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Understand Slope of a Line
Use geoboards to model slope
Use Geoboards to Model Slope
A geoboard is a board with a grid of pegs on it. Using rubber bands on a geoboard gives us a concrete way to model lines on a coordinate grid. By stretching a rubber band between two pegs on a geoboard, we can discover how to find the slope of a line.
We’ll start by stretching a rubber band between two pegs
Doesn’t it look like a line?
Now we stretch one part of the rubber band straight up from the left peg and around a third peg to make the sides of a right triangle, as shown in
We carefully make a 90º angle around the third peg, so one of the newly formed lines is vertical and the other is horizontal.
To find the slope of the line, we measure the distance along the vertical and horizontal sides of the triangle. The vertical distance is called the rise and the horizontal distance is called the run,
If our geoboard and rubber band look just like the one shown in , the rise is 2. The rubber band goes up 2 units. (Each space is one unit.)
\[\begin{array}{lll}m & = & \frac{\text{rise}}{\text{run}} \\ m & = & \frac{2}{3}\end{array}\]Example
Try it.
What is the slope of the line on the geoboard shown?
Solution
Use the definition of slope: \(m=\frac{\text{rise}}{\text{run}}.\)
Start at the left peg and count the spaces up and to the right to reach the second peg.
| The rise is 3. | \(m=\frac{3}{\text{run}}\) |
| The run is 4. | \(m=\frac{3}{4}\) |
| The slope is \(\frac{3}{4}\). |
This means that the line rises 3 units for every 4 units of run.
Example
Try it.
What is the slope of the line on the geoboard shown?
Solution
Use the definition of slope: \(m=\frac{\text{rise}}{\text{run}}.\)
Start at the left peg and count the units down and to the right to reach the second peg.
| The rise is −1. | \(=\frac{-1}{\text{run}}\) |
| The run is 3. | \(\begin{array}{l}m=\frac{-1}{3} \\ m=-\frac{1}{3}\end{array}\) |
| The slope is \(-\frac{1}{3}\). |
This means that the line drops 1 unit for every 3 units of run.
Condensed: the full section is in OpenStax Elementary Algebra 2e.
Use
Now, we’ll look at some graphs on the \(xy\)-coordinate plane and see how to find their slopes. The method will be very similar to what we just modeled on our geoboards.
To find the slope, we must count out the rise and the run. But where do we start?
We locate two points on the line whose coordinates are integers. We then start with the point on the left and sketch a right triangle, so we can count the rise and run.
How to Use
Try it.
Find the slope of the line shown.
Solution
Example
Try it.
Find the slope of the line shown.
Solution
| Locate two points on the graph whose coordinates are integers. | \((0,5)\) and \((3,3)\) |
| Which point is on the left? | \((0,5)\) |
| Starting at \((0,5)\), sketch a right triangle to \((3,3)\). | |
| Count the rise, it is negative. | The rise is \(-2\). |
| Count the run. | The run is 3. |
| Use the slope formula. | \(m=\frac{\text{rise}}{\text{run}}\) |
| Substitute the values of the rise and run. | \(m=\frac{-2}{3}\) |
| Simplify. | \(m=-\frac{2}{3}\) |
| The slope of the line is \(-\frac{2}{3}\). |
So \(y\) decreases by 2 units as \(x\) increases by 3 units.
What if we used the points \((-3,7)\) and \((6,1)\) to find the slope of the line?
The rise would be \(-6\) and the run would be 9. Then \(m=\frac{-6}{9}\), and that simplifies to \(m=-\frac{2}{3}\). Remember, it does not matter which points you use, the slope of the line is always the same.
In the last two examples, the lines had y-intercepts with integer values, so it was convenient to use the y-intercept as one of the points to find the slope. In the next example, the y-intercept is a fraction. Instead of using that point, we’ll look for two other points whose coordinates are integers. This will make the slope calculations easier.
Condensed: the full section is in OpenStax Elementary Algebra 2e.
Find the Slope of Horizontal and Vertical Lines
Do you remember what was special about horizontal and vertical lines? Their equations had just one variable.
\[\begin{array}{llll}\text{Horizontal line}\ y=b & & & \text{Vertical line}\ x=a \\ \\ \text{y}\text{-coordinates are the same.} & & & \text{x}\text{-coordinates are the same.}\end{array}\]So how do we find the slope of the horizontal line \(y=4\)? One approach would be to graph the horizontal line, find two points on it, and count the rise and the run. Let’s see what happens when we do this.
| What is the rise? | The rise is \(0\). |
| Count the run. | The run is \(3\). |
| What is the slope? | \(\begin{array}{l}m=\frac{\text{rise}}{\text{run}} \\ m=\frac{0}{3} \\ m=0\end{array}\) |
| The slope of the horizontal line \(y=4\) is \(0\). |
All horizontal lines have slope 0. When the y-coordinates are the same, the rise is 0.
The floor of your room is horizontal. Its slope is 0. If you carefully placed a ball on the floor, it would not roll away.
Now, we’ll consider a vertical line, the line.
| What is the rise? | The rise is \(2\). |
| Count the run. | The run is \(0\). |
| What is the slope? | \(\begin{array}{l}m=\frac{\text{rise}}{\text{run}} \\ m=\frac{2}{0}\end{array}\) |
But we can’t divide by 0. Division by 0 is not defined. So we say that the slope of the vertical line \(x=3\) is undefined.
