  ### Home > CCA > Chapter 9 > Lesson 9.3.1 > Problem9-72

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.

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 $y=3.5·2^x$

$a=3$