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Linear Function 1:
A coordinate plane shows the x-axis ranging from negative 1 to 4 in increments of 1 and the y-axis ranging from negative 1 to 13 in increments of 1. A line with arrows at both ends joins two plotted points. The coordinates of the plotted points are as follows: left parenthesis 1 comma 9 right parenthesis and left parenthesis 2 comma 12 right parenthesis.

Values x and y for a Linear Function
x y
0 2
5 42
Compare Linear Function 1 with Linear Function 2. Which one has the greater rate of change?

(1 point)
Responses

Linear Function 1 has the greater rate of change, with a rate of change equal to 6 while Linear Function 2 has a rate of change equal to 2.
Linear Function 1 has the greater rate of change, with a rate of change equal to 6 while Linear Function 2 has a rate of change equal to 2.

Linear Function 2 has the greater rate of change, with a rate of change equal to 8 while Linear Function 1 has a rate of change equal to 3.
Linear Function 2 has the greater rate of change, with a rate of change equal to 8 while Linear Function 1 has a rate of change equal to 3.

Linear Function 2 has the greater rate of change, with a rate of change equal to 8 while Linear Function 1 has a rate of change equal to 6.
Linear Function 2 has the greater rate of change, with a rate of change equal to 8 while Linear Function 1 has a rate of change equal to 6.

Linear Function 1 has the greater rate of change, with a rate of change equal to 3

1 answer

To determine which linear function has a greater rate of change, we need to calculate the rate of change for both Linear Function 1 and Linear Function 2.

For Linear Function 1: We have the two points (1, 9) and (2, 12). The formula for the rate of change (slope) is:

\[ \text{Rate of Change} = \frac{y_2 - y_1}{x_2 - x_1} \]

Using the points:

  • \( (x_1, y_1) = (1, 9) \)
  • \( (x_2, y_2) = (2, 12) \)

Substituting into the formula:

\[ \text{Rate of Change} = \frac{12 - 9}{2 - 1} = \frac{3}{1} = 3 \]

For Linear Function 2: We have the two points (0, 2) and (5, 42). Using the same formula:

\[ \text{Rate of Change} = \frac{y_2 - y_1}{x_2 - x_1} \]

Using the points:

  • \( (x_1, y_1) = (0, 2) \)
  • \( (x_2, y_2) = (5, 42) \)

Substituting into the formula:

\[ \text{Rate of Change} = \frac{42 - 2}{5 - 0} = \frac{40}{5} = 8 \]

Comparison:

  • Rate of Change of Linear Function 1 = 3
  • Rate of Change of Linear Function 2 = 8

Thus, Linear Function 2 has the greater rate of change. Therefore, the correct response is:

Linear Function 2 has the greater rate of change, with a rate of change equal to 8 while Linear Function 1 has a rate of change equal to 3.