The slope of any vertical line is undefined. When the x-coordinates of a line are all the same, the run is 0.
Example
Try it.
Find the slope of each line:
ⓐ \(x=8\) ⓑ \(y=-5\).
Solution
- ⓐ \(x=8\)
This is a vertical line.
Its slope is undefined. - ⓑ \(y=-5\)
This is a horizontal line.
It has slope 0.
Condensed: the full section is in OpenStax Elementary Algebra 2e.
Use the Slope Formula to find the Slope of a Line Between Two Points
Sometimes we’ll need to find the slope of a line between two points when we don’t have a graph to count out the rise and the run. We could plot the points on grid paper, then count out the rise and the run, but as we’ll see, there is a way to find the slope without graphing. Before we get to it, we need to introduce some algebraic notation.
We have seen that an ordered pair \((x,y)\) gives the coordinates of a point. But when we work with slopes, we use two points. How can the same symbol \((x,y)\) be used to represent two different points? Mathematicians use subscripts to distinguish the points.
\[\begin{array}{lll}({x}_{1},{y}_{1}) & & \text{read ‘}\ x\ \text{sub 1,}\ y\ \text{sub 1’} \\ ({x}_{2},{y}_{2}) & & \text{read ‘}\ x\ \text{sub 2,}\ y\ \text{sub 2’}\end{array}\]The use of subscripts in math is very much like the use of last name initials in elementary school. Maybe you remember Laura C. and Laura M. in your third grade class?
We will use \(({x}_{1},{y}_{1})\) to identify the first point and \(({x}_{2},{y}_{2})\) to identify the second point.
If we had more than two points, we could use \(({x}_{3},{y}_{3})\), \(({x}_{4},{y}_{4})\), and so on.
Let’s see how the rise and run relate to the coordinates of the two points by taking another look at the slope of the line between the points \((2,3)\) and \((7,6)\).
Since we have two points, we will use subscript notation, \((\overset{{x}_{1},}{2,}\overset{{y}_{1}}{3})\)\((\overset{{x}_{2},{y}_{2}}{7,6})\).
\[3=6-3\]\[5=7-2\]Example
Try it.
Use the slope formula to find the slope of the line between the points \((1,2)\) and \((4,5)\).
Solution
| We'll call \((1,2)\) point #1 and \((4,5)\) point #2. | \((\overset{{x}_{1},{y}_{1}}{1,2})\ {(\overset{{x}_{2},{y}_{2}}{4,5})}_{}\) |
| Use the slope formula. | \(m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\). |
| Substitute the values. | |
| \(y\) of the second point minus \(y\) of the first point | \(m=\frac{5-2}{{x}_{2}-{x}_{1}}\). |
| \(x\) of the second point minus \(x\) of the first point | \(m=\frac{5-2}{4-1}\). |
| Simplify the numerator and the denominator. | \(m=\frac{3}{3}\). |
| Simplify. | \(m=1\). |
Let’s confirm this by counting out the slope on a graph using \(m=\frac{\text{rise}}{\text{run}}\).
It doesn’t matter which point you call point #1 and which one you call point #2. The slope will be the same. Try the calculation yourself.
Condensed: the full section is in OpenStax Elementary Algebra 2e.
Graph a Line Given a Point and the Slope
Up to now, in this chapter, we have graphed lines by plotting points, by using intercepts, and by recognizing horizontal and vertical lines.
One other method we can use to graph lines is called the point, slope method. We will use this method when we know one point and the slope of the line. We will start by plotting the point and then use the definition of slope to draw the graph of the line.
How To Graph a Line Given a Point and The Slope
Try it.
Graph the line passing through the point \((1,-1)\) whose slope is \(m=\frac{3}{4}\).
Solution
Example
Try it.
Graph the line with y-intercept 2 whose slope is \(m=-\frac{2}{3}\).
Solution
Plot the given point, the y-intercept, \((0,2)\).
| Identify the rise and the run. | \(m=-\frac{2}{3}\) |
| \(\frac{\text{rise}}{\text{run}}=\frac{-2}{3}\) | |
| \(\text{rise}=-2\) | |
| \(\text{run}=3\) |
Count the rise and the run. Mark the second point.
Connect the two points with a line.
You can check your work by finding a third point. Since the slope is \(m=-\frac{2}{3}\), it can be written as \(m=\frac{2}{-3}\). Go back to \((0,2)\) and count out the rise, 2, and the run, \(-3\).
Example
Try it.
Graph the line passing through the point \((-1,-3)\) whose slope is \(m=4.\)
Solution
Plot the given point.
| Identify the rise and the run. | \(m=4\) |
| Write 4 as a fraction. | \(\frac{\text{rise}}{\text{run}}=\frac{4}{1}\) |
| \(\text{rise}=4,\text{run}=1\) |
Count the rise and run and mark the second point.
Connect the two points with a line.
You can check your work by finding a third point. Since the slope is \(m=4\), it can be written as \(m=\frac{-4}{-1}\). Go back to \((-1,-3)\) and count out the rise, \(-4\), and the run, \(-1\).
Solve Slope Applications
At the beginning of this section, we said there are many applications of slope in the real world. Let’s look at a few now.
Example
Try it.
