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.

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.

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 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.

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.

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.