Sunday, February 5, 2017

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...

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...

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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