Calculate this average rate of change to show that it is the same as the slope m.
f(x2) − f(x1)
x2 − x1
 = 	
(mx2 + b) − 

 
  
Correct: Your answer is correct.
 

x2 − x1

 = 	
mx2 − 

 
  
Correct: Your answer is correct.
 

 + b − b
x2 − x1

 = 	
m 

 
  
Incorrect: Your answer is incorrect.
 

x2 − x1

 = 	
  
Correct: Your answer is correct.
 ,


To show that the average rate of change of the function \( f(x) = mx + b \) is equal to the slope \( m \), we can calculate the average rate of change between two points \( x_1 \) and \( x_2 \).

The average rate of change is given by:

\[
\frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]

Substitute \( f(x) = mx + b \) into the formula:

\[
\frac{f(x_2) - f(x_1)}{x_2 - x_1} = \frac{(mx_2 + b) - (mx_1 + b)}{x_2 - x_1}
\]

Simplify the numerator:

\[
(mx_2 + b) - (mx_1 + b) = mx_2 + b - mx_1 - b
\]

Combine like terms:

\[
mx_2 + b - mx_1 - b = mx_2 - mx_1
\]

Factor out \( m \):

\[
mx_2 - mx_1 = m(x_2 - x_1)
\]

Substitute back into the formula for the average rate of change:

\[
\frac{m(x_2 - x_1)}{x_2 - x_1}
\]

Simplify by canceling \( x_2 - x_1 \) (assuming \( x_2 \neq x_1 \)):

\[
m
\]

Thus, the average rate of change of the function \( f(x) = mx + b \) over any interval \( [x_1, x_2] \) is equal to the slope \( m \).

If you have any further questions or need more details, feel free to ask!

Here are some related questions to explore further:

1. How does the average rate of change differ for a quadratic function compared to a linear function?
2. Can the average rate of change be zero? What does that signify?
3. What happens to the average rate of change when \( x_1 = x_2 \)?
4. How can we interpret the average rate of change geometrically?
5. How would you find the instantaneous rate of change for a function?

**Tip:** The average rate of change of a function over an interval is geometrically the slope of the secant line connecting the two endpoints of the interval on the function's graph.

Calculate this average rate of change to show that it is the same as the slope m. f(x2) − f(x1) / (x2 − x1) = (mx2 + b) − x2 − x1 = mx2 − + b − b / (x2 − x1) = m.

Discover how to calculate the average rate of change for a linear function and verify that it matches the slope m. This problem explains the key concepts of linear functions, slope, and rate of change, providing a step-by-step solution. Ideal for students in Grades 8-10.

Math Problem Statement

Calculate this average rate of change to show that it is the same as the slope m. f(x2) − f(x1) x2 − x1

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation