CPM Homework Help : CCA Problem 9-72

If the graph of an exponential function passes through the points $\left(1, 6\right)$ and $\left(4, 48\right)$ , find an equation of the function. Refer to the Math Notes box in this lesson if you need help.

Hint:

To find an exponential function that goes through two given points, create a system of equations by substituting one ( $x,y$ ) point into $y=ab^x$ , then substituting the other point. Rewrite both equations in “ $a=$ ” form. Solve the system with the equal values method to find $a$ and $b$ and now you can write the equation.

For example, find an exponential function that passes through $(2,14)$ and $(5,112)$ . Create a system of equations by substituting ( $x,y$ ) $=(2,14)$ into $y=ab^x$ , and then substituting again with ( $x,y$ ) $=(5,112)$ :

$14=ab^2$ and $112=ab^5$ Rewrite as $a=\frac{14}{b^2}$ and $a=\frac{112}{b^5}$ .

Use the equal values method to find $b$ :

$\left. \begin{array}{l}{ \frac { 14 } { b ^ { 2 } } = \frac { 112 } { b ^ { 5 } } }\\{ 14 b ^ { 5 } = 112 b ^ { 2 } }\\{ 14 \cdot \frac { b ^ { 5 } } { b ^ { 2 } } = 112 }\\{ 14 b ^ { 3 } = 112 }\\{ b ^ { 3 } = \frac { 112 } { 14 } = 8}\\{ b = 2 }\end{array} \right.$

$\Huge\nearrow$

Use either original equation to find $a$ :

$\left. \begin{array} { l } { 14 = a b ^ { 2 } } \\ { 14 = a ( 2 ) ^ { 2 } } \\ { \frac { 14 } { 4 } = a } \\ { 3.5 = a } \end{array} \right.$

The equation of the exponential function that passes through the two given points is $y=3.5·2^x$ .

Step (1):

$a=3$