To find the rate of change, the y-intercept, and the equation of the line in slope-intercept form (y = mx + b) using the provided points, we follow these steps:
Step 1: Rate of Change (Slope)
The rate of change (slope, m) can be calculated using two points from the table. The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
We can use the points (-2, 80) and (5, 10):
- Point 1: (-2, 80) \(\Rightarrow x_1 = -2\), \(y_1 = 80\)
- Point 2: (5, 10) \(\Rightarrow x_2 = 5\), \(y_2 = 10\)
Plugging values into the slope formula:
\[ m = \frac{10 - 80}{5 - (-2)} = \frac{-70}{7} = -10 \]
The rate of change (slope) = -10
Step 2: Y-Intercept
To find the y-intercept (b), we can use the slope-intercept form of the equation \(y = mx + b\) and one of the points. Let's use the point (5, 10):
Using the formula:
\[ 10 = -10(5) + b \]
Calculating the right-hand side:
\[ 10 = -50 + b \implies b = 10 + 50 = 60 \]
The y-intercept = 60
Step 3: Slope-Intercept Form Equation
Now that we have the slope \(m = -10\) and the y-intercept \(b = 60\), we can write the slope-intercept form of the equation as:
\[ y = -10x + 60 \]
The equation of the line in slope-intercept form is:
\[ y = -10x + 60 \]
Summary of Answers:
- The rate of change = -10
- The y-intercept = 60
- The equation of the line in slope-intercept form is y = -10x + 60