Chapter EP, Problem 8.2.2EP

To determine

The exponential function for the graph that passes through the two points.

Explanation of Solution

Given information:

The given two points are $(0,-4)\&(-4,-64)$

Calculations:

The standard form of exponential function is, $f(x)=a{(b)}^x$

Now for, $f(x)=y$

The equation can be written as,

  $y=a{(b)}^x$

Now taking different values of x and y we get,

  $\begin{array}{}$ -4=a{(b)}^0\text{ Where, }y=y_1=-4\&x=x_1=0$$ \text{Now solving the equations,}$$ \begin{array}{l}a=-4.\\ Again,\end{array}$$ -64=-4{(b)}^{-4}\text{ Where, }y=y_2=-64\&x=x_2=-4$$ \text{Now solving the equations,}$$ -64=-4{(b)}^{-4}$$ {(b)}^{-4}=\frac{-64}{-4}$$ {(b)}^{-4}=16$$ {\left(\frac{1}{b}\right)}^4=2^4$$ \frac{1}{b}=2$$ b=\frac{1}{2}=0.5$\end{array}$

Now putting the values of a and b in the standard form of equation, we get the desired equation.

  $f(x)=-4.{(0.5)}^x$