Chapter 7.7, Problem 1GP

To determine

An exponential function for the given graph.

Answer to Problem 1GP

Exponential function for given graph is $y=2\cdot 4^x.$

Explanation of Solution

Given information: We have given two points (0, 2) and (2, 32) on an exponential growth graph.

Calculation: We can see that graph is going up from left to right. Therefore, the shown graph is the exponential growth graph.

We have given two points (0, 2) and (2, 32) on the exponential growth graph.

We need to find an exponential function in the form $y=a\cdot b^x$ , where a is the initial value and b is the growth factor.

From the first point (0, 2), we get the initial value ( a ) = 2.

Now, plugging second point (2, 32) for ( x , y ) and a =2 in exponential function equation $y=a\cdot b^x.$

$32=2\cdot b^2$

Switch sides.

$2b^2=32$

Divide both sides by 2.

$\frac{2b^2}{2}=\frac{32}{2}$

Simplify

$b^2=16$

Taking square root on both sides

$\sqrt{b^2}=\sqrt{16}$

  $b=4.$

Plugging value of $a=2$ and $b=4$ in exponential function

  $y=a\cdot b^x$ .

$y=2\cdot 4^x$

Therefore, exponential function for give graph is $y=2\cdot 4^x.$