Math Reflection 2

1. The area of a rectangle can illustrate the Distributive Property. For example, if the rectangle is split into four parts, the sides of the original square would both be x and the two additional sides might be 2 and 4. You multiply all the values to each other to find the area of each of the four sections. One would be x^2, one would be 2x, one would be 4x, and the fourth would be 8. You combine the four areas and the equation would be x^2+8x+8.

2a. If a quadratic expression is in factored form, you can change it to expanded form using the Distributive Property. If the equation was (x+3)(x+4), you multiply each term by the other three: x(x), x(3), x(4), 3(4). You get x^2+3x+4x+12, and you combine like terms to get the final equation: x^2+7x+12.

b. If a qaudratic expression is in expanded form, you can change it to factored form using the Distributive Property. If the equation was x^2+11x+30, you have to break up all the terms. x^2 becomes x and x. To figure out the other two terms, you have to find a pair of values that add up to 11 and multiply together to 30. The pair is 5 and 6 because 5+6=11 and 5(6)=30. The new equation is (x+5)(x+6).

3. You can recognize a quadratic function from its equation if it has an exponent that's not a variable (expanded form) or if it has two quantities containing x that are being multiplied (factored form).

4. The graph of a quadratic function is a parabola. The x-intercepts are the opposite of the two numbers in the equation. If the equation is (x+9)(x-2) the x-intercepts are -9 and 2. The y-intercept is the last term in the expanded form equation. If the equation is x^2+5x+6 the y-intercept is 6.

Math Reflection 2- p.39

1. The area of a rectangle illustrates the Distributive Property because it allows you to see how the different parts of the side legnth affect the overall area. When you have an equation for the area of a rectangle, you often need to distribute to find another form or a value. For example, with the equation A=(x+3)(x+4), you would distribute the x to both the numbers in the second parenthesis, and then the 3, to get the expanded form of the equation.

2.a. If an equation is in factored form, you use the Distributive Property to find the equation in expanded form. For the equation y=(x+2)(x+4) you would distribute the numbers in the first set of parenthesis to both the numbers in the second parenthesis.
b. If an equation is in expanded form and you need to put it in factored form you can use an almost reverse distributive property. If the equation is y=x^2+7x+10, you first find two numbers that add up to 7 and multiply to 10, 5 and 2. You rewrite the equation as y=x^2+5x+2x+10. You then look at the numbers and find, for every set of two numbers, the thing in common. For this equation you have x in common for x^2 and 5x, so you write x(x+5) then you add to that what you get from the other two numbers. In common from 2x and 10, you get 2. So you write, 2(x+5). In both the parts you have something in common, (x+5). You take this as one of the sets in parenthesis, and for the other, you take the things that were multiplied into those parenthesis and get (x+2), so you have y= (x+5)(x+2).

3. You can recognize a quadratic function from its equation if it has the right form, either the expanded form or the factored form.

4. The graph of a quadratic function is a parabola, an symmetrical ark-shape. Some important features on these graphs are the x and y-axis, the minimum\maximum point, and the line of symmetry. You find the x-intercepts by finding the two numbers added to x in the factored form of the equation. You find the y-intercept by if there is a number added\subtracted in the expanded form. You can find the minimum\maximum point by adding together the x-intercepts and dividing by 2 to get the x coordinate. To find the y coordinate you plug the x coordinate into the equation and solve. The line of symmetry is the x-coordinate of the minimum\maximum point.

by Allie :)

Mathematical Reflection 2

1) You can use a rectangle to model the distributive property. The rectangle shown at the bottom of the post illustrates the expression (x+3)(x+4). When simplifying this expression using the distributive property, you would distribute the x and the 3 separately, so it would look like this: x(x+4)+3(x+4). You would solve this by adding up all the products of your distribution. The model shows all the separate products as parts of a whole. You distribute the 3 and x to the x and the 4, which gives you four separate values, which are represented by the four parts of the rectangle. 2)a. To put an expression in factored form into expanded form, you could distribute. You would take one quantity and distribute the value(s) in it separately to the other quantity. For (x+4)(x+2), you would distribute like this: x(x+2)+4(x+2). b. It is generally easier to work with quadratic expressions when they are in factored form, but that means that there will be times when you have to change from expanded to factored. Expanded form usually looks like this: x²+5x+6. There is sometimes more than one way to write the expression in factored form. First, you have to split up the coefficient of x, and the two values have to multiply to the constant in the expression. 3) A function's equation is quadratic if the highest power of the variable is 2 in standard form, and in factored form the equation must have exactly two linear factors to be quadratic. 4) A quadratic function's graph is always a parabola. In factored form, the opposites of the two constants are the x-intercepts, and the line of symmetry is the value between them. The difference between the constants will tell you if it is opening up or down on the coordinate plane.

