Question

Find the exponential function \(f\left( x \right) = C{b^x}\) whose graph is given.

19.

Short answer

The exponential equation is \(y = 3 \cdot {\left( 2 \right)^x}\).

Step-by-step solution

Obtain the value of constant

From the graph, the points are \(\left( {1,6} \right)\), and \(\left( {3,24} \right)\).

Consider the equation\(y = f\left( x \right) = C{b^x}\).

Use the point\(\left( {1,6} \right)\)to get the value of constant as shown below:

\(\begin{aligned}6 &= C{b^1}\\C &= \frac{6}{b}\end{aligned}\)

Thus, \(C = \frac{6}{b}\).

Obtain the value of constant

Consider the equation \(y = f\left( x \right) = C{b^x}\).

Use the point\(\left( {3,24} \right)\)to get the value of constant as shown below:

\(24 = C{b^3}\)

Substitute\(\frac{6}{b}\)for \(C\) in the equation \(24 = C{b^3}\)to get the value of\(b\)as shown below:

\(\begin{aligned}C{b^3} &= 24\\\left( {\frac{6}{b}} \right) \cdot {b^3} &= 24\\6{b^2} &= 24\\{b^2} &= 4\\b &= \pm 2\end{aligned}\)

Since\(b > 0\), so\(b = 2\).

The value of\(C\)is shown below:

\(\begin{aligned}C &= \frac{6}{2}\\ &= 3\end{aligned}\)

So, the exponential equation becomes \(y = 3 \cdot {\left( 2 \right)^x}\).

Thus, the exponential equation is \(y = 3 \cdot {\left( 2 \right)^x}\).