# Graphs

## Understand Slope of a Line

As we’ve been graphing linear equations, we’ve seen that some lines slant up as they go from left to right and some lines slant down. Some lines are very steep and some lines are flatter. What determines whether a line slants up or down, and if its slant is steep or flat?
The steepness of the slant of a line is called the slope of the line. The concept of slope has many applications in the real world. The pitch of a roof and the grade of a highway or wheelchair ramp are just some examples in which you literally see slopes. And when you ride a bicycle, you feel the slope as you pump uphill or coast downhill.

### 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. And when you ride a bicycle, you feel the slope as you pump uphill or coast downhill. 
We’ll start by stretching a rubber band between two pegs to make a line as shown in .
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 as shown in . We carefully make a  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  and the horizontal distance is called the , as shown in .
To help remember the terms, it may help to think of the images shown in .

On our geoboard, the rise is  units because the rubber band goes up  spaces on the vertical leg. See .
What is the run? Be sure to count the spaces between the pegs rather than the pegs themselves! The rubber band goes across  spaces on the horizontal leg, so the run is  units.
The slope of a line is the ratio of the rise to the run. So the slope of our line is  In mathematics, the slope is always represented by the letter 
What is the slope of the line on the geoboard in ?
When we work with geoboards, it is a good idea to get in the habit of starting at a peg on the left and connecting to a peg to the right. Then we stretch the rubber band to form a right triangle.
If we start by going up the rise is positive, and if we stretch it down the rise is negative. We will count the run from left to right, just like you read this paragraph, so the run will be positive.
Since the slope formula has rise over run, it may be easier to always count out the rise first and then the run.
Notice that in the first example, the slope is positive and in the second example the slope is negative. Do you notice any difference in the two lines shown in .

As you read from left to right, the line in Figure A, is going up; it has positive slope.  The line Figure B is going down; it has negative slope.


### 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.
The lines in the previous examples had -intercepts with integer values, so it was convenient to use the  as one of the points we used to find the slope. In the next example, the -intercept is a fraction. The calculations are easier if we use two points with integer coordinates.

### 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.
1. horizontal line  all the -coordinates are the same.
2. vertical line  all the -coordinates are the same.
So how do we find the slope of the horizontal line  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  and  to count the rise and run.
The slope of the horizontal line  is 
All horizontal lines have slope . When the -coordinates are the same, the rise is .
Now we’ll consider a vertical line, such as the line , shown in . We’ll use the two points  and  to count the rise and run.
But we can’t divide by  Division by  is undefined. So we say that the slope of the vertical line  is undefined. The slope of all vertical lines is undefined, because the run is 

### 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  gives the coordinates of a point. But when we work with slopes, we use two points. How can the same symbol  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.
1. 
2. 
We will use  to identify the first point and  to identify the second point. If we had more than two points, we could use  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  and  in .
Since we have two points, we will use subscript notation.
On the graph, we counted the rise of  The rise can also be found by subtracting the  of the points.
We counted a run of  The run can also be found by subtracting the 
We’ve shown that  is really another version of  We can use this formula to find the slope of a line when we have two points on the line.
Say the formula to yourself to help you remember it:
Doing the Manipulative Mathematics activity  “Slope of Lines Between Two Points” will help you develop a better understanding of how to find the slope of a line between two points.
How do we know which point to call #1 and which to call #2? Let’s find the slope again, this time switching the names of the points to see what happens. Since we will now be counting the run from right to left, it will be negative.
The slope is the same no matter which order we use the points.

### 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.

### 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.
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.

### Key Concepts

1. Find the slope from a graph
2. Slope of a Horizontal Line
3. Slope of a Vertical Line
4. Slope Formula
5. Graph a line given a point and a slope.

### Section Exercises


### Practice Makes Perfect

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.
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.
Graph a Line Given a Point and the Slope 
In the following exercises, graph the line given a point and the slope.
Solve Slope Applications 
In the following exercises, solve these slope applications.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ 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?


### Chapter Review Exercises


### Use the Rectangular Coordinate System

Plot Points in a Rectangular Coordinate System
In the following exercises, plot each point in a rectangular coordinate system.
In the following exercises, plot each point in a rectangular coordinate system and identify the quadrant in which the point is located.
Identify Points on a Graph 
In the following exercises, name the ordered pair of each point shown in the rectangular coordinate system.
Verify Solutions to an Equation in Two Variables 
In the following exercises, find the ordered pairs that are solutions to the given equation.
Complete a Table of Solutions to a Linear Equation in Two Variables 
In the following exercises, complete the table to find solutions to each linear equation.
Find Solutions to a Linear Equation in Two Variables 
In the following exercises, find three solutions to each linear equation.

### Graphing Linear Equations

Recognize the Relation Between the Solutions of an Equation and its Graph 
In each of the following exercises, an equation and its graph is shown. For each ordered pair, decide

Graph a Linear Equation by Plotting Points 
In the following exercises, graph by plotting points.
Graph Vertical and Horizontal lines 
In the following exercises, graph the vertical or horizontal lines.

### Graphing with Intercepts

Identify the Intercepts on a Graph 
In the following exercises, find the  and 
Find the Intercepts from an Equation of a Line 
In the following exercises, find the intercepts.
Graph a Line Using the Intercepts 
In the following exercises, graph using the intercepts.
Choose the Most Convenient Method to Graph a Line 
In the following exercises, identify the most convenient method to graph each line.

### 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.
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.
Graph a Line Given a Point and the Slope 
In the following exercises, graph the line given a point and the slope.
Solve Slope Applications 
In the following exercise, solve the slope application.


### 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.