Question

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(1/2)^x

Option #2: y=49(2/7)^x

Option #3: y=7(2/7)^x

Option #4: y=49(7/2)^x

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

Answer 1

To determine the correct exponential equation for the given curve that passes through the points (1, 14) and (2, 4), we can start by substituting the x-values of these points into each option to check for consistency.

Option 1: \( y = 7\left(\frac{1}{2}\right)^x \)

• For \( x = 1 \):
  \( y = 7\left(\frac{1}{2}\right)^1 = 7 \cdot \frac{1}{2} = 3.5 \) (not equal to 14)

Option 2: \( y = 49\left(\frac{2}{7}\right)^x \)

• For \( x = 1 \):
  \( y = 49\left(\frac{2}{7}\right)^1 = 49 \cdot \frac{2}{7} = 14 \) (matches)
• For \( x = 2 \):
  \( y = 49\left(\frac{2}{7}\right)^2 = 49 \cdot \frac{4}{49} = 4 \) (matches)

Option 3: \( y = 7\left(\frac{2}{7}\right)^x \)

• For \( x = 1 \):
  \( y = 7\left(\frac{2}{7}\right)^1 = 7 \cdot \frac{2}{7} = 2 \) (not equal to 14)

Option 4: \( y = 49\left(\frac{7}{2}\right)^x \)

• For \( x = 1 \):
  \( y = 49\left(\frac{7}{2}\right)^1 = 49 \cdot \frac{7}{2} = 171.5 \) (not equal to 14)

From this analysis, only Option 2 satisfies both points (1, 14) and (2, 4).

Therefore, the correct equation for the graph is:

Option #2: \( y = 49\left(\frac{2}{7}\right)^x \).