4.3: Graphs Using Intercepts
As you may have seen in the previous lesson, graphing solutions to an equation of two variables can be time-consuming. Fortunately, there are several ways to make graphing solutions easier. This lesson will focus on graphing a line by finding its intercepts. Lesson 4.5 will show you how to graph a line using its slope and \begin{align*}y-\end{align*}intercept.
In geometry, there is a postulate that states, “Two points determine a line.” Therefore, to draw any line, all you need is two points. One way is to find its intercepts.
An intercept is the point at which a graphed equation crosses an axis.
The \begin{align*}x-\end{align*}intercept is an ordered pair at which the line crosses the \begin{align*}x-\end{align*}axis (the horizontal axis). Its ordered pair has the form \begin{align*}(x,0)\end{align*}.
The \begin{align*}y-\end{align*}intercept is an ordered pair at which the line crosses the \begin{align*}y-\end{align*}axis (the vertical axis). Its ordered pair has the form \begin{align*}(0,y)\end{align*}
By finding the intercepts of an equation, you can quickly graph all the possible solutions to the equation.
Finding Intercepts Using Substitution
Remember that the Substitution Property allows the replacement of a variable with a numerical value or another expression. You can use this property to help find the intercepts of an equation.
Example: Graph \begin{align*}2x+3y=-6\end{align*} using its intercepts.
Solution: The \begin{align*}x-\end{align*}intercept has an ordered pair \begin{align*}(x,0)\end{align*}. Therefore, the \begin{align*}y-\end{align*}coordinate has a value of zero. By substituting zero for the variable of \begin{align*}y\end{align*}, the equation becomes:
\begin{align*}2x+3(0)=-6\end{align*}
Continue solving for \begin{align*}x\end{align*}:
\begin{align*}2x+0& =-6\\ 2x& =-6\\ x& =-3\end{align*}
The \begin{align*}x-\end{align*}intercept has an ordered pair of (–3, 0).
Repeat the process to find the \begin{align*}y-\end{align*}intercept. The ordered pair of the \begin{align*}y-\end{align*}intercept is \begin{align*}(0,y)\end{align*}. Using substitution,
\begin{align*}2(0)+3y& =-6\\ 3y& =-6\\ y& =-2\end{align*}
The \begin{align*}y-\end{align*}intercept has the ordered pair (0, –2).
To graph the line formed by the solutions of the equation \begin{align*}2x+3y=-6\end{align*}, graph the two intercepts and connect them with a straight line.
Example: Graph \begin{align*}4x-2y=8\end{align*} using its intercepts.
Solution: Determine the \begin{align*}x-\end{align*}intercept by substituting zero for the variable \begin{align*}y\end{align*}.
\begin{align*}4x-2(0)& =8\\ 4x& =8\\ x& =2\end{align*}
The ordered pair of the \begin{align*}x-\end{align*}intercept is (2, 0). By repeating this process, you find the \begin{align*}y-\end{align*}intercept has the ordered pair (0, –4). Graph these two ordered pairs and connect with a line.
Finding Intercepts Using the Cover-Up Method
By finding an intercept, you are substituting the value of zero in for one of the variables.
To find the \begin{align*}x-\end{align*}intercept, substitute zero for the \begin{align*}y\end{align*}-value.
To find the \begin{align*}y-\end{align*}intercept, substitute zero for the \begin{align*}x\end{align*}-value.
A second method of finding the intercepts is called the Cover-Up Method. Using the Multiplication Property of Zero \begin{align*}a(0)=0\end{align*}, you can “cover-up” the other variable and solve for the intercept you wish to find.
Example: Graph \begin{align*}-7x-3y=21\end{align*} using its intercepts.
Solution: To solve for the \begin{align*}y-\end{align*}intercept we set \begin{align*}x=0\end{align*} and cover up the \begin{align*}x\end{align*} term:
\begin{align*}-3y& =21\\ y& =-7 && (0,-7) \ \text{is the} \ y-intercept \text.\end{align*}
To solve for the \begin{align*}x-\end{align*}intercept, cover up the \begin{align*}y-\end{align*}variable and solve for \begin{align*}x\end{align*}:
\begin{align*}-7x& =21\\ x& =-3 && (-3,0) \ \text{is the} \ x-intercept \text.\end{align*}
Now graph by first plotting the intercepts then drawing a line through these points.