The ‘pitch’ of a building’s roof is the slope of the roof. Knowing the pitch is important in climates where there is heavy snowfall. If the roof is too flat, the weight of the snow may cause it to collapse. What is the slope of the roof shown?
Solution
| Use the slope formula. | \(m=\frac{\text{rise}}{\text{run}}\) |
| Substitute the values for rise and run. | \(m=\frac{9}{18}\) |
| Simplify. | \(m=\frac{1}{2}\) |
| The slope of the roof is \(\frac{1}{2}\). | |
| The roof rises 1 foot for every 2 feet of horizontal run. |
Example
Try it.
Have you ever thought about the sewage pipes going from your house to the street? They must slope down \(\frac{1}{4}\) inch per foot in order to drain properly. What is the required slope?
Solution
| Use the slope formula. | \(\begin{array}{l}m=\frac{\text{rise}}{\text{run}} \\ m=\frac{-\frac{1}{4}\text{inch}}{\text{1 foot}} \\ m=\frac{-\frac{1}{4}\text{inch}}{\text{12 inches}}\end{array}\) |
| Simplify. | \(m=-\frac{1}{48}\) |
| The slope of the pipe is \(-\frac{1}{48}\). |
The pipe drops 1 inch for every 48 inches of horizontal run.
Key Concepts
- Find the Slope of a Line from its Graph using \(m=\frac{\text{rise}}{\text{run}}\)
- Locate two points on the line whose coordinates are integers.
- Starting with the point on the left, sketch a right triangle, with the hypotenuse going from the first point to the second point.
- Count the rise and the run on the legs of the triangle.
- Take the ratio of rise to run to find the slope.
- Graph a Line Given a Point and the Slope
- Plot the given point.
- Use the slope formula \(m=\frac{\text{rise}}{\text{run}}\) to identify the rise and the run.
- Starting at the given point, count out the rise and run to mark the second point.
- Connect the points with a line.
- Slope of a Horizontal Line
- The slope of a horizontal line, \(y=b\), is 0.
- Slope of a vertical line
- The slope of a vertical line, \(x=a\), is undefined
Understand Slope of a Line
Use Geoboards to Model Slope
In the following exercises, find the slope modeled on each geoboard.
In the following exercises, model each slope. Draw a picture to show your results.
Try it.
\(\frac{2}{3}\)
Solution
Try it.
\(\frac{3}{4}\)
Try it.
\(\frac{1}{4}\)
Solution
Try it.
\(\frac{4}{3}\)
Try it.
\(-\frac{1}{2}\)
Solution
Try it.
\(-\frac{3}{4}\)
Try it.
\(-\frac{2}{3}\)
Solution
Try it.
\(-\frac{3}{2}\)
Use \(m=\frac{\text{rise}}{\text{run}}\) to find the Slope of a Line from its Graph
In the following exercises, find the slope of each line shown.
Find the Slope of Horizontal and Vertical Lines
In the following exercises, find the slope of each line.
Use the Slope Formula to find the Slope of a Line between Two Points
In the following exercises, use the slope formula to find the slope of the line between each pair of points.
Try it.
\((1,4),(3,9)\)
Solution
\(\frac{5}{2}\)
Try it.
\((2,3),(5,7)\)
Try it.
\((0,3),(4,6)\)
Solution
\(\frac{3}{4}\)
Try it.
\((0,1),(5,4)\)
Try it.
\((2,5),(4,0)\)
Solution
\(-\frac{5}{2}\)
Try it.
\((3,6),(8,0)\)
Try it.
\((-3,3),(4,-5)\)
Solution
\(-\frac{8}{7}\)
Try it.
\((-2,4),(3,-1)\)
Try it.
\((-1,-2),(2,5)\)
Solution
\(\frac{7}{3}\)
Try it.
\((-2,-1),(6,5)\)
Try it.
\((4,-5),(1,-2)\)
Solution
\(-1\)
Try it.
\((3,-6),(2,-2)\)
Graph a Line Given a Point and the Slope
In the following exercises, graph each line with the given point and slope.
Try it.
\((1,-2)\); \(m=\frac{3}{4}\)
Solution
Try it.
\((1,-1)\); \(m=\frac{2}{3}\)
Try it.
\((2,5)\); \(m=-\frac{1}{3}\)
Solution
Try it.
\((1,4)\); \(m=-\frac{1}{2}\)
Try it.
\((-3,4)\); \(m=-\frac{3}{2}\)
Solution
Try it.
\((-2,5)\); \(m=-\frac{5}{4}\)
Try it.
\((-1,-4)\); \(m=\frac{4}{3}\)
Solution
Try it.
\((-3,-5)\); \(m=\frac{3}{2}\)
Try it.
y-intercept 3; \(m=-\frac{2}{5}\)
Solution
Try it.
y-intercept 5; \(m=-\frac{4}{3}\)
Try it.
x-intercept \(-2\); \(m=\frac{3}{4}\)
Solution
Try it.
x-intercept \(-1\); \(m=\frac{1}{5}\)
Try it.
\((-3,3)\); \(m=2\)
Solution
Try it.
\((-4,2)\); \(m=4\)
Try it.
\((1,5)\); \(m=-3\)
Solution
Try it.
\((2,3)\); \(m=-1\)
Try it.
What does the sign of the slope tell you about a line?
