Chapter 7.6, Problem 55PPE

a.

To determine

To calculate: value of constant $a$

Answer to Problem 55PPE

  $a=4$

Explanation of Solution

Given information: $\left(0,4\right),\left(2,36\right)$ are on the graph of an exponential function.

Calculation:

Given exponential function is $y=a\cdot b^x$

Put $x=0,y=4$ in $y=a\cdot b^x$

  $\begin{array}{l}4=a\cdot b^0\\ \Rightarrow 4=a(1)\\ \Rightarrow 4=a\end{array}$

b.

To determine

To calculate: value of constant $b$

Answer to Problem 55PPE

  $b=3$

Explanation of Solution

Calculation:

Given exponential function is $y=4b^x$

Put $x=2,y=36$ in $y=4b^x$

  $\begin{array}{l}36=4b^2\\ \Rightarrow 9=b^2\\ \Rightarrow \sqrt{9}=b\\ \Rightarrow 3=b\end{array}$

c.

To determine

To find:rule for the function

Explanation of Solution

The function rule is $y=a\cdot b^x$ .

d.

To determine

To calculate: value of the function at $x=-2,4$

Answer to Problem 55PPE

  $\begin{array}{l}y\left(-2\right)=\frac{4}{9}\\ y\left(4\right)=324\end{array}$

Explanation of Solution

Calculation:

From parts $a,b$ ,

  $y=4\left(3^x\right)$

Put $x=-2$

  $y=4\left(3^{-2}\right)=4\left(\frac{1}{3^2}\right)=\frac{4}{9}$

Put $x=4$

  $y=4\left(3^4\right)=4\left(81\right)=324$

  $\begin{array}{l}y\left(-2\right)=\frac{4}{9}\\ y\left(4\right)=324\end{array}$