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 based on the points given, we can analyze the two points plotted on the curve: (1, 14) and (2, 4).

The general form of an exponential equation is \( y = a \cdot b^x \), where:
- \( a \) is the initial value (when \( x = 0 \))
- \( b \) is the base (the growth or decay factor)

Using the point (1, 14), we can plug it into the equation:
\[ 14 = a \cdot b^1 \]
\[ 14 = a \cdot b \] (1)

Using the point (2, 4):
\[ 4 = a \cdot b^2 \] (2)

Now, we can express \( a \) in terms of \( b \) from equation (1):
\[ a = \frac{14}{b} \]

Substituting this into equation (2):
\[ 4 = \left(\frac{14}{b}\right) \cdot b^2 \]
\[ 4 = 14b \]
\[ b = \frac{4}{14} = \frac{2}{7} \]

Now substitute \( b \) back to find \( a \):
\[ a = \frac{14}{\frac{2}{7}} = 14 \cdot \frac{7}{2} = 49 \]

Thus, the exponential equation is:
\[ y = 49 \cdot \left(\frac{2}{7}\right)^x \]

Expressing \(\frac{2}{7}\) as \( \left(27\right)^{-1} \):
\[ y = 49 \cdot (7^{-1})^x = 49 \cdot 72^{-x} \]

Hence, in standard exponential form the equation looks like \( y = 49 \cdot \left(\frac{1}{27}\right)^x \).

Comparing this with the options provided, the form of the equation matches:

**Option #4: \( y = 49(72)^{-x} \)**

Therefore, the correct answer is:

**Option #4** is the correct equation for the graph.