4.5: Intercepts and the CoverUp Method
What if you were given the equation of a line like ? How could you find the x  and y intercepts to help you graph the line? After completing this Concept, you'll be able to find the intercepts of linear equations like this one.
Watch This
CK12 Foundation: 0405S Graphing Using Intercepts (H264)
Try This
To learn more about equations in standard form, try the Java applet at http://www.analyzemath.com/line/line.htm (scroll down and click the “click here to start” button.) You can use the sliders to change the values of and and see how that affects the graph.
Guidance
Sanjit’s office is 25 miles from home, and in traffic he expects the trip home to take him an hour if he starts at 5 PM. Today he hopes to stop at the post office along the way. If the post office is 6 miles from his office, when will Sanjit get there?
If you know just one of the points on a line, you’ll find that isn’t enough information to plot the line on a graph. As you can see in the graph above, there are many lines—in fact, infinitely many lines—that pass through a single point. But what if you know two points that are both on the line? Then there’s only one way to graph that line; all you need to do is plot the two points and use a ruler to draw the line that passes through both of them.
There are a lot of options for choosing which two points on the line you use to plot it. In this lesson, we’ll focus on two points that are rather convenient for graphing: the points where our line crosses the and axes, or intercepts. We’ll see how to find intercepts algebraically and use them to quickly plot graphs.
Look at the graph above. The intercept occurs at the point where the graph crosses the axis. The value at this point is 8, and the value is 0.
Similarly, the intercept occurs at the point where the graph crosses the axis. The value at this point is 6, and the value is 0.
So we know the coordinates of two points on the graph: (0, 8) and (6, 0). If we’d just been given those two coordinates out of the blue, we could quickly plot those points and join them with a line to recreate the above graph.
Note: Not all lines will have both an and a intercept, but most do. However, horizontal lines never cross the axis and vertical lines never cross the axis.
For examples of these special cases, see the graph below.
Finding Intercepts by Substitution
Example A
Find the intercepts of the line and use them to graph the function.
Solution
The first intercept is easy to find. The intercept occurs when . Substituting gives us , so the intercept is (0, 13).
Similarly, the intercept occurs when . Plugging in 0 for gives us , and adding to both sides gives us . So (13, 0) is the intercept.
To draw the graph, simply plot these points and join them with a line.
Example B
Graph the following functions by finding intercepts.
a)
b)
c)
Solution
a) Find the intercept by plugging in
Find the intercept by plugging in
b) Find the intercept by plugging in
Find the intercept by plugging in
c) Find the intercept by plugging in
Find the intercept by plugging in
Finding Intercepts for Standard Form Equations Using the CoverUp Method
Look at the last two equations in example 2. These equations are written in standard form. Standard form equations are always written “ coefficient times plus (or minus) coefficient times equals value ”. In other words, they look like this:
where has to be positive, but and do not.
There is a neat method for finding intercepts in standard form, often referred to as the coverup method.
Example C
Find the intercepts of the following equations:
a)
b)
Solution
To solve for each intercept, we realize that at the intercepts the value of either or is zero, and so any terms that contain that variable effectively drop out of the equation. To make a term disappear, simply cover it (a finger is an excellent way to cover up terms) and solve the resulting equation.
a) To solve for the intercept we set and cover up the term:
Now we solve for the intercept:
b) To solve for the intercept , cover up the term:
Now solve for the intercept :
Solving RealWorld Problems Using Intercepts of a Graph
Example D
Jesus has $30 to spend on food for a class barbecue. Hot dogs cost $0.75 each (including the bun) and burgers cost $1.25 (including the bun). Plot a graph that shows all the combinations of hot dogs and burgers he could buy for the barbecue, without spending more than $30.
This time we will find an equation first, and then we can think logically about finding the intercepts.
If the number of burgers that Jesus buys is , then the money he spends on burgers is
If the number of hot dogs he buys is , then the money he spends on hot dogs is
So the total cost of the food is .
The total amount of money he has to spend is $30, so if he is to spend it ALL, we can use the following equation:
We can solve for the intercepts using the coverup method. First the intercept :
Then the intercept :
Now we plot those two points and join them to create our graph, shown here:
We could also have created this graph without needing to come up with an equation. We know that if John were to spend ALL the money on hot dogs, he could buy hot dogs. And if he were to buy only burgers he could buy burgers. From those numbers, we can get 2 intercepts: (0 burgers, 40 hot dogs) and (24 burgers, 0 hot dogs). We could plot these just as we did above and obtain our graph that way.
As a final note, we should realize that Jesus’ problem is really an example of an inequality. He can, in fact, spend any amount up to $30. The only thing he cannot do is spend more than $30. The graph above reflects this: the line is the set of solutions that involve spending exactly $30, and the shaded region shows solutions that involve spending less than $30. We’ll work with inequalities some more in Chapter 6.
Watch this video for help with the Examples above.
CK12 Foundation: Graphing Using Intercepts
Vocabulary
 A intercept occurs at the point where a graph crosses the axis (where ) and an intercept occurs at the point where a graph crosses the axis (where ).
 The intercept can be found by substituting into the equation and solving for . Likewise, the intercept can be found by substituting into the equation and solving for .
 A linear equation is in standard form if it is written as “positive coefficient times plus coefficient times equals value”. Equations in standard form can be solved for the intercepts by covering up the (or ) term and solving the equation that remains.
Guided Practice
1. Graph by finding intercepts.
2. Find the intercepts of using the coverup method.
Solution
1. Find the intercept by plugging in
Find the intercept by plugging in
The graph of this line is the line labeled d, the two intercepts are marked by dots.
2. To solve for the intercept , cover up the term:
Solve for the intercept:
The graph of this function and the intercepts is line c:
Practice
For 18, find the intercepts for the following equations using substitution.
For 916, find the intercepts of the following equations using the coverup method.
For 1720, use any method to find the intercepts and then graph the following equations.

At the local grocery store strawberries cost $3.00 per pound and bananas cost $1.00 per pound.
 If I have $10 to spend on strawberries and bananas, draw a graph to show what combinations of each I can buy and spend exactly $10.
 Plot the point representing 3 pounds of strawberries and 2 pounds of bananas. Will that cost more or less than $10?
 Do the same for the point representing 1 pound of strawberries and 5 pounds of bananas.
 A movie theater charges $7.50 for adult tickets and $4.50 for children. If the theater takes in $900 in ticket sales for a particular screening, draw a graph which depicts the possibilities for the number of adult tickets and the number of child tickets sold.
 Why can't we use the intercept method to graph the following equation?
 Name two more equations that we can’t use the intercept method to graph.
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Description
Learning Objectives
Here you'll learn how to find the intercepts of an equation. You'll then use those intercepts to graph the equation.