Question

What is the rate of change in the graph?
points:
0,5 1,7 2,9 3,11 4,13

Answer 1

To find the rate of change between points on the graph, we typically calculate the slope between consecutive points. The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Let's calculate the slope between each pair of consecutive points in the dataset \( (0, 5) \), \( (1, 7) \), \( (2, 9) \), \( (3, 11) \), and \( (4, 13) \):

1. Between \( (0, 5) \) and \( (1, 7) \):
\[
m = \frac{7 - 5}{1 - 0} = \frac{2}{1} = 2
\]

2. Between \( (1, 7) \) and \( (2, 9) \):
\[
m = \frac{9 - 7}{2 - 1} = \frac{2}{1} = 2
\]

3. Between \( (2, 9) \) and \( (3, 11) \):
\[
m = \frac{11 - 9}{3 - 2} = \frac{2}{1} = 2
\]

4. Between \( (3, 11) \) and \( (4, 13) \):
\[
m = \frac{13 - 11}{4 - 3} = \frac{2}{1} = 2
\]

In all cases, the rate of change (slope) between the points is consistently \( 2 \). This means the graph is a straight line with a constant rate of change, indicating that for every increase of 1 in \( x \), \( y \) increases by \( 2 \).