
Standard notation is just a number or answer to a problem in number form. Scientific notation is another way of writing the answer to a problem with exponents. Its written in the form a times ten to the power of b. If the answer is too big they use scientific notation to simplify by earasing zeros.
During the proble we have to figure out how many rubas will be on the last tile and how many will be on the whole board. The king kept trying to change the proposal but, was really bad in math.
Standard notation is just a number or answer to a problem in number form. Scientific notation is another way of writing the answer to a problem with exponents. Its written in the form a times ten to the power of b. If the answer is too big they use scientific notation to simplify by earasing zeros.
During the proble we have to figure out how many rubas will be on the last tile and how many will be on the whole board. The king kept trying to change the proposal but, was really bad in math.
Tuesday, March 9, 2010
Problems 5.3 and 5.4
Today in class we did problems 5.3 and 5.4.
In problem 5.3, there were various problems about graphing inequalities. We learned that when y is greater than or equal to mx+b, the shaded region is above the line of the inequality, and when y is less than or equal to mx+b, the shaded region is below the line. We also learned that when the sign in the inequality is greater than/less than or equal to, the line is solid and when y cannot be equal to mx+b, the line is dashed.
In problem 5.4 we learned how to graph two inequalities, or a system of linear inequalities, on the same axis. The region where both shaded areas overlap contains all the points that satisfy both inequalities.
In problem 5.3, there were various problems about graphing inequalities. We learned that when y is greater than or equal to mx+b, the shaded region is above the line of the inequality, and when y is less than or equal to mx+b, the shaded region is below the line. We also learned that when the sign in the inequality is greater than/less than or equal to, the line is solid and when y cannot be equal to mx+b, the line is dashed.
In problem 5.4 we learned how to graph two inequalities, or a system of linear inequalities, on the same axis. The region where both shaded areas overlap contains all the points that satisfy both inequalities.
Thursday, March 4, 2010
Monday, March 1, 2010
Elimination Method
In class on Thursday, we reviewed the three ways to solve systems we knew; graphing, setting equal, and substitution. Then we learned the 4th method, Elimination.
8x+2y=38
You would multiply the first equation by 2
2(4x+5y=47)
8x+10y=94
Then you would subtract the second equation from the first, cancelling out the x
8x+10y=94
- 8x+2y=38
8y=56
y=7
Then you would substitute in 4 for x
4x+5(7)=47
4x+35=47
4x=12
x=3
The answer to the system is (3,7).
TIPS
You can solve for either the x or the y.
Sometimes you may have to multiply both equations to be able to cancel out a variable.
You can add or subtract the equations to get an answer.
For example,
4x+5y=478x+2y=38
You would multiply the first equation by 2
2(4x+5y=47)
8x+10y=94
Then you would subtract the second equation from the first, cancelling out the x
8x+10y=94
- 8x+2y=38
8y=56
y=7
Then you would substitute in 4 for x
4x+5(7)=47
4x+35=47
4x=12
x=3
The answer to the system is (3,7).
TIPS
You can solve for either the x or the y.
Sometimes you may have to multiply both equations to be able to cancel out a variable.
You can add or subtract the equations to get an answer.
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