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x:0,1 y: 24,14,4 Question 3 Which option is the correct exponential equation for the graph above?(1 point) Responses y=49(2/7)^X, y is equal to 49 times 2 sevenths to the cap x th power, y=49(7/2)^X, y is equal to 49 times 7 halves to the cap x th power, y=7(2/7)^X

1 answer

To determine the correct exponential equation from the options given, we need to analyze the data points provided: \( (0, 24) \), \( (1, 14) \), and \( (2, 4) \).

Exponential functions generally have the form:

\[ y = a(b^x) \]

where \( a \) is the initial value (the value of \( y \) when \( x = 0 \)) and \( b \) is the base of the exponential decay or growth.

Step 1: Identify the initial value \( a \)

From the point \( (0, 24) \), we see that when \( x = 0 \), \( y = 24 \). Therefore, \( a = 24 \).

Step 2: Calculate the ratio between successive \( y \) values

Next, let's find the common ratio between the points.

  • From \( y = 24 \) (when \( x = 0 \)) to \( y = 14 \) (when \( x = 1 \)): \[ \text{Ratio} = \frac{14}{24} = \frac{7}{12} \]

  • From \( y = 14 \) (when \( x = 1 \)) to \( y = 4 \) (when \( x = 2 \)): \[ \text{Ratio} = \frac{4}{14} = \frac{2}{7} \]

Step 3: Analyze options given

Now, let’s consider the options provided:

  1. \( y = 49 \left(\frac{2}{7}\right)^x \)
  2. \( y = 49 \left(\frac{7}{2}\right)^x \)
  3. \( y = 7 \left(\frac{2}{7}\right)^x \)

Step 4: Testing the options

  • Testing Option 1: \( y = 49 \left(\frac{2}{7}\right)^x \)

    • At \( x = 0 \):
      • \( y = 49 \left(\frac{2}{7}\right)^0 = 49 \) (not correct, as it should be 24)
    • At \( x = 1 \):
      • \( y = 49 \left(\frac{2}{7}\right)^1 = 49 \cdot \frac{2}{7} = 14 \)
    • At \( x = 2 \):
      • \( y = 49 \left(\frac{2}{7}\right)^2 = 49 \cdot \frac{4}{49} = 4 \)

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

    • This will give increasing values which do not match the data (should be decreasing).

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

    • At \( x = 0 \):
      • \( y = 7 \left(\frac{2}{7}\right)^0 = 7 \) (not correct)
    • It does not match the required points.

Conclusion

After testing the equations, we find that Option 1: \( y = 49(\frac{2}{7})^x \) is the only one that correctly gives the outputs \( y = 24 \), \( y = 14 \), and \( y = 4 \) for the respective values of \( x = 0, 1, 2 \).

Thus, the correct exponential equation for the graph is:

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