Question

Use the graph to answer the question.

The graph shows an exponentially rising line passing through (-2, 1.28) and (1, 2.5)

Write the exponential equation shown in the graph.

Answer 1

To derive the exponential equation of the form \( y = ab^x \) based on the points given, we'll use the two points \((-2, 1.28)\) and \( (1, 2.5) \).

1. **Set up equations using the two points.**

For the point \((-2, 1.28)\):
\[
1.28 = ab^{-2}
\]

For the point \( (1, 2.5) \):
\[
2.5 = ab^1 = ab
\]

2. **Rearranging the equations**:
From the second equation, we can express \( a \):
\[
a = \frac{2.5}{b}
\]

Now, substitute \( a \) in the first equation:
\[
1.28 = \left(\frac{2.5}{b}\right) b^{-2}
\]
\[
1.28 = \frac{2.5}{b^3}
\]

3. **Solving for \( b \)**:
Rearranging gives:
\[
b^3 = \frac{2.5}{1.28}
\]
\[
b^3 \approx 1.953125
\]
Taking the cube root:
\[
b \approx \sqrt[3]{1.953125} \approx 1.26
\]

4. **Substituting back to find \( a \)**:
Now substitute \( b \) back into \( a = \frac{2.5}{b} \):
\[
a \approx \frac{2.5}{1.26} \approx 1.98
\]

5. **Write the final equation**:
Therefore, the exponential equation is:
\[
y \approx 1.98 \cdot (1.26)^x
\]

To create an accurate model, further adjustment with exact calculations might be necessary, but the estimated values yield:
\[
y \approx 1.98 \cdot (1.26)^x
\]