Solution
When the slope is a positive number the line goes up from left to right. When the slope is a negative number the line goes down from left to right.
Try it.
How does the graph of a line with slope \(m=\frac{1}{2}\) differ from the graph of a line with slope \(m=2\)?
Try it.
Why is the slope of a vertical line “undefined”?
Solution
A vertical line has 0 run and since division by 0 is undefined the slope is undefined.
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?
Condensed: the full section is in OpenStax Elementary Algebra 2e.
Use Geoboards to Model Slope
In this section, we will explore the concepts of slope.
Using rubber bands on a geoboard gives a concrete way to model lines on a coordinate grid. By stretching a rubber band between two pegs on a geoboard, we can discover how to find the slope of a line.
We’ll start by stretching a rubber band between two pegs to make a line
Does it look like a line?
Now we stretch one part of the rubber band straight up from the left peg and around a third peg to make the sides of a right triangle We carefully make a \(90^{\circ}\) angle around the third peg, so that one side is vertical and the other is horizontal.
To find the slope of the line, we measure the distance along the vertical and horizontal legs of the triangle. The vertical distance is called the rise and the horizontal distance is called the run,
To help remember the terms, it may help to think of the images
\[\ m=\frac{\text{rise}}{\text{run}}\]\[m=\frac{2}{3}\]\[\text{The line has slope}\ \frac{2}{3}.\]Example
Try it.
What is the slope of the line on the geoboard shown?
Solution
Use the definition of slope.
\(m=\frac{\text{rise}}{\text{run}}\)
Start at the left peg and make a right triangle by stretching the rubber band up and to the right to reach the second peg.
Count the rise and the run as shown.
\(\begin{array}{llll}\text{The rise is}\ 3\ \text{units}. & & & m=\frac{3}{\text{run}} \\ \text{The run is}\ 4\ \text{units}. & & & m=\frac{3}{4} \\ & & & \text{The slope is}\ \frac{3}{4}.\end{array}\)
Example
Try it.
What is the slope of the line on the geoboard shown?
Solution
Use the definition of slope.
\(m=\frac{\text{rise}}{\text{run}}\)
Start at the left peg and make a right triangle by stretching the rubber band to the peg on the right. This time we need to stretch the rubber band down to make the vertical leg, so the rise is negative.
\(\begin{array}{llll}\text{The rise is}\ -1. & & & m=\frac{-1}{\text{run}} \\ \text{The run is}\ 3. & & & m=\frac{-1}{3} \\ & & & m=-\frac{1}{3} \\ & & & \text{The slope is}\ -\frac{1}{3}.\end{array}\)
Condensed: the full section is in OpenStax Prealgebra 2e.
Find the Slope of a Line from its Graph
Now we’ll look at some graphs on a coordinate grid to find their slopes. The method will be very similar to what we just modeled on our geoboards.
To find the slope, we must count out the rise and the run. But where do we start?
We locate any two points on the line. We try to choose points with coordinates that are integers to make our calculations easier. We then start with the point on the left and sketch a right triangle, so we can count the rise and run.
Example
Try it.
Find the slope of the line shown:
Solution
Locate two points on the graph, choosing points whose coordinates are integers. We will use \((0,-3)\) and \((5,1).\)
Starting with the point on the left, \((0,-3),\) sketch a right triangle, going from the first point to the second point, \((5,1).\)
| Count the rise on the vertical leg of the triangle. | The rise is 4 units. |
| Count the run on the horizontal leg. | The run is 5 units. |
| Use the slope formula. | \(m=\frac{\text{rise}}{\text{run}}\) |
| Substitute the values of the rise and run. | \(m=\frac{4}{5}\) |
| The slope of the line is \(\frac{4}{5}\). |
Notice that the slope is positive since the line slants upward from left to right.
The lines in the previous examples had \(y\)-intercepts with integer values, so it was convenient to use the y-intercept as one of the points we used to find the slope. In the next example, the \(y\)-intercept is a fraction. The calculations are easier if we use two points with integer coordinates.
Example
Try it.
Find the slope of the line shown:
Solution
| Locate two points on the graph whose coordinates are integers. | \((2,3)\) and \((7,6)\) |
| Which point is on the left? | \((2,3)\) |
| Starting at \((2,3)\), sketch a right angle to \((7,6)\) as shown below. |
| Count the rise. | The rise is 3. |
| Count the run. | The run is 5. |
| Use the slope formula. | \(m=\frac{\text{rise}}{\text{run}}\) |
| Substitute the values of the rise and run. | \(m=\frac{3}{5}\) |
| The slope of the line is \(\frac{3}{5}.\) |
Condensed: the full section is in OpenStax Prealgebra 2e.
Find the Slope of Horizontal and Vertical Lines
Do you remember what was special about horizontal and vertical lines? Their equations had just one variable.
- horizontal line \(y=b;\) all the \(y\)-coordinates are the same.
- vertical line \(x=a;\) all the \(x\)-coordinates are the same.
