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Precalculus for Team-Based Inquiry Learning 2025 Edition — Instructor Version

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Section 3.4 Parallel and Perpendicular Lines (LF4)

Subsection 3.4.1 Activities

Activity 3.4.1.

Let’s revisit Activity 3.2.1 to investigate special types of lines.
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(e)
What is the same about the two lines?
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Answer.
Both lines have the same slope (\(m=2\)).
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(f)
What is different about the two lines?
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Answer.
The lines have different \(y\)-intercepts.
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Definition 3.4.3.

Parallel lines are lines that always have the same distance apart (equidistant) and will never meet. Parallel lines have the same slope, but different \(y\)-intercepts.
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Activity 3.4.4.

Suppose you have the function,
\begin{equation*} f(x)=-\dfrac{1}{2}x-1 \end{equation*}
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(a)
What is the slope of \(f(x)\text{?}\)
  1. \(\displaystyle -1\)
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  2. \(\displaystyle 2\)
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  3. \(\displaystyle 1\)
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  4. \(\displaystyle -\dfrac{1}{2}\)
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Answer.
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(c)
Find the equation of a line parallel to \(f(x)\) that passes through the point \((-4,2)\text{.}\)
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Answer.
\(y-2=-\dfrac{1}{2}(x+4)\) or
\(y=-\dfrac{1}{2}x\)
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Activity 3.4.5.

Consider the graph of the two lines below.
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(e)
If you were to think of slope as "rise over run," how would you write the slope of each line?
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Answer.
Line A could be written as \(-\dfrac{1}{2}\) and Line B could be written as \(\dfrac{2}{1}\text{.}\)
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(f)
How would you compare the slopes of the two lines?
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Answer.
Students might notice that when writing the slopes of Line A and Line B, the slopes are negative reciprocals of each other.
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Remark 3.4.6.

Notice in Activity 3.4.5, that even though the two lines have different slopes, the slopes are somewhat similar. For example, if you take the slope of Line A \(\left(-\dfrac{1}{2}\right)\) and flip and negate it, you will get the slope of Line B \(\left(\dfrac{2}{1}\right)\text{.}\)
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Definition 3.4.7.

Perpendicular lines are two lines that meet or intersect each other at a right angle. The slopes of two perpendicular lines are negative reciprocals of each other (given that the slope exists!).
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Activity 3.4.8.

Suppose you have the function,
\begin{equation*} f(x)=3x+5 \end{equation*}
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(a)
What is the slope of \(f(x)\text{?}\)
  1. \(\displaystyle -\dfrac{1}{3}\)
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  2. \(\displaystyle 3\)
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  3. \(\displaystyle 5\)
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  4. \(\displaystyle -\dfrac{1}{5}\)
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Answer.
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(b)
Applying Definition 3.4.7, what would the slope of a line perpendicular to \(f(x)\) be?
  1. \(\displaystyle -\dfrac{1}{3}\)
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  2. \(\displaystyle 3\)
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  3. \(\displaystyle 5\)
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  4. \(\displaystyle -\dfrac{1}{5}\)
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Answer.
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(c)
Find an equation of the line perpendicular to \(f(x)\) that passes through the point \((3,6)\text{.}\)
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Answer.
\(y-6=-\dfrac{1}{3}(x-3)\) or
\(y=-\dfrac{1}{3}x+7\)
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Activity 3.4.9.

For each pair of lines, determine if they are parallel, perpendicular, or neither.
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(a)
\begin{equation*} f(x)=-3x+4 \end{equation*}
\begin{equation*} g(x)=5-3x \end{equation*}
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Answer.
Parallel. The slope of \(f(x)\) is \(-3\) and the slope of \(g(x)\) is \(-3\text{.}\)
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(b)
\begin{equation*} f(x)=2x-5 \end{equation*}
\begin{equation*} g(x)=6x-5 \end{equation*}
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Answer.
Neither. The slope of \(f(x)\) is \(2\) and the slope of \(g(x)\) is \(6\text{.}\) These lines do, however, have the same \(y\)-intercept.
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(c)
\begin{equation*} f(x)=6x-5 \end{equation*}
\begin{equation*} g(x)=\dfrac{1}{6}x+8 \end{equation*}
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Answer.
Neither. The slope of \(f(x)\) is \(6\) and the slope of \(g(x)\) is \(\dfrac{1}{6}\text{.}\) Although they are reciprocals of one another, they are not negative reciprocals.
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(d)
\begin{equation*} f(x)=\dfrac{4}{5}x+3 \end{equation*}
\begin{equation*} g(x)=-\dfrac{5}{4}x-1 \end{equation*}
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Answer.
Perpendicular. The slope of \(f(x)\) is \(\dfrac{4}{5}\) and the slope of \(g(x)\) is \(-\dfrac{5}{4}\) (and are negative reciprocals of one another).
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Activity 3.4.10.

Consider the linear equation, \(f(x)=-\dfrac{2}{3}x-4\) and the point A: \((-6,4)\text{.}\)
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(a)
Find an equation of the line that is parallel to \(f(x)\) and passes through the point A.
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Answer.
\(y-4=-\dfrac{2}{3}(x+6)\) or
\(y=-\dfrac{2}{3}x\)
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(b)
Find an equation of the line that is perpendicular to \(f(x)\) and passes through the point A.
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Answer.
\(y-4=\dfrac{3}{2}(x+6)\) or
\(y=\dfrac{3}{2}x+13\)
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Activity 3.4.11.

Consider the line, \(y=2\text{,}\) as shown in the graph below.
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(a)
What is the slope of the line \(y=2\text{?}\)
  1. undefined
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  2. \(\displaystyle 0\)
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  3. \(\displaystyle 1\)
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  4. \(\displaystyle -\dfrac{1}{2}\)
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Answer.
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(b)
What is the slope of a line that is parallel to \(y=2\text{?}\)
  1. undefined
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  2. \(\displaystyle 0\)
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  3. \(\displaystyle 1\)
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  4. \(\displaystyle -\dfrac{1}{2}\)
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Answer.
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(c)
Find an equation of the line that is parallel to \(y=2\) and passes through the point \((-1,-4)\text{.}\)
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Answer.
\(y=-4\text{.}\) Students might need the graph to help them visualize why the equation is in the form \(y=\) number.
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(d)
What is the slope of a line that is perpendicular to \(y=2\text{?}\)
  1. undefined
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  2. \(\displaystyle 0\)
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  3. \(\displaystyle 1\)
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  4. \(\displaystyle -\dfrac{1}{2}\)
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Answer.
A. You might need to help students see why the slope is undefined by showing that \(-\dfrac{1}{0}\) is not defined.
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(e)
Find an equation of the line that is perpendicular to \(y=2\) and passes through the point \((-1,2)\text{.}\)
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Answer.
\(x=-1\text{.}\) Students might need the graph to help them visualize why the equation is in the form \(x=\) number.
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Subsection 3.4.2 Exercises

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