Two Points Determine an Exponential Function I

Task

The graph of a function of the form $f(x)=ab^x$ is shown below. Find the values of $a$ and $b$.

IM Commentary

A more sophisticated version of this problem is F.LE Two Points Determine an Exponential Function 2.

Solution

The value of the function decreases from 2 to $\frac{1}{2}$ by multiplying 2 twice by $b$.

$$2 \times b \times b = \frac{1}{2}.$$

Writing this as $2b^2 = \frac{1}{2}$, we divide by 2 to obtain $b^2=\frac{1}{4}$, so $b=\pm\frac{1}{2}$.  Since the base $b$ must be positive, we conclude $b = \frac{1}{2}$.

Now since the point $(0, 2)$ lies on the graph of $f$, we know $f(0)=2$, and substituting that input-output pair into the expression for $f(x)$ yields

$$ 2=a\left(\frac{1}{2}\right)^0.$$

Thus $a=2$ and so $f(x) = 2\left(\frac{1}{2}\right)^x$.