Question

an exponential function contains ordered pairs as shown in the table below.

\

$$\begin{tabular}{|c|c|} \\hline x & y \\\\ \\hline 0 & 15 \\\\ \\hline 1 & 60 \\\\ \\hline 2 & 240 \\\\ \\hline 3 & 960 \\\\ \\hline 4 & 3840 \\\\ \\hline \\end{tabular}$$

what is the rate of growth for the function? 5 points

a. 15
b. 45
c. 4
d. 60

Explanation:

Identify the given data points

We are given a table of values representing an exponential function:

• When \(x = 0\), \(y = 15\)
• When \(x = 1\), \(y = 60\)
• When \(x = 2\), \(y = 240\)
• When \(x = 3\), \(y = 960\)
• When \(x = 4\), \(y = 3840\)

Recall the exponential function model

Using the Exponential Functions concept, a standard exponential function has the form:

$$y = a \cdot b^x$$

where:

• \(a\) is the initial value (the \(y\)-intercept, where \(x = 0\))
• \(b\) is the growth factor (or rate of growth)

Determine the initial value

Using the point \((0, 15)\):

$$15 = a \cdot b^0 \implies a = 15$$

Calculate the growth factor

Using the Exponential Growth concept, we find the ratio of consecutive \(y\)-values as \(x\) increases by \(1\):

$$b = \frac{y_{x+1}}{y_x}$$

Let's calculate this ratio for the first few steps:

$$b = \frac{60}{15} = 4$$

$$b = \frac{240}{60} = 4$$

$$b = \frac{960}{240} = 4$$

$$b = \frac{3840}{960} = 4$$

The constant ratio, which represents the rate of growth (growth factor), is \(4\).

Answer:

• A. 15
• B. 45
• C. 4 (Correct answer)
• D. 60