Chapter 0.3, Problem 48E

To determine

The values of $a$ and $k$ for which the graph of exponential function $f\left(x\right)=k\cdot a^x$ passes through the two given points.

Answer to Problem 48E

  $k=1.5,a=3$

Explanation of Solution

Given:

The two points: $\left(1,4.5\right)$ and $\left(-1,0.5\right)$ .

Concepts Used:

Points $\left(a,b\right)$ lying on the graph of $y=f\left(x\right)$ satisfy the equation $b=f\left(a\right)$ .

Solving system of equations by substitutions.

Calculations:

The point $\left(1,4.5\right)$ is given to lie on the graph of $y=k\cdot a^x$ . Thus:

  $\begin{array}{l}4.5=k\cdot a^1\\ 4.5=ka\end{array}$

The point $\left(-1,0.5\right)$ is also given to lie on the graph of $y=k\cdot a^x$ . Thus:

  $\begin{array}{l}0.5=k\cdot a^{-1}\\ 0.5=\frac{k}{a}\end{array}$

Multiply the expression on the corresponding sides of the equations $4.5=ka$ and $0.5=\frac{k}{a}$ . Then solve the resulting equation for $k$ :

  $\begin{array}{c}4.5\times 0.5=ka\times \frac{k}{a}\\ 0.5\times 9\times 0.5=k^2\\ \sqrt{0.5\times 0.5\times 3\times 3}=k\\ k=0.5\times 3\\ k=1.5\end{array}$

Substitute the value $k=1.5$ in the equation $4.5=ka$ and then solve the resulting equation for $a$ :

  $\begin{array}{c}4.5=ka\\ 4.5=1.5\times a\\ \frac{4.5}{1.5}=a\\ a=3\end{array}$

Conclusion:

For $k=1.5$ and $a=3$ , the graph of exponential function $f\left(x\right)=k\cdot a^x$ passes through the two given points.