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 1 Pg. 18

1a. The graphs I have observed have all been parabolas or upside down U's. Also the graph is symmetrical. The highest point on the graph is always the square. The tables show the shapes facing a different direction. The are on the table increase to a certain point then decrease at the same pattern.

1b. The tables display the pattern on the graph by using coordinate points that go together in table form.

2. One way to find the maximum area for a rectangle with a fixed perimeter is to square the number on the x axis below the highest point on the graph. Another way is to find the largest length on the table and square that.

3. The rate is unsteady in quadratics. The graph is a parabola instead of a line. Also it goes up and down in the same equation.


By, Joe P., Nick S., Harry K. and Mike K.

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.

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.

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.

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 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.

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.

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.