Question

Use the graph to answer the question.

The graph shows an exponentially rising line passing through (-1, 1/3) and (1, 3/4)

Write the exponential equation shown in the graph.

Answer 1

To find the exponential equation of the form \(y = ab^x\) that passes through the points (-1, 1/3) and (1, 3/4), we can set up a system of equations using the given points.

1. From the point (-1, 1/3):
\[
y = ab^{-1} = \frac{1}{3}
\]
This can be rewritten as:
\[
\frac{a}{b} = \frac{1}{3} \quad \text{(Equation 1)}
\]

2. From the point (1, 3/4):
\[
y = ab^1 = \frac{3}{4}
\]
This gives us:
\[
ab = \frac{3}{4} \quad \text{(Equation 2)}
\]

Now, we can express \(a\) in terms of \(b\) using Equation 1:
\[
a = \frac{1}{3}b
\]

Substituting \(a\) into Equation 2:
\[
\left(\frac{1}{3}b\right)b = \frac{3}{4}
\]
\[
\frac{1}{3}b^2 = \frac{3}{4}
\]

Next, we can multiply both sides by 3 to eliminate the fraction:
\[
b^2 = \frac{3 \times 3}{4} = \frac{9}{4}
\]

Now, taking the square root of both sides:
\[
b = \frac{3}{2} \quad \text{(using the positive root since we are modeling growth)}
\]

Now we can substitute \(b = \frac{3}{2}\) back into Equation 1 to find \(a\):
\[
\frac{a}{\frac{3}{2}} = \frac{1}{3}
\]
Multiplying both sides by \(\frac{3}{2}\):
\[
a = \frac{1}{3} \cdot \frac{3}{2} = \frac{1}{2}
\]

Thus, we have found \(a\) and \(b\):
\[
a = \frac{1}{2}, \quad b = \frac{3}{2}
\]

Finally, the exponential equation is:
\[
y = \frac{1}{2} \left(\frac{3}{2}\right)^x
\]

This is the exponential equation that corresponds to the rising line in the graph given the specified points.