Find the exponential equation whose graph passes through the points (-3,1/2) and (1,8)

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 \(a\).

Find the exponential equation whose graph passes through the points (-3,1/2) and (1,8).

Learn how to find the exponential equation passing through the points (-3, 1/2) and (1, 8). This step-by-step solution covers solving exponential functions using algebraic methods and applying key formulas such as y = ab^x. Ideal for students in Grades 9-12.

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