How can you use a table, a graph, and an equation that represent an exponential function to find the y intercept and growth factor for the function? Explain. 

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Minor addition to the answer:

Finding Growth Factor from the graph:

From the graph `y_1=6, y_2=12, y_3=24`

Growth factor(b)=`y_2/y_1=y_3/y_2`

Growth factor=`12/6=24/12=2`

So, Growth Factor will be constant.

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Let `y=f(x)=ab^x` represents the general form of the exponential function, 

When a`>0` and b`>0` , b represents the growth factor,

When x=0 , then y=a which is the y-intercept.

2) Finding Grwoth factor and y-intercept from the table

Now let's have a sample table stated below to find out the y-intercept and growth factor.

If there is x value in the table having x=0 , then the corresponding y value represents the y-intercept, otherwise y-intercept can be found as follows:

 x         y

 1         6

 2        12

 3        24

 4         48

Since the exponential function passes through the above points say (x_1,y_1) and (x_2,y_2), the points will satisfy the equation of the function,

`:.y_1=ab^(x_1)` -----equation 1

and `y_2=ab^(x_2)`

Dividing the above equations will yield,

`y_2/y_1=b^(x_2)/b^(x_1)`

`y_2/y_1=b^(x_2-x_1)`

or `b=(y_2/y_1)^(1/(x_2-x_1))`

Now from the table , plug in the values of the points to find the b,

`b=(12/6)^(1/(2-1))`

`b=2^(1/1)`

`b=2`

Now y-intercept (a) can be found by plugging in any of the equation,

Let's plug b in the equation 1

`y_1=ab^(x_1)`  

`6=a(2)^1`

`6=2a`

`a=6/2`

`a=3`

Therefore y-intercept=3

3)  Finding y-intercept and growth rate from the graph

Pl see the attached graph.

Look at the graph, y-intercept will be the y-coordinate where the graph of the function intersects the y-axis.

From the graph y-intercept=3

Growth factor can be found by noting the x and y-coordinates and then plugging them in the equation as follows:

From the graph,

`x_2=2,y_2=12`

`x_1=1,y_1=6`

So growth factor=`(y_2/y_1)^(1/(x_2-x_1))`

`b=(12/6)^(1/(2-1))`

`b=2^1`

b=2

 

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