Math Problem Statement

The function y = f(x) is graphed below. What is the average rate of change of the function f(x) on the interval -9 ≤ x ≤ 1?

Solution

To find the average rate of change of the function f(x)f(x) on the interval [9,1][-9, 1], we use the formula for the average rate of change:

Average Rate of Change=f(b)f(a)ba\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}

where a=9a = -9 and b=1b = 1.

Step 1: Determine f(9)f(-9) and f(1)f(1)

From the graph:

  • f(9)=16f(-9) = -16 (this is the y-value corresponding to x=9x = -9),
  • f(1)=8f(1) = -8 (this is the y-value corresponding to x=1x = 1).

Step 2: Apply the formula

Now, substitute these values into the formula:

Average Rate of Change=f(1)f(9)1(9)=8(16)1+9=8+1610=810=0.8\text{Average Rate of Change} = \frac{f(1) - f(-9)}{1 - (-9)} = \frac{-8 - (-16)}{1 + 9} = \frac{-8 + 16}{10} = \frac{8}{10} = 0.8

Conclusion:

The average rate of change of f(x)f(x) on the interval [9,1][-9, 1] is 0.80.8.

Would you like further details or have any questions?

Relative Questions:

  1. How do you interpret the average rate of change of a function geometrically?
  2. What does a positive average rate of change imply about the function over the given interval?
  3. How is the average rate of change related to the slope of a secant line?
  4. Can the average rate of change be negative? Under what conditions?
  5. How does the concept of the derivative relate to the average rate of change?

Tip:

The average rate of change between two points on a function is essentially the slope of the line connecting those two points (the secant line).

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Calculus
Average Rate of Change
Functions

Formulas

Average Rate of Change = (f(b) - f(a)) / (b - a)

Theorems

Secant Line Slope Theorem

Suitable Grade Level

Grades 10-12