So how do we find the slope of the horizontal line \(y=4?\) One approach would be to graph the horizontal line, find two points on it, and count the rise and the run. Let’s see what happens in . We’ll use the two points \((0,\ 4)\) and \((3,\ 4)\) to count the rise and run.
| What is the rise? | The rise is 0. |
| What is the run? | The run is 3. |
| What is the slope? | \(m=\frac{\text{rise}}{\text{run}}\) |
| \(m=\frac{0}{3}\) | |
| \(m=0\) |
The slope of the horizontal line \(y=4\) is \(0.\)
All horizontal lines have slope \(0\). When the \(y\)-coordinates are the same, the rise is \(0\).
Now we’ll consider a vertical line, such as the line \(x=3\), We’ll use the two points \((3,\ 0)\) and \((3,\ 2)\) to count the rise and run.
| What is the rise? | The rise is 2. |
| What is the run? | The run is 0. |
| What is the slope? | \(m=\frac{\text{rise}}{\text{run}}\) |
| \(m=\frac{2}{0}\) |
But we can’t divide by \(0.\) Division by \(0\) is undefined. So we say that the slope of the vertical line \(x=3\) is undefined. The slope of all vertical lines is undefined, because the run is \(0.\)
Example
Try it.
Find the slope of each line:
- ⓐ \(\ x=8\)
- ⓑ \(\ y=-5\)
Solution
ⓐ \(\ x=8\)
This is a vertical line, so its slope is undefined.
ⓑ \(\ y=-5\)
This is a horizontal line, so its slope is \(0.\)
Use the Slope Formula to find the Slope of a Line between Two Points
Sometimes we need to find the slope of a line between two points and we might not have a graph to count out the rise and the run. We could plot the points on grid paper, then count out the rise and the run, but there is a way to find the slope without graphing.
Before we get to it, we need to introduce some new algebraic notation. We have seen that an ordered pair \((x,y)\) gives the coordinates of a point. But when we work with slopes, we use two points. How can the same symbol \((x,y)\) be used to represent two different points?
Mathematicians use subscripts to distinguish between the points. A subscript is a small number written to the right of, and a little lower than, a variable.
- \(({x}_{1},{y}_{1})\ \text{read}\ x\ \text{sub}\ 1,y\ \text{sub}\ 1\)
- \(({x}_{2},{y}_{2})\ \text{read}\ x\ \text{sub}\ 2,y\ \text{sub}\ 2\)
We will use \(({x}_{1},{y}_{1})\) to identify the first point and \(({x}_{2},{y}_{2})\) to identify the second point. If we had more than two points, we could use \(({x}_{3},{y}_{3}),({x}_{4},{y}_{4}),\) and so on.
To see how the rise and run relate to the coordinates of the two points, let’s take another look at the slope of the line between the points \((2,3)\) and \((7,6)\) in .
Since we have two points, we will use subscript notation.
\[\overset{{x}_{1},{y}_{1}}{(2,3)}\ \overset{{x}_{2},{y}_{2}}{(7,6)}\]On the graph, we counted the rise of \(3.\) The rise can also be found by subtracting the \(y\text{-coordinates}\) of the points.
\[\begin{array}{l}{y}_{2}-{y}_{1} \\ 6-3 \\ 3\end{array}\]\[\begin{array}{l}{x}_{2}-{x}_{1} \\ 7-2 \\ 5\end{array}\]| We know | \(m=\frac{\text{rise}}{\text{run}}\) |
| So | \(m=\frac{3}{5}\) |
| We rewrite the rise and run by putting in the coordinates. | \(m=\frac{6-3}{7-2}\) |
| But 6 is the \(y\)-coordinate of the second point, \({y}_{2}\) and 3 is the \(y\)-coordinate of the first point \({y}_{1}\). So we can rewrite the rise using subscript notation. | \(m=\frac{{y}_{2}-{y}_{1}}{7-2}\) |
| Also 7 is the \(x\)-coordinate of the second point, \({x}_{2}\) and 2 is the \(x\)-coordinate of the first point \({x}_{2}\). So we rewrite the run using subscript notation. | \(m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\) |
Condensed: the full section is in OpenStax Prealgebra 2e.
Graph a Line Given a Point and the Slope
In this chapter, we graphed lines by plotting points, by using intercepts, and by recognizing horizontal and vertical lines.
Another method we can use to graph lines is the point-slope method. Sometimes, we will be given one point and the slope of the line, instead of its equation. When this happens, we use the definition of slope to draw the graph of the line.
Example
Try it.
Graph the line passing through the point \((1,-1)\) whose slope is \(m=\frac{3}{4}.\)
Solution
Plot the given point, \((1,-1).\)
Use the slope formula \(m=\frac{\text{rise}}{\text{run}}\) to identify the rise and the run.
\[\begin{array}{l} \\ \\ m=\frac{3}{4} \\ \frac{\text{rise}}{\text{run}}=\frac{3}{4} \\ \\ \\ \text{rise}=3 \\ \text{run}=4\end{array}\]Starting at the point we plotted, count out the rise and run to mark the second point. We count \(3\) units up and \(4\) units right.
Then we connect the points with a line and draw arrows at the ends to show it continues.
We can check our line by starting at any point and counting up \(3\) and to the right \(4.\) We should get to another point on the line.
Example
Try it.
Graph the line with \(y\)-intercept \((0,2)\) and slope \(m=-\frac{2}{3}.\)
Solution
Plot the given point, the \(y\)-intercept \((0,2).\)
Use the slope formula \(m=\frac{\text{rise}}{\text{run}}\) to identify the rise and the run.
\[\begin{array}{l} \\ \\ m=-\frac{2}{3} \\ \frac{\text{rise}}{\text{run}}=\frac{-2}{3} \\ \\ \\ \text{rise}=-2 \\ \text{run}=3\end{array}\]Starting at \((0,2),\) count the rise and the run and mark the second point.
