Question
find the rate of change and the initial value of the linear function given
(-3,2.5) (0,1) (2,0)
(-3,2.5) (0,1) (2,0)
Answers
GPT-4o mini
To find the rate of change (slope) and the initial value (y-intercept) of the linear function given the points (-3, 2.5), (0, 1), and (2, 0), we can follow these steps:
1. **Determine the slope (rate of change)** between two points. We can use any two points; however, let's use (0, 1) and (2, 0) for this calculation.
The formula for the slope \( m \) between two points (x₁, y₁) and (x₂, y₂) is:
\[
m = \frac{y₂ - y₁}{x₂ - x₁}
\]
Using points (0, 1) and (2, 0):
\[
m = \frac{0 - 1}{2 - 0} = \frac{-1}{2} = -0.5
\]
2. **Determine the y-intercept (initial value)**. The y-intercept is the value of y when x = 0. From the given coordinates (0, 1), we can see that the y-intercept is 1.
Putting it together, the rate of change (slope) of the linear function is \(-0.5\), and the initial value (y-intercept) is \(1\).
Additionally, the equation of the line in slope-intercept form (y = mx + b) would be:
\[
y = -0.5x + 1
\]
This confirms that the initial value (b) is 1. We can also check that this line passes through all three points provided, confirming consistency across the given data points.
1. **Determine the slope (rate of change)** between two points. We can use any two points; however, let's use (0, 1) and (2, 0) for this calculation.
The formula for the slope \( m \) between two points (x₁, y₁) and (x₂, y₂) is:
\[
m = \frac{y₂ - y₁}{x₂ - x₁}
\]
Using points (0, 1) and (2, 0):
\[
m = \frac{0 - 1}{2 - 0} = \frac{-1}{2} = -0.5
\]
2. **Determine the y-intercept (initial value)**. The y-intercept is the value of y when x = 0. From the given coordinates (0, 1), we can see that the y-intercept is 1.
Putting it together, the rate of change (slope) of the linear function is \(-0.5\), and the initial value (y-intercept) is \(1\).
Additionally, the equation of the line in slope-intercept form (y = mx + b) would be:
\[
y = -0.5x + 1
\]
This confirms that the initial value (b) is 1. We can also check that this line passes through all three points provided, confirming consistency across the given data points.