Math Snap

STEP 1

1. The function is exponential and can be expressed as $y = a b^x$.
2. The table provides specific values of $x$ and corresponding $y$.

STEP 2

1. Use the initial value to determine $a$.
2. Use the ratio of consecutive $y$-values to determine $b$.
3. Verify the values of $a$ and $b$ using additional points.

STEP 3

The value of $y$ when $x = 0$ is 4. Since $y = a b^0 = a$, we have:
$$a = 4$$

STEP 4

To find $b$, use the ratio of consecutive $y$-values. For example, when $x = 1$, $y = 2$ and when $x = 0$, $y = 4$. Therefore:
$$b = \frac{y_1}{y_0} = \frac{2}{4} = \frac{1}{2}$$

Verify the values of $a$ and $b$ using additional points. Check $x = 2$, where $y = 1$:
$$y = a b^2 = 4 \left(\frac{1}{2}\right)^2 = 4 \times \frac{1}{4} = 1$$ The values are consistent with the table, confirming $a = 4$ and $b = \frac{1}{2}$.
The exponential function is:
$$a = 4$$ $$b = \frac{1}{2}$$