Connect the points with a line.
Example
Try it.
Graph the line passing through the point \((-1,-3)\) whose slope is \(m=4.\)
Solution
Plot the given point.
| Identify the rise and the run. | \(m=4\) |
| Write 4 as a fraction. | \(\frac{\text{rise}}{\text{run}}=\frac{4}{1}\) |
| \(\text{rise}=4\ \text{run}=1\) |
Count the rise and run.
Mark the second point. Connect the two points with a line.
Condensed: the full section is in OpenStax Prealgebra 2e.
Solve Slope Applications
At the beginning of this section, we said there are many applications of slope in the real world. Let’s look at a few now.
Example
Try it.
The pitch of a building’s roof is the slope of the roof. Knowing the pitch is important in climates where there is heavy snowfall. If the roof is too flat, the weight of the snow may cause it to collapse. What is the slope of the roof shown?
Solution
| Use the slope formula. | \(m=\frac{\text{rise}}{\text{run}}\) |
| Substitute the values for rise and run. | \(m=\frac{\text{9 ft}}{\text{18 ft}}\) |
| Simplify. | \(m=\frac{1}{2}\) |
| The slope of the roof is \(\frac{1}{2}\). |
Have you ever thought about the sewage pipes going from your house to the street? Their slope is an important factor in how they take waste away from your house.
Example
Try it.
Sewage pipes must slope down \(\frac{1}{4}\) inch per foot in order to drain properly. What is the required slope?
Solution
| Use the slope formula. | \(m=\frac{\text{rise}}{\text{run}}\) |
| \(m=\frac{-\frac{1}{4}\ \text{in}\text{.}}{1\ \text{ft}}\) | |
| \(m=\frac{-\frac{1}{4}\ \text{in}\text{.}}{1\ \text{ft}}\) | |
| Convert 1 foot to 12 inches. | \(m=\frac{-\frac{1}{4}\ \text{in}\text{.}}{12\ \text{in.}}\) |
| Simplify. | \(m=-\frac{1}{48}\) |
| The slope of the pipe is \(-\frac{1}{48}.\) |
Key Concepts
- Find the slope from a graph
- Locate two points on the line whose coordinates are integers.
- Starting with the point on the left, sketch a right triangle, with the hypotenuse going from the first point to the second point.
- Count the rise and the run on the legs of the triangle.
- Take the ratio of rise to run to find the slope, \(m=\frac{\text{rise}}{\text{run}}\)
- Slope of a Horizontal Line
- The slope of a horizontal line, \(y=b\), is 0.
- Slope of a Vertical Line
- The slope of a vertical line, \(x=a\), is undefined.
- Slope Formula
- The slope of the line between two points \(({x}_{1},{y}_{1})\) and \(({x}_{2},{y}_{2})\) is \(m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\)
- Graph a line given a point and a slope.
- Plot the given point.
- Use the slope formula to identify the rise and the run.
- Starting at the given point, count out the rise and run to mark the second point.
- Connect the points with a line.
Chapter Practice Test
In the following exercises, find three solutions to each equation and then graph each line.
In the following exercises, find the slope of each line.
Nu jij. Geen rekenmachine regelt deze, maar de stukken ervan zijn berekenbaar. Probeer er een hieronder, of typ je eigen.
Praktijk (40)
Probeer ze eerst op papier. Onthul het antwoord om te controleren; geverifieerde kunnen worden geopend in de oplossing voor elke stap.
-
Simplify: \(\frac{1-4}{8-2}\).
Onthul het antwoord
\(-\frac{1}{2}\)
-
Divide: \(\frac{0}{4},\frac{4}{0}\).
Onthul het antwoord
0, undefined
-
Simplify: \(\frac{15}{-3},\frac{-15}{3},\frac{-15}{-3}\).
Onthul het antwoord
\(-5,-5,5\)
-
What is the slope of the line on the geoboard shown?
Onthul het antwoord
Use the definition of slope: \(m=\frac{\text{rise}}{\text{run}}.\)
Start at the left peg and count the spaces up and to the right to reach the second peg.
The rise is 3. \(m=\frac{3}{\text{run}}\) The run is 4. \(m=\frac{3}{4}\) The slope is \(\frac{3}{4}\). This means that the line rises 3 units for every 4 units of run.
-
What is the slope of the line on the geoboard shown?
Onthul het antwoord
\(\frac{4}{3}\)
-
What is the slope of the line on the geoboard shown?
Onthul het antwoord
\(\frac{1}{4}\)
-
What is the slope of the line on the geoboard shown?
Onthul het antwoord
Use the definition of slope: \(m=\frac{\text{rise}}{\text{run}}.\)
Start at the left peg and count the units down and to the right to reach the second peg.
The rise is −1. \(=\frac{-1}{\text{run}}\) The run is 3. \(\begin{array}{l}m=\frac{-1}{3} \\ m=-\frac{1}{3}\end{array}\) The slope is \(-\frac{1}{3}\). This means that the line drops 1 unit for every 3 units of run.
