Question

Use the image to answer the question.

A descending curve is graphed in quadrant 1 of a coordinate plane. The points left parenthesis 1 comma 14 right parenthesis and left parenthesis 2 comma 4 right parenthesis are plotted on the curve. The curve descends to but does not cross the x-axis.

Given the graph of the exponential equation, which option is the correct exponential equation for the graph?

Option #1: y=7(12)x

Option #2: y=49(27)x

Option #3: y=7(27)x

Option #4: y=49(72)x

(1 point)
Option #
is the correct equation for the graph.

Answer 1

To determine the correct exponential equation for the graph described, we can utilize the points provided: (1, 14) and (2, 4).

In general, an exponential equation can be represented in the form \( y = ab^x \). We can use the two points to set up a system of equations.

1. Using the point (1, 14):
\[
14 = ab^1 \implies ab = 14 \quad \text{(Equation 1)}
\]

2. Using the point (2, 4):
\[
4 = ab^2 \quad \text{(Equation 2)}
\]

Now, we can express \( ab^2 \) in terms of Equation 1:
- From Equation 1, we know \( a = \frac{14}{b} \).
- Substitute \( a \) in Equation 2:
\[
4 = \left(\frac{14}{b}\right)b^2 \implies 4 = 14b \implies b = \frac{4}{14} = \frac{2}{7}
\]

Now, substituting \( b \) back into Equation 1 to find \( a \):
- \( ab = 14 \) becomes:
\[
a \left(\frac{2}{7}\right) = 14 \implies a = 14 \cdot \frac{7}{2} = 49
\]

Putting it all together, we have:
\[
y = 49\left(\frac{2}{7}\right)^x
\]

Now, we can look at the options provided to find an equivalent expression. Rewriting \(\frac{2}{7}\) as \( \frac{27}{49} \) gives:
\[
y = 49 \cdot \left(\frac{2}{7}\right)^x = 49 \cdot \left(\frac{1}{\frac{7}{2}}\right)^x = 49(27)^{-x}
\]

After some consideration, we realize that since the curve descends but does not cross the x-axis, it indicates a decay (which the fraction represents). Looking at the options:

1. None of the options provide a clearly correct representation since \(27^{-x}\) wasn't listed; however, based on the decay,
2. The most logical option that aligns with our value of \(49\) for \(a\) is:

**Option #2: \(y=49(27^{-x})\), which matches closely the form derived.**

Since the exact negative exponent was absent, we rely on the shape and derived values.

So, the answer is **Option #2: y=49(27)^x** is the closest representation of the trend represented in the question.