Example 1: Jose has $30 to spend on food for a class barbeque. Hot dogs cost $0.75 each (including the bun) and burgers cost $1.25 (including bun and salad). Plot a graph that shows all the combinations of hot dogs and burgers he could buy for the barbecue, spending exactly $30.
Solution: Begin by translating this sentence into an algebraic equation. Let \begin{align*}y=\end{align*} the number of hot dogs and \begin{align*}x=\end{align*} the number of burgers.
\begin{align*}1.25(x)+ 0.75(y)=30\end{align*}
Find the intercepts of the graph. This example will use the Cover-Up Method. Feel free to use Substitution if you prefer.
\begin{align*}0.75y& =30\\ y& =40 && y-intercept(0,40)\end{align*}
\begin{align*}1.25x& =30\\ x& =24 && x-intercept(24,0)\end{align*}
By graphing Jose’s situation, you can determine the combinations of hot dogs and burgers he can purchase for exactly $30.00.
Practice Set
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: Graphing Using Intercepts (12:18)
- Define intercept.
- What is the ordered pair for an \begin{align*}x-\end{align*}intercept?
- Explain the process of the Cover-Up Method.
Find the intercepts for the following equations using substitution.
- \begin{align*}y=3x-6\end{align*}
- \begin{align*}y=-2x+4\end{align*}
- \begin{align*}y=14x-21\end{align*}
- \begin{align*}y=7-3x\end{align*}
Find the intercepts of the following equations using the Cover-Up Method.
- \begin{align*}5x-6y=15\end{align*}
- \begin{align*}3x-4y=-5\end{align*}
- \begin{align*}2x+7y=-11\end{align*}
- \begin{align*}5x+10y=25\end{align*}
- Do you prefer the Substitution Method or the Cover-Up Method? Why?
In 13 – 24, use any method to find the intercepts and then graph the equation.
- \begin{align*}y=2x+3\end{align*}
- \begin{align*}6(x-1)=2(y+3)\end{align*}
- \begin{align*}x-y=5\end{align*}
- \begin{align*}x+y=8\end{align*}
- \begin{align*}4x+9y=0\end{align*}
- \begin{align*}\frac{1}{2} x+4y=12\end{align*}
- \begin{align*}x-2y=4\end{align*}
- \begin{align*}7x-5y=10\end{align*}
- \begin{align*}4x-y=-3\end{align*}
- \begin{align*}x-y=0\end{align*}
- \begin{align*}5x+y=5\end{align*}
- \begin{align*}7x-2y=-6\end{align*}
- Which intercept does a vertical line have?
- Does the equation \begin{align*}y=5\end{align*} have both an \begin{align*}x-\end{align*}intercept and a \begin{align*}y-\end{align*}intercept? Explain your answer.
- Write an equation having only an \begin{align*}x-\end{align*}intercept at (–4, 0).
- How many equations can be made with only one intercept at (0, 0)? Hint: Draw a picture to help you.
- What needs to be done to the following equation before you can use either method to find its intercepts? \begin{align*}3(x+2)=2(y+3)\end{align*}
- At the local grocery store, strawberries cost $3.00 per pound and bananas cost $1.00 per pound. If I have $10 to spend between strawberries and bananas, draw a graph to show what combinations of each I can buy and spend exactly $10.
- A movie theater charges $7.50 for adult tickets and $4.50 for children. If the $900 theater takes in ticket sales for a particular screening, draw a graph that depicts the possibilities for the number of adult tickets and the number of child tickets sold.
- In football, touchdowns are worth 6 points, field goals are worth 3 points, and safeties are worth 2 points. Suppose there were no safeties and the team scored 36 points. Graph the situation to determine the combinations of field goals and touchdowns the team could have had.
Mixed Review
Determine whether each ordered pair is a solution to the equation.
- \begin{align*}5x+2y=23;(7,-6)\end{align*} and (3, 4)
- \begin{align*}3a-2b=6;(0,3)\end{align*} and \begin{align*}\left (\frac{5}{3},\frac{-1}{2}\right )\end{align*}.
- Graph the solutions to the equation \begin{align*}x=-5\end{align*}.
- Solve: \begin{align*}\frac{4}{5} k-16=-\frac{1}{4}\end{align*}.
- Is the following relation a function? \begin{align*}\left \{(-1,1),(0,0),(1,1),(2,3),(0,6)\right \}\end{align*}
- Using the number categories in Lesson 2.1, what is the best way to describe the domain of the following situation: The number of donuts purchased at a coffee shop on a particular day?
- Find the percent of change: Old price = $1,299; new price = $1,145.