Question

write an exponential function for the graph. write the function in the form $y = a(b)^x$.

Explanation:

Step1: Identify y-intercept (a)

The graph crosses the y-axis at $(0,1)$. For $y=a(b)^x$, when $x=0$, $y=a(b)^0=a$. So $a=1$.

Step2: Use another point to find b

Take the point $(-2, \frac{1}{4})$. Substitute $x=-2$, $y=\frac{1}{4}$, $a=1$ into the function:
$$\frac{1}{4}=1(b)^{-2}$$
Rewrite $b^{-2}$ as $\frac{1}{b^2}$:
$$\frac{1}{4}=\frac{1}{b^2}$$
Cross-multiply: $b^2=4$, so $b=2$ (since the function is increasing, $b>1$).

Step3: Build the function

Substitute $a=1$ and $b=2$ into $y=a(b)^x$.

Answer:

$y=2^x$