Suppose you have a toy airplane that rises 5 feet for every 6 feet that it travels along the horizontal after takeoff. What would be the slope of its ascent? Would it be a positive value or a negative value?

### Slope

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 slope.

The slope 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.

Formally, the **slope** 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 formal definition of slope, \begin{align*}\frac{\text{vertical change}}{\text{horizontal change}}\end{align*} the slope of this hill can be written as \begin{align*}\frac{3 \text{ meters}}{4 \text{ meters}}=\frac{3}{4}\end{align*}. Also, since \begin{align*}\frac{3}{4}=75\%\end{align*}, we can say this hill has a 75% positive slope.

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

\begin{align*}\text{Slope}=\frac{\text{vertical change}}{\text{horizontal change}}=\frac{\text{-}3}{4}\end{align*}

The slope in this instance is negative because the car is traveling downhill.

Another way to think of slope is: \begin{align*}\text{slope}=\frac{\text{rise}}{\text{run}}\end{align*}.

When graphing an equation, slope is a very powerful tool. It provides the directions on how to get from one ordered pair to another. To determine slope, it is often helpful to draw a **slope-triangle**.

For example, take the graph below. Choose two ordered pairs on a graph of a that have integer values such as (–3, 0) and (0, –2). Now draw in the slope 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.

\begin{align*}Remember: \text{slope}=\frac{\text{rise}}{\text{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.

\begin{align*}\text{slope}=\frac{\text{rise}}{\text{run}}=\frac{\text{-}2}{+3}=\text{-}\frac{2}{3}\end{align*}

#### Let's use the information from above to solve the following problems:

- Find the slope of the line graphed below.

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

Draw in the slope 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.

\begin{align*}\text{Slope}=\frac{\text{rise}}{\text{run}}=\frac{+3}{+1}=\frac{3}{1}\end{align*}

A more algebraic way to determine a slope is by using a formula. The formula for slope is:

The slope between any two points \begin{align*}(x_1,y_1 )\end{align*} and \begin{align*}(x_2,y_2)\end{align*} is: \begin{align*}slope=\frac{y_2-y_1}{x_2-x_1}\end{align*}.

\begin{align*}(x_1,y_1)\end{align*} represents one of the two ordered pairs and \begin{align*}(x_2,y_2)\end{align*} represents the other. The following example helps show this formula.

- Using the slope formula, determine the slope of the equation graphed in the previous problem.

Use the integer ordered pairs used to form the slope triangle: (1, 1) and (0, –2). Since (1, 1) is written first, it can be called \begin{align*}(x_1,y_1)\end{align*}. That means \begin{align*}(0,\text{-}2)=(x_2,y_2)\end{align*}

Use the formula: \begin{align*}\text{slope}=\frac{y_2-y_1}{x_2-x_1}=\frac{\text{-}2-1}{0-1}=\frac{\text{-}3}{\text{-}1}=\frac{3}{1}\end{align*}

As you can see, the slope is the same regardless of the method you use. If the ordered pairs are fractional or spaced very far apart, it is easier to use the formula than to draw a slope triangle.

#### Types of Slopes

Slopes come in four different types: negative, zero, positive, and undefined. The first graph of this Concept had a negative slope. The second graph had a positive slope. Slopes with zero slopes are lines without any steepness, and undefined slopes cannot be computed.

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

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

#### Let's find the slope of the two lines shown in the graph below:

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

(-4, 3) and (1, 3). Choose one ordered pair to represent \begin{align*}(x_1,y_1)\end{align*} and the other to represent \begin{align*}(x_2,y_2)\end{align*}.

Now apply the formula: \begin{align*}slope=\frac{y_2-y_1}{x_2-x_1}=\frac{3-3}{1-(\text{-}4)}=\frac{0}{1+4}=0\end{align*}.

- To determine the slope of line \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)

\begin{align*}\text{slope}=\frac{y_2-y_1}{x_2-x_1}=\frac{\text{-}6-1}{5-5}=\frac{\text{-}7}{0}=\text{Undefined}\end{align*}

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

### Examples

#### Example 1

Earlier, you were told that you have a toy airplane that rises 5 feet for every 6 feet it travels along the horizontal after takeoff. What would be the slope of its ascent? Would it be positive or negative?

Slope is defined as the rise divided by the run of an object or line. In this case, the rise is 5 feet and the run is 6 feet. Therefore, the slope is \begin{align*}\frac{5}{6}\end{align*}. The value is positive because the plane is ascending and moving into the sky, not descending.

#### Example 2

Find the slope of each line in the graph below:

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

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

\begin{align*} \text{slope}=\frac{0-2}{5-0}=-\frac{2}{5}\end{align*}

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

\begin{align*} \text{slope}=\frac{1-1}{7-6}=\frac{0}{1}=0\end{align*}

The slopes can be seen in this graph:

### Review

- Define slope.
- Describe the two methods used to find slope. Which one do you prefer and why?
- What is the slope of all vertical lines? Why is this true?
- What is the slope of all horizontal lines? Why is this true?

Using the graphed coordinates, find the slope of each line.

In 8 – 20, find the slope between the two given points.

- (–5, 7) and (0, 0)
- (–3, –5) and (3, 11)
- (3, –5) and (–2, 9)
- (–5, 7) and (–5, 11)
- (9, 9) and (–9, –9)
- (3, 5) and (–2, 7)
- \begin{align*}\left (\frac{1}{2},\frac{3}{4}\right )\end{align*} and (–2, 6)
- (–2, 3) and (4, 8)
- (–17, 11) and (4, 11)
- (31, 2) and (31, –19)
- (0, –3) and (3, –1)
- (2, 7) and (7, 2)
- (0, 0) and \begin{align*}\left (\frac{2}{3},\frac{1}{4}\right )\end{align*}
- Determine the slope of \begin{align*}y=16\end{align*}.
- Determine the slope of \begin{align*}x=-99\end{align*}.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 4.7.