How to find the average rate of change between two points?
Understand the Problem
The question is asking how to calculate the average rate of change of a function between two specified points. This typically involves using the formula for average rate of change, which is (f(b) - f(a)) / (b - a) where a and b are the two points on the function.
Answer
The average rate of change is $4$.
Answer for screen readers
Let’s assume the function is $f(x) = x^2$, and you are evaluating it between the points $a = 1$ and $b = 3$.
- Find $f(1) = 1^2 = 1$.
- Find $f(3) = 3^2 = 9$.
Now substitute these values into the formula:
$$ \text{Average Rate of Change} = \frac{9 - 1}{3 - 1} = \frac{8}{2} = 4 $$
Thus, the average rate of change is $4$.
Steps to Solve
- Identify the Function and Points
Determine the function $f(x)$ you are working with and identify the two points $a$ and $b$ between which you want to calculate the average rate of change.
- Evaluate the Function at the Points
Calculate the function values at the identified points. This means finding $f(a)$ and $f(b)$.
- Apply the Average Rate of Change Formula
Use the average rate of change formula:
$$ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} $$
Substitute the values of $f(a)$ and $f(b)$ along with the values of $a$ and $b$ into the formula.
- Simplify the Expression
Perform the arithmetic operations to simplify the expression and find the average rate of change.
Let’s assume the function is $f(x) = x^2$, and you are evaluating it between the points $a = 1$ and $b = 3$.
- Find $f(1) = 1^2 = 1$.
- Find $f(3) = 3^2 = 9$.
Now substitute these values into the formula:
$$ \text{Average Rate of Change} = \frac{9 - 1}{3 - 1} = \frac{8}{2} = 4 $$
Thus, the average rate of change is $4$.
More Information
This average rate of change indicates that, on average, the function $f(x) = x^2$ increases by $4$ units for each unit increase in $x$ between the points $1$ and $3$.
Tips
- Not correctly substituting the values of $f(a)$ or $f(b)$ into the formula.
- Forgetting to simplify the final fraction.