To find the exponential equation of the form:

$y = ab^x$

that passes through the points $(-3, \frac{1}{2})$ and $(1, 8)$, follow these steps:

Step 1: Plug in the points into the general equation

From $(-3, \frac{1}{2})$: $\frac{1}{2} = ab^{-3} \quad \text{(1)}$

From $(1, 8)$: $8 = ab^1 \quad \text{(2)}$

Step 2: Solve for $a$ and $b$

Equation (2):

$8 = ab \implies a = \frac{8}{b} \quad \text{(3)}$

Substitute equation (3) into equation (1):

$\frac{1}{2} = \frac{8}{b}b^{-3} = \frac{8}{b^4}$

Multiply both sides by $b^4$:

$\frac{1}{2}b^4 = 8 \quad \Rightarrow \quad b^4 = 16 \quad \Rightarrow \quad b = 2$

Step 3: Solve for $a$

Substitute $b = 2$ into equation (3):

$a = \frac{8}{2} = 4$

Final Equation

Thus, the exponential equation is:

$y = 4(2)^x$

Verification

Let's verify by plugging in the given points:

• For $(-3, \frac{1}{2})$: $y = 4(2)^{-3} = 4 \times \frac{1}{8} = \frac{4}{8} = \frac{1}{2}$

• For $(1, 8)$: $y = 4(2)^1 = 4 \times 2 = 8$

Both points satisfy the equation.

Would you like more details or have any questions?

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Here are 5 related questions to consider:

1. How can you find an exponential equation with more than two points?
2. What happens if the base $b$ is a fraction instead of an integer?
3. How can you determine if a graph represents an exponential function?
4. How do transformations affect the graph of an exponential equation?
5. Can you find an exponential equation passing through negative $y$-values?

Tip: When solving exponential equations, solving for the base $b$ first helps simplify finding the constant $a$.