-
What is the slope of the line on the geoboard?
Onthul het antwoord
\(-\frac{2}{3}\)
-
What is the slope of the line on the geoboard?
Onthul het antwoord
\(-\frac{4}{3}\)
-
Use a geoboard to model a line with slope \(\frac{1}{2}\).
Onthul het antwoord
To model a line on a geoboard, we need the rise and the run.
Use the slope formula. \(m=\frac{\text{rise}}{\text{run}}\) Replace \(m\) with \(\frac{1}{2}\). \(\frac{1}{2}=\frac{\text{rise}}{\text{run}}\) So, the rise is 1 and the run is 2.
Start at a peg in the lower left of the geoboard.
Stretch the rubber band up 1 unit, and then right 2 units.
The hypotenuse of the right triangle formed by the rubber band represents a line whose slope is \(\frac{1}{2}\).
-
Model the slope \(m=\frac{1}{3}\). Draw a picture to show your results.
-
Model the slope \(m=\frac{3}{2}\). Draw a picture to show your results.
-
Use a geoboard to model a line with slope \(\frac{-1}{4}.\)
Onthul het antwoord
Use the slope formula. \(m=\frac{\text{rise}}{\text{run}}\) Replace \(m\) with \(\frac{-1}{4}\). \(\frac{-1}{4}=\frac{\text{rise}}{\text{run}}\) So, the rise is \(-1\) and the run is 4.
Since the rise is negative, we choose a starting peg on the upper left that will give us room to count down.
We stretch the rubber band down 1 unit, then go to the right 4 units, as shown.
The hypotenuse of the right triangle formed by the rubber band represents a line whose slope is \(\frac{-1}{4}\).
-
Model the slope \(m=\frac{-2}{3}\). Draw a picture to show your results.
-
Model the slope \(m=\frac{-1}{3}\). Draw a picture to show your results.
-
Find the slope of the line shown.
-
Find the slope of the line shown.
Onthul het antwoord
\(\frac{2}{5}\)
-
Find the slope of the line shown.
Onthul het antwoord
\(\frac{3}{4}\)
-
Find the slope of the line shown.
Onthul het antwoord
Locate two points on the graph whose coordinates are integers. \((0,5)\) and \((3,3)\) Which point is on the left? \((0,5)\) Starting at \((0,5)\), sketch a right triangle to \((3,3)\). Count the rise, it is negative. The rise is \(-2\). Count the run. The run is 3. Use the slope formula. \(m=\frac{\text{rise}}{\text{run}}\) Substitute the values of the rise and run. \(m=\frac{-2}{3}\) Simplify. \(m=-\frac{2}{3}\) The slope of the line is \(-\frac{2}{3}\). So \(y\) decreases by 2 units as \(x\) increases by 3 units.
What if we used the points \((-3,7)\) and \((6,1)\) to find the slope of the line?
The rise would be \(-6\) and the run would be 9. Then \(m=\frac{-6}{9}\), and that simplifies to \(m=-\frac{2}{3}\). Remember, it does not matter which points you use, the slope of the line is always the same.
-
Find the slope of the line shown.
Onthul het antwoord
\(-\frac{4}{3}\)
-
Find the slope of the line shown.
Onthul het antwoord
\(-\frac{3}{5}\)
-
Find the slope of the line shown.
Onthul het antwoord
Locate two points on the graph whose coordinates are integers. \((2,3)\) and \((7,6)\) Which point is on the left? \((2,3)\) Starting at \((2,3)\), sketch a right triangle to \((7,6)\). Count the rise. The rise is 3. Count the run. The run is 5. Use the slope formula. \(m=\frac{\text{rise}}{\text{run}}\) Substitute the values of the rise and run. \(m=\frac{3}{5}\) The slope of the line is \(\frac{3}{5}\). This means that \(y\) increases 5 units as \(x\) increases 3 units.
When we used geoboards to introduce the concept of slope, we said that we would always start with the point on the left and count the rise and the run to get to the point on the right. That way the run was always positive and the rise determined whether the slope was positive or negative.
What would happen if we started with the point on the right?
Let’s use the points \((2,3)\) and \((7,6)\) again, but now we’ll start at \((7,6)\).
Count the rise. The rise is \(-3\). Count the run. It goes from right to left, so it is negative. The run is \(-5\). Use the slope formula. \(m=\frac{\text{rise}}{\text{run}}\) Substitute the values of the rise and run. \(m=\frac{-3}{-5}\) The slope of the line is \(\frac{-3}{-5}\). It does not matter where you start, the slope of the line is always the same.
-
Find the slope of the line shown.
Onthul het antwoord
\(\frac{5}{4}\)
-
Find the slope of the line shown.
Onthul het antwoord
\(\frac{3}{2}\)
-
Find the slope of each line:
ⓐ \(x=8\) ⓑ \(y=-5\).
Onthul het antwoord
- ⓐ \(x=8\)
This is a vertical line.
Its slope is undefined. - ⓑ \(y=-5\)
This is a horizontal line.
It has slope 0.
- ⓐ \(x=8\)
-
Find the slope of the line: \(x=-4.\)
Onthul het antwoord
undefined
-
Find the slope of the line: \(y=7.\)
Onthul het antwoord
0
-
Use the slope formula to find the slope of the line between the points \((1,2)\) and \((4,5)\).
