4.7: Linear Function Graphs
Learning Objectives
At the end of this lesson, students will be able to:
- Recognize and use function notation.
- Graph a linear function.
- Change slope and intercepts of function graphs.
- Analyze graphs of real-world functions.
Vocabulary
Terms introduced in this lesson:
- vertical line test
- slope-intercept form
- arithmetic progression
- common difference
- discrete
- continuous
Teaching Strategies and Tips
Students learn to use function notation for the first time.
- \begin{align*}f(x)\end{align*} is pronounced “\begin{align*}f\end{align*} of \begin{align*}x\end{align*}".
- \begin{align*}y = f(x)\end{align*}, “\begin{align*}y\end{align*} is a function of \begin{align*}x\end{align*}”, and “\begin{align*}y\end{align*} depends on \begin{align*}x\end{align*}” are synonymous. Also, the input or independent variable is \begin{align*}x\end{align*}.
- \begin{align*}“f(x)”\end{align*} and \begin{align*}“y=”\end{align*} are interchangeable. This means that the graphing techniques students have learned previously can be used to graph functions.
Point out in Example 1 that the \begin{align*}y\end{align*} variable depends on \begin{align*}x\end{align*}; that’s the only reason for solving for \begin{align*}y\end{align*} in each case.
Additional Example:
Rewrite the following equation using function notation if cost depends on the number of pounds purchased.
\begin{align*} C - 3 - 2n = 0\end{align*}
Solution: Solve for \begin{align*}C\end{align*} and replace \begin{align*}C\end{align*} with \begin{align*}C(n)\end{align*}.
\begin{align*} C & = 2n + 3\\ C(n) & = 2n + 3\end{align*}
Use Example 2 to show how function notation is used. There is nothing new computationally; students evaluate expressions as they before. Remind students to use order of operations.
\begin{align*}x-\end{align*}intercepts can be found by solving \begin{align*}f(x) = 0\end{align*} and \begin{align*}y-\end{align*}intercepts by computing \begin{align*}f(0)\end{align*}. See Example 4.
Use Example 5 to show how linear functions and arithmetic sequences are related.
- In an arithmetic sequence, terms are found by adding the same constant to the previous term.
- For linear functions, when the input variable is increased by \begin{align*}1\end{align*}, the output variable changes by the value of the slope.
Use Example 5 to distinguish between discrete and continuous. Use examples from daily life: The number of pumpkin seeds in a pumpkin is discrete. The number of people that can fill a football stadium is discrete. The waiting time for a bus at a bus stop is continuous. The weights of newborn babies are continuous.
Error Troubleshooting
General Tip. \begin{align*}f(x)\end{align*} is function notation and should not be confused with multiplication (\begin{align*}f\end{align*} times \begin{align*}x\end{align*} is not correct).
Example 5c and Problems 6b and 6c of the Review Questions have no consecutive terms to subtract to determine the common difference. In such an event, use unknowns for the sequence terms starting with the first given term. Solve the resulting equation. See the hint at the end of Example 5c.
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