To find the rate of change (slope) and to write the equation of the line, we need to examine the points provided in the graph: (2, 80), (4, 60), and (6, 240).
Step 1: Calculate the Slope (Rate of Change)
The slope (m) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
We can use the points (2, 80) and (6, 240) to calculate the slope.
Using point (2, 80) as \((x_1, y_1)\) and (6, 240) as \((x_2, y_2)\):
\[ m = \frac{240 - 80}{6 - 2} = \frac{160}{4} = 40 \]
Step 2: Write the Equation of the Line
Using the slope-intercept form of the equation of a line, which is:
\[ y = mx + b \]
We have \(m = 40\). We need to find \(b\), the y-intercept. We can use one of the points provided; let's use (2, 80):
\[ 80 = 40(2) + b \]
\[ 80 = 80 + b \]
\[ b = 80 - 80 = 0 \]
So, the y-intercept \(b = 0\).
Now we can write the equation:
\[ y = 40x \]
Summary
- The slope of the function is \(m = 40\).
- The equation can be written as \(y = 40x\), where \(x\) represents time in hours and \(y\) represents distance.