Math Reflection p. 39

1. If you have a square, the area is x^2. If you add to the length and width (ex. 3 and 5), you get a new rectangle and therefore a new equation for the area. (x+3)(x+5). The Distributive Property is when you distribute an equation in factored form, (x+3)(x+5) to get a simplified expression in expanded form, x^2+8x+15. The rectangle that has a length of x+3 and a width of x+5 has those two equations: (x+3)(x+5) and x^2+8x+15. You can find the area by using either of these two equations.

2. a) If a quadratic expression is in factored form, you can change it to expanded form by distributing.

factored form: (x+3)(x+5)

You have to multiply the first x by everything in the second set of parenthesis. So you have x*x or x^2 and x*5 or 5x. Now you have to multiply the number in the first set of parenthesis (in this case 3) by everything in the second set of parenthesis. So you have 3*x or 3x and 3*5 or 15. Now your equation in expanded form is x^2+5x+3x+15 or x^2+8x+15.

2. b) If a quadratic expression is in expanded form, you can change it to factored form by splitting each term into smaller terms that will then be able to be distributed back into the same equation in expanded form.

expanded form: x^2+8x+15

The x^2 is easily split into x*x so you know that the factored equation will be (x+?)(x+?). The 8x can be split into 1 and 7, 2 and 6, 3 and 5, or 4 and 4. To know which one it is, you have to figure out the numbers that multiply to 15. 15 can be split into 1 and 15 or 3 and 5. To determine which pair of numbers is correct you have to figure out which one is in both sets. 3 and 5 are in both sets so you know that they will fit into the equation. Therefore, you know what numbers will fit the ?'s. (x+3)(x+5). That is the factored form of the equation x^2+8x+15.

3. You can recognize a quadratic function from its equation by noticing if there is an exponent and if the exponent is a number or a variable. If the exponent is a number then it is a quadratic equation, and if it is a variable then it isn't quadratic. If it is an equation that you can distribute you will have to do that to figure out if there is an exponent and if it is a number or variable.

4. The shape of a quadratic function is always a parabola. Whether it opens up or down, that depends on the equation. A quadratic function usually has two x-intercepts and one y-intercept. To find the x-intercepts, the equation has to be in factored form. You take the opposite of the numbers that are added to x. For example, the equation (x+3)(x+5) would have two x-intercepts, -3 and -5. To find the y-intercept, the equation has to be in expanded form. It is the number added without being multiplied by x. For example, the equation x^2+8x+15 has a y-intercept of 15. Quadratic functions also have a line of symmetry and a maximum or minimum point depending on whether the parabola opens up or down. The equation for the line of symmetry can be found by averaging the two x-intercepts. The x-intercepts of the equation (x+3)(x+5) are -3 and -5 so -3-5=-8/2=-4. Therefore, the line of symmetry is x=-4. To know if the graph has a maximum or minimum point you have to figure out if it opens up or down. The equation x^2+8x+15 has a graph of a parabola opening up. That means it has a minimum point. To find it you can plug the equation for the line of symmetry into the equation to find the y-value of the minimum point. The x-value of the minimum point is the line of symmetry (x=-4 in this case). The line of symmetry is x=-4 so you plug in -4 for the x-values in the equation, x^2+8x+15. -4^2+8(-4)+15=16-32+15=-16+15=-1. Therefore, the minimum point is (-4,-1).

Math Reflection

1a. All of the graphs have been parabolas and the tables increased in a pattern and decreased in the same pattern after the middle point.

b. With both, it increases in the same pattern and then decreases in that pattern when it reaches the middle point.

2. You can multiply the x-value of the middle point by 4 or double the number in the equation.

3. For tables of quadratic functions, the y-value increases until it reaches the middle point, then decreases in the same pattern. For tables of linear functions, both values increase at a steady rate. For tables of exponential functions, the y-value doesn't increase at a steady rate or with a pattern. The graphs of quadratic functions are parabolas and the middle point is called the vertex. The graphs of linear functions are straight lines. The graphs of exponential functions start at a slow increase and then increase rapidly. The equations all have different parts that make them that certain kind of equation. In a quadratic function, the equation has a variable multiplied by a quantity. A=l(1/2P-l). In a linear function, the equation multiplies two variables and adds another. y=mx+b. In an exponential function, the equation raises a variable to a certain power and multiplies another number to it. y=ab^x.
Emily L. and Alex S.

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.

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 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.

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.

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.

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.

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.

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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.

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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.

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.

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.

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.

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.

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.

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.

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.

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 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.