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### Course: Integrated math 1>Unit 8

Lesson 12: Average rate of change word problems

# Average rate of change review

Review average rate of change and how to apply it to solve problems.

## What is average rate of change?

The average rate of change of function $f$ over the interval $a\le x\le b$ is given by this expression:
$\frac{f\left(b\right)-f\left(a\right)}{b-a}$
It is a measure of how much the function changed per unit, on average, over that interval.
It is derived from the slope of the straight line connecting the interval's endpoints on the function's graph.
Want to learn more about average rate of change? Check out this video.

## Finding average rate of change

### Example 1: Average rate of change from graph

Let's find the average rate of change of $f$ over the interval $0\le x\le 9$:
We can see from the graph that $f\left(0\right)=-7$ and $f\left(9\right)=3$.
$\begin{array}{rl}\text{Average rate of change}& =\frac{f\left(9\right)-f\left(0\right)}{9-0}\\ \\ & =\frac{3-\left(-7\right)}{9}\\ \\ & =\frac{10}{9}\end{array}$

### Example 2: Average rate of change from equation

Let's find the rate of change of $g\left(x\right)={x}^{3}-9x$ over the interval $1\le x\le 6$.
$g\left(1\right)={1}^{3}-9\cdot 1=-8$
$g\left(6\right)={6}^{3}-9\cdot 6=162$
$\begin{array}{rl}\text{Average rate of change}& =\frac{g\left(6\right)-g\left(1\right)}{6-1}\\ \\ & =\frac{162-\left(-8\right)}{5}\\ \\ & =34\end{array}$
Problem 1
What is the average rate of change of $g$ over the interval $-8\le x\le -2$?

Want to try more problems like this? Check out this exercise.

## Want to join the conversation?

• Over which interval does h have a negative average rate of change? Can I ask for a some help please? because I looked at the problems above but it still seems a little confusing to me.
(15 votes)
• Remember that the rate of change is just the slope of the function. Look back at some of those problems to identify intervals with positive and negative slopes.

Hope this helps. <|:)
(27 votes)
• I need help to solve this and I don't know how to solve this.

Here is the question and the problem:
Solve the system of equations.
−9y+4x−20=0
-7y+16x-80=0
(3 votes)
• First, it will simplify things if we convert everything to standard form (Ax+By=C) such that the terms without a variable are on the other side of the equation.

In this way, we get:

4x-9y=20 and 16x-7y=80

Then, we look for a way to get one of the variables to cancel out with the other equation. Thus, we multiply the entirety of the first equation by 4:

16x-36y=80 and 16x-7y=80

Since we have identical coefficients for the x-variable in both equations, we can subtract one equation from the other so that the x-terms cancel out.

16x-36y=80
-16x-7y =80
-----------------
-43y=0

We have successfully isolated y. From here, we can divide both sides of the equation by -43 to get the value of y:

y=0

From here, we can plug the y-value back into one of the previous equations to determine the x-value:

4x-9y=20
4x-9(0)=20
4x=20

This yields the solution:

x=5

In these system of equations problems, your strategy should be as follows: choose one variable and eliminate it, solve for the other variable, and then plug the value of the solved variable into the original equation to solve for the unsolved variable.
(9 votes)
• Find the rate change between f(3)=10 and f(7)=18
(2 votes)
• You were given two ordered pairs, but in the function notation. Hint: f(x)=y. Use this to identify the (x, y) ordered pairs. Then, use the slope formula to calculate the rate of change.

Hope this helps.
(6 votes)
• What interval should I use if I was given 0<t<10?
(3 votes)
• That is the interval or inputs so you should find the corresponding OUTPUTS.
(3 votes)
• can there be no solution to this type of problem?
(2 votes)
• Finding an average rate of change is just finding the slope between 2 points. You can always find the slope.
m = (y2-y1)/(x2-x1)
The slope could be 0. It could be a number/0 = undefined. Or, it could be an integer or fraction.
(4 votes)
• Why are we doing the rate of change with these equations instead of how we were taught in the videos?
(2 votes)
• The formula in these equations is more applicable to the work you will be doing in maths, especially when functions get involved. It still means the same thing, with 'f(b) - f(a)/ b - a' being a different way of writing 'the change in y/ the change in x'.

Why the formula is not written in the videos is unclear; I assume it's because writing the same formula over and over again will make the videos feel drawn out.
(3 votes)
• what if the dots on your graft land on the number line itself. Like (-6) and (-4). Would that mean that the other number is "0" for both?
(2 votes)
• Yes. If the dot lands on the horizontal line, it is called the x-intercept. While the dot which lands on a vertical line is called the y-intercept. Basically, x-intercept is when x > 0 and y = 0. While y-intercept is when y > 0 and x = 0.
(2 votes)
• is it just (y1-y2)/(x1-x2) to find the Rate of change
(2 votes)
• Yep. It is also the formula for the slope.
Hope this was helpful! This is my first real answer :)
(1 vote)
• f(x)=x
2
−x−1f, left parenthesis, x, right parenthesis, equals, x, squared, minus, x, minus, 1
Over which interval does fff have an average rate of change of zero?
(1 vote)
• Your function creates a parabola when graphed. So have an average rate of change = 0, your interval would need 2 points on direct opposite sides of the parabola. A line thru those 2 points would be a horizontal line and have a slope of 0.
(3 votes)
• What is the average rate of change of F over the interval -7≤x≤2?
(2 votes)