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# Slope

## Understand slope as the steepness of a line.

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Introduction to Rate of Change

Learning Goal:

By the end of this lesson I can determine the rate of change using rise over run.

Suppose you have a toy airplane, and upon takeoff, it rises 5 feet for every 6 feet that it travels along the horizontal. What would be the steepness of its ascent? Would it be a positive value or a negative value? In this Concept, you'll learn how to determine the rate of change of a line by analyzing vertical change and horizontal change so that you can handle problems such as this one.

### Guidance

The pitch of a roof, the slant of a ladder against a wall, the incline of a road, and even your treadmill incline are all examples of rate of change.

The rate of change of a line measures its steepness (either negative or positive).

For example, if you have ever driven through a mountain range, you may have seen a sign stating, “10% incline.” The percent tells you how steep the incline is. You have probably seen this on a treadmill too. The incline on a treadmill measures how steep you are walking uphill. Below is a more formal definition of rate of change.

The rate of change of a line is the vertical change divided by the horizontal change.

In the figure below, a car is beginning to climb up a hill. The height of the hill is 3 meters and the length of the hill is 4 meters. Using the definition above, the rate of change of this hill can be written as 3 meters4 meters=34\begin{align*}\frac{3 \ meters}{4 \ meters}=\frac{3}{4}\end{align*}. Because 34=75%\begin{align*}\frac{3}{4}=75\%\end{align*}, we can say this hill has a 75% positive rate of change.

Similarly, if the car begins to descend down a hill, you can still determine the rate of change.

Rate of Change=vertical changehorizontal change=34\begin{align*}Rate \ of \ Change=\frac{vertical \ change}{horizontal \ change}=\frac{-3}{4}\end{align*}

The rate of change in this instance is negative because the car is traveling downhill.

It is helpful to think of the rate of change of the line as the “rise-over-run.” That is, the rate of change is the ratio of the vertical (up and down) rise of a line to its horizontal (left to right) run.

To help us understand this ratio, let's look at line AB\begin{align*}AB\end{align*} on the coordinate plane below.

Imagine placing your finger on point A\begin{align*}A\end{align*}. To move from point A\begin{align*}A\end{align*} to point B\begin{align*}B\end{align*}, your finger would need to move 5 units up and then move 6 units to the right. That is because the line has a rise of 5 units up and a run of 6 units to the right.

So, Rate of Change=RiseRun\begin{align*}Rate \ of \ Change = \frac{Rise}{Run}\end{align*}.

The rate of change of line AB\begin{align*}AB\end{align*} is 56\begin{align*}\frac{5}{6}\end{align*}.

Line AB\begin{align*}AB\end{align*} has a rate of change of 56\begin{align*}\frac{5}{6}\end{align*}, which is a positive rate of change.

Notice that line AB\begin{align*}AB\end{align*} slants up from left to right!!

Knowing some basic information about the rate of change of a line can tell you about its slant.

• A line that slants up from left to right has a positive rate of change.
• A line that slants down from left to right has a negative rate of change.

Determine if the rate of change of each line shown below is positive or negative.

a.

b.

Consider the line in a\begin{align*}a\end{align*}.

The line slants down from left to right, so its rate of change is negative.

Consider the line in b\begin{align*}b\end{align*}.

The line slants up from left to right, so its rate of change is positive.

When graphing a linear relation, rate of change is a very powerful tool. It provides the directions on how to get from one ordered pair to another. To determine rate of change, it is helpful to use a rate-triangle.

Using the following graph, choose two ordered pairs that have integer values such as (–3, 0) and (0, –2). Now draw in the rate triangle by connecting these two points as shown.

The vertical leg of the triangle represents the rise of the line and the horizontal leg of the triangle represents the run of the line. A third way to represent rate of change is:

Rate of Change=RiseRun\begin{align*}Rate \ of \ Change = \frac{Rise}{Run}\end{align*}

Starting at the left-most coordinate, count the number of vertical units and horizontal units it took to get to the right-most coordinate.

Rate of Change=RiseRun=2+3=23\begin{align*}Rate \ of \ Change = \frac{Rise}{Run} = \frac{-2}{+3} = - \frac{2}{3}\end{align*}

Example A

Find the rate of change of the line graphed below.

