Writing an Exponential Function Given Its Graph

Find an equation for the exponential function graphed in Figure 5.

Figure 5

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
We can choose the y-intercept of the graph, $(0, 3),$ as our first point. This gives us the initial value, $a = 3.$ Next, choose a point on the curve some distance away from $(0, 3)$ that has integer coordinates. One such point is $(2, 12) .$
$\begin{array}{l} y = a b^{x} & Write the general form of an exponential equation . \\ y = 3 b^{x} & Substitute the initial value 3 for a . \\ 12 = 3 b^{2} & Substitute in 12 for y and 2 for x . \\ 4 = b^{2} & Divide by 3 . \\ b = \pm 2 \begin{matrix} \end{matrix} & Take the square root . \end{array}$
Because we restrict ourselves to positive values of $b,$ we will use $b = 2.$ Substitute $a$ and $b$ into the standard form to yield the equation $f (x) = 3 {(2)}^{x} .$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Find an equation for the exponential function graphed in Figure 6.

Figure 6

How did it go?