Onthul het antwoord
We'll call \((1,2)\) point #1 and \((4,5)\) point #2. \((\overset{{x}_{1},{y}_{1}}{1,2})\ {(\overset{{x}_{2},{y}_{2}}{4,5})}_{}\) Use the slope formula. \(m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\). Substitute the values. \(y\) of the second point minus \(y\) of the first point \(m=\frac{5-2}{{x}_{2}-{x}_{1}}\). \(x\) of the second point minus \(x\) of the first point \(m=\frac{5-2}{4-1}\). Simplify the numerator and the denominator. \(m=\frac{3}{3}\). Simplify. \(m=1\). Let’s confirm this by counting out the slope on a graph using \(m=\frac{\text{rise}}{\text{run}}\).
It doesn’t matter which point you call point #1 and which one you call point #2. The slope will be the same. Try the calculation yourself.
-
Use the slope formula to find the slope of the line through the points: \((8,5)\) and \((6,3)\).
Onthul het antwoord
1
-
Use the slope formula to find the slope of the line through the points: \((1,5)\) and \((5,9)\).
Onthul het antwoord
1
-
Use the slope formula to find the slope of the line through the points \((-2,-3)\) and \((-7,4)\).
Onthul het antwoord
We'll call \((-2,-3)\) point #1 and \((-7,4)\) point #2. \((\overset{{x}_{1},{y}_{1}}{-2,-3})\ {(\overset{{x}_{2},{y}_{2}}{-7,4})}_{}\) Use the slope formula. \(m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\). Substitute the values. \(y\) of the second point minus \(y\) of the first point \(m=\frac{4-(-3)}{{x}_{2}-{x}_{1}}\). \(x\) of the second point minus \(x\) of the first point \(m=\frac{4-(-3)}{-7-(-2)}\). Simplify. \(\begin{array}{l}m=\frac{7}{-5} \\ m=-\frac{7}{5}\end{array}\) Let’s verify this slope on the graph shown.
\[\begin{array}{lll}m & = & \frac{\text{rise}}{\text{run}} \\ m & = & \frac{-7}{5} \\ m & = & -\frac{7}{5}\end{array}\] -
Use the slope formula to find the slope of the line through the points: \((-3,4)\) and \((2,-1).\)
Onthul het antwoord
\(-1\)
-
Use the slope formula to find the slope of the line through the pair of points: \((-2,6)\) and \((-3,-4)\).
Onthul het antwoord
10
-
Graph the line passing through the point \((1,-1)\) whose slope is \(m=\frac{3}{4}\).
-
Graph the line passing through the point \((2,-2)\) with the slope \(m=\frac{4}{3}\).
-
Graph the line passing through the point \((-2,3)\) with the slope \(m=\frac{1}{4}\).
-
Graph the line with y-intercept 2 whose slope is \(m=-\frac{2}{3}\).
Onthul het antwoord
Plot the given point, the y-intercept, \((0,2)\).
Identify the rise and the run. \(m=-\frac{2}{3}\) \(\frac{\text{rise}}{\text{run}}=\frac{-2}{3}\) \(\text{rise}=-2\) \(\text{run}=3\) Count the rise and the run. Mark the second point.
Connect the two points with a line.
You can check your work by finding a third point. Since the slope is \(m=-\frac{2}{3}\), it can be written as \(m=\frac{2}{-3}\). Go back to \((0,2)\) and count out the rise, 2, and the run, \(-3\).
-
Graph the line with the y-intercept 4 and slope \(m=-\frac{5}{2}.\)
-
Graph the line with the x-intercept \(-3\) and slope \(m=-\frac{3}{4}\).
-
Graph the line passing through the point \((-1,-3)\) whose slope is \(m=4.\)
Onthul het antwoord
Plot the given point.
Identify the rise and the run. \(m=4\) Write 4 as a fraction. \(\frac{\text{rise}}{\text{run}}=\frac{4}{1}\) \(\text{rise}=4,\text{run}=1\) Count the rise and run and mark the second point.
Connect the two points with a line.
You can check your work by finding a third point. Since the slope is \(m=4\), it can be written as \(m=\frac{-4}{-1}\). Go back to \((-1,-3)\) and count out the rise, \(-4\), and the run, \(-1\).
Hier gebruikte symbolen
Tik op elk symbool voor de volledige definitie, een afbeelding en wat elke letter erin betekent.
Hoe: Understand Slope of a Line
- Use geoboards to model slope
- Find the slope of a line from its graph
- Find the slope of horizontal and vertical lines
- Use the slope formula to find the slope of a line between two points
- Graph a line given a point and the slope Solve slope applications
- Locate two points on the line whose coordinates are integers.
- Starting with the point on the left, sketch a right triangle, with the hypotenuse going from the first point to the second point.
- Count the rise and the run on the legs of the triangle.
Vragen die mensen stellen
What does it mean to solve an equation?
To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.
Why do I sometimes get two answers?
A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.
How do I know whether to factor or use the quadratic formula?
Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.
Delen van deze pagina zijn aangepast van OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0), OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Gecondenseerd en opnieuw verklaard, fouten zijn van ons. Omdat de bron ervan een licentie heeft voor niet-commercieel gebruik, is deze les gratis, verkoopt niets en wordt gedeeld onder dezelfde licentie.
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