Solution: Begin by finding two pairs of ordered pairs with integer values: (1, 1) and (0, –2).

Draw in the rate triangle.

Count the number of vertical units to get from the left ordered pair to the right.

Count the number of horizontal units to get from the left ordered pair to the right.

Rate of Change=RiseRun=+3+1=31=3\begin{align*}Rate \ of \ Change = \frac{Rise}{Run} = \frac{+3}{+1} = \frac{3}{1} = 3\end{align*}

#### Rates of Change come in four different types: negative, zero, positive, and undefined. The first graph of this Concept had a negative rate of change. The second graph had a positive rate of change. Rates of Change equal to zero are lines without any steepness, and if they are undefined then the rate of change  cannot be computed.

Any line with a rate of change of zero will be a horizontal line with equation y=some number\begin{align*}y = some \ number\end{align*}.

Any line with an undefined rate of change will be a vertical line with equation x=some number\begin{align*}x = some \ number\end{align*}.

We will use the next two graphs to illustrate the previous definitions.

#### Example B

To determine the rate of change of line A\begin{align*}A\end{align*}, you need to find two ordered pairs with integer values.

(–4, 3) and (1, 3).

Now apply:   Rate of Change=RiseRun\begin{align*}Rate \ of \ Change = \frac{Rise}{Run}\end{align*}.

Rate of Change=RiseRun=05=0\begin{align*}Rate \ of \ Change = \frac{Rise}{Run} = \frac{0}{5} = 0\end{align*}

To determine the rate of change of line B\begin{align*}B\end{align*}, you need to find two ordered pairs on this line with integer values and apply the formula.

(5, 1) and (5, –6)

Rate of Change=RiseRun=70=undefined\begin{align*}Rate \ of \ Change = \frac{Rise}{Run} = \frac{7}{0} = undefined\end{align*}

It is impossible to divide by zero, so the rate of change of line B\begin{align*}B\end{align*} cannot be determined and is called undefined.

### Guided Practice

Find the rate of change of each line in the graph below:

Solution:

For each line, identify two coordinate pairs on the line and use them to calculate the rate of change.

For the green line, one choice is (0,2)\begin{align*} (0, 2)\end{align*} and (5,0)\begin{align*} (5, 0)\end{align*}. This results in a rate of change of:

Rate of Change=RiseRun=25=25\begin{align*}Rate \ of \ Change = \frac{Rise}{Run} = \frac{-2}{5} = -\frac{2}{5}\end{align*}

For the blue line, one choice is (6,1)\begin{align*} (6, 1)\end{align*} and (7,1)\begin{align*} (7, 1)\end{align*}. This results in a rate of change of

Rate of Change=RiseRun=01=0\begin{align*}Rate \ of \ Change = \frac{Rise}{Run} = \frac{0}{1} = 0\end{align*}:

The rates of change can be seen in this graph:

### Practice

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Slope and Rate of Change (13:42)

1. Define rate of change.
2. Describe how you can determine the rate of change. What must you remember when choosing your points on the graph?
3. What is the rate of change of all vertical lines? Why is this true?
4. What is the rate of change of all horizontal lines? Why is this true?

Using the graphed coordinates, find the rate of change of each line.

5.

In #9 to #19,

a) plot the points to graph the line

b) determine the rate of change between the two given points.

1. (–5, 7) and (0, 0)
2. (–3, –5) and (3, 11)
3. (3, –5) and (–2, 9)
4. (–5, 7) and (–5, 11)
5. (9, 9) and (–9, –9)
6. (3, 5) and (–2, 7)
7. (–2, 3) and (4, 8)
8. (–17, 11) and (4, 11)
9. (31, 2) and (31, –19)
10. (0, –3) and (3, –1)
11. (2, 7) and (7, 2)
12. Determine the rate of change of the line  y=16\begin{align*}y=16\end{align*}.
13. Determine the rate of change of the line  x=99\begin{align*}x=-99\end{align*}.
14. Does a positive rate of change have to contain positive numbers?

15. For each graph, tell if the rate of change of the line shown is positive, negative, zero, or undefined

1. b.

c.

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