Questions: Determine whether the table represents a linear or an exponential function. Explain. x y 0 3 1 7 2 11 3 15 Danielle responded to her teacher: This is an exponential sequence because the numbers are increasing. The numbers are doubling. Is Danielle correct? Why or why not? Write your response in complete sentences using appropriate academic vocabulary. Include terms "linear function" "exponential function" and "rate" in your response.

Solution Steps

To determine whether the table represents a linear or an exponential function, we need to analyze the rate of change between the values of \( y \) as \( x \) increases. For a linear function, the rate of change (difference between consecutive \( y \) values) should be constant. For an exponential function, the ratio between consecutive \( y \) values should be constant.

1. Calculate the differences between consecutive \( y \) values to check for a constant rate of change (linear function).
2. Calculate the ratios between consecutive \( y \) values to check for a constant ratio (exponential function).
3. Compare the results to determine the type of function.

We calculate the differences between consecutive \( y \) values: \[ \begin{align_} y(1) - y(0) & = 7 - 3 = 4, \\ y(2) - y(1) & = 11 - 7 = 4, \\ y(3) - y(2) & = 15 - 11 = 4. \end{align_} \] The differences are \( [4, 4, 4] \), which are constant.

Next, we calculate the ratios between consecutive \( y \) values: \[ \begin{align_} \frac{y(1)}{y(0)} & = \frac{7}{3} \approx 2.3333, \\ \frac{y(2)}{y(1)} & = \frac{11}{7} \approx 1.5714, \\ \frac{y(3)}{y(2)} & = \frac{15}{11} \approx 1.3636. \end{align_} \] The ratios are approximately \( [2.3333, 1.5714, 1.3636] \), which are not constant.

Since the differences between consecutive \( y \) values are constant, we conclude that the function is linear. The constant rate of change confirms this. The ratios, however, are not constant, indicating that the function is not exponential.

Final Answer