4.3: Graphing Using Intercepts
Learning Objectives
At the end of this lesson, students will be able to:
- Find intercepts of the graph of an equation.
- Use intercepts to graph an equation.
- Solve real-world problems using intercepts of a graph.
Vocabulary
Terms introduced in this lesson:
- intercepts
- \begin{align*}x-\end{align*}intercept, \begin{align*}y-\end{align*}intercept
- standard form
- coefficients
- initial cost
Teaching Strategies and Tips
One way to graph lines is by plotting intercepts. Additional questions to ask students in the introduction:
- Do all lines have \begin{align*}x\end{align*} and \begin{align*}y-\end{align*}intercepts?
Solution: No.
- Which lines have only one intercept?
Solution: Horizontal and vertical lines.
- Is it possible for lines not to have intercepts?
Solution: No, every line must have at least one intercept.
- Is it possible for lines to have more than \begin{align*}2\end{align*} intercepts?
Solution: Yes, but infinitely many intercepts. The lines \begin{align*}x = 0\end{align*} and \begin{align*}y = 0\end{align*} cross the axes infinitely may times.
In Examples 2a and 2b, have students convert their fractions into decimals before plotting.
Use the cover-up method as a quick way to find intercepts algebraically.
In Examples 4 and 5, encourage students to interpret the meaning of the intercepts in context of the problem.
Error Troubleshooting
General Tip: Students will often attempt to find an \begin{align*}x-\end{align*}intercept by setting \begin{align*}x = 0\end{align*}; and a \begin{align*}y-\end{align*}intercept by plugging in \begin{align*}0\end{align*} for \begin{align*}y\end{align*}. The opposite is true. To find an intercept, set the other variable to \begin{align*}0\end{align*}.
In Example 4, distinguish between the independent and dependent variables first. To find an appropriate scale for the axes, determine the domain of the independent variable next.
In Example 5, although “without spending more than \begin{align*}\$30\end{align*}” implies an inequality (translates as \begin{align*}\le 30\end{align*}), solve the problem as if it were an equality (“spending exactly \begin{align*}\$30\end{align*}”) and then shade the triangular region. For more on systems of inequalities, see chapter Solving Systems of Equations and Inequalities.
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