Chapter 6.7, Problem 13E

To determine

To write : An exponential function $y=a{b}^x$ whose graph passes through the given points.

Answer to Problem 13E

The exponential function is $y=5{\left(0.5\right)}^x$ .

Explanation of Solution

Given information :

The given points are $\left(-1,\text{ 10}\right)$ and $\left(4,\text{ 0}\text{.31}\right)$ .

Calculation :

Substitute the coordinate of the two points into

  $y=a{b}^x$ …… (1)

The graph passes through the point $\left(-1,\text{ 10}\right)$ . Substitute $x=-1$ and $y=10$ in equation (1) and solve:

  $10=a{b}^{-1}$ …… (2)

Similarly, Substitute $x=4$ and $y=0.31$ in equation (1) and solve:

  $0.31=a{b}^4$ …… (3)

Divide equation (3) by (2) and solve for b:

  $\begin{array}{l}\frac{0.31}{10}=\frac{a{b}^4}{a{b}^{-1}}\\ 0.031=b^5\\ 0.5=b\text{ (Take }5th\text{ root on both sides)}\end{array}$

Substitute $b=0.5$ in equation (2) and solve for a:

  $\begin{array}{l}10=a{\left(0.5\right)}^{-1}\\ 10=2a\\ \frac{10}{2}=a\\ 5=a\end{array}$

Substitute the value of a and b in equation (1) to get the exponential function:

  $y=5{\left(0.5\right)}^x$

Hence,

The exponential function is $y=5{\left(0.5\right)}^x$ .