To analyze the data in the provided table, we need to determine the rate of change, the y-intercept, and the equation of the line in slope-intercept form.
Given Points:
- (-2, 80)
- (-1, 70)
- (2, 40)
- (5, 10)
1. Calculate the rate of change (slope):
To find the slope (rate of change), we can use two of the points. Let's use the points (-2, 80) and (5, 10):
The formula for the slope \( m \) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Using the points (-2, 80) as \((x_1, y_1)\) and (5, 10) as \((x_2, y_2)\): \[ m = \frac{10 - 80}{5 - (-2)} = \frac{-70}{7} = -10 \]
The rate of change = -10.
2. Find the y-intercept (b):
The y-intercept is the value of \( y \) when \( x = 0 \). We can use the point-slope form, using the slope we just calculated (-10) and one of the points to find the y-intercept.
Using the point (-2, 80): \[ y - y_1 = m(x - x_1) \] Substituting: \[ y - 80 = -10(x + 2) \] Expanding this: \[ y - 80 = -10x - 20 \] \[ y = -10x + 60 \]
The y-intercept = \( b = 60 \).
3. The equation of the line in slope-intercept form is: \[ y = -10x + 60 \]
So, summarizing:
- The rate of change = -10
- The y-intercept = 60
- The equation of the line in slope-intercept form is \( y = -10x + 60 \).
3 answers