4.6.4: Families of Lines
- Page ID
- 4324
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Families of Lines
Think about the members of your family. You probably all have some things in common, but you're definitely not all identical. The same is true of a family of lines. What could a family of lines have in common? What might be different?
Families of Lines
A straight line has two very important properties, its slope and its y−intercept. The slope tells us how steeply the line rises or falls, and the y−intercept tells us where the line intersects the y−axis. In this Concept, we will look at two families of lines.
A family of lines is a set of lines that have something in common with each other. Straight lines can belong to two types of families: where the slope is the same and where the y−intercept is the same.
Family 1: The slope is the same
Remember that lines with the same slope are parallel. Each line on the Cartesian plane below has an identical slope with different y−intercepts. All the lines look the same but they are shifted up and down the y−axis as shown in the graph below. As b gets larger the line rises on the y−axis and as b gets smaller the line goes lower on the y−axis. This behavior is often called a vertical shift.
Let's write the equation for the red line in the image above:
We can see from the graph that the equation has a y-intercept of 1. Since all the lines have the same slope, we can look at any line to determine the slope, so the slope is −2. Therefore, the equation of the red line is:
y=−2x+1.
Family 2: The y−intercept is the same
The graph below shows several lines with the same y−intercept but with varying slopes.
Let's write the equation for the brown line in the image above:
All the lines share the same y-intercept, which is 2. Looking at the graph, the slope is -1. Thus, the equation is:
y=−x+2.
Now, let's write a general equation for each family of lines shown in the images in this Concept.
- For family 1, the red line has the equation y=−2x+1. Since all the lines share the same slope, we keep the slope of -2. But they all have different y-intercepts, so we will use b:
y=−2x+b.
- For family 2, the brown line has the equation y=−x+2. Since all the lines share the same y-intercept but have different slopes:
y=mx+2.
Examples
Example 4.6.4.1
Earlier, you were asked what a family of lines could have in common and what could be different.
Solution
As shown in this concept, there are two important parts of a line, the y−intercept and the slope, that a family of lines can have in common. A family of lines does not need to have both the y−intercept and slope in common, just one.
Example 4.6.4.2
Write the equation of the family of lines perpendicular to 6x+2y=24.
Solution
First we must find the slope of 6x+2y=24:
slope=−6/2=−3.
Now we find the slope of any line perpendicular to our original line:
−3⋅m=−1
(−3⋅m)/−3=−1/−3
m=1/3
The family of lines perpendicular to 6x+2y=24 will have a slope of m=1/3. They will all have different y-intercepts:
y=1/3x+b.
Review
- What is a family of lines?
- Find the equation of the line parallel to 5x−2y=2 that passes through the point (3, –2).
- Find the equation of the line perpendicular to y=−25x−3 that passes through the point (2, 8).
- Find the equation of the line parallel to 7y+2x−10=0 that passes through the point (2, 2).
- Find the equation of the line perpendicular to y+5=3(x−2) that passes through the point (6, 2).
- Find the equation of the line through (2, –4) perpendicular to y=2/7x+3.
- Find the equation of the line through (2, 3) parallel to y=3/2x+5.
In 8–11, write the equation of the family of lines satisfying the given condition.
- All lines pass through point (0, 4).
- All lines are perpendicular to 4x+3y−1=0.
- All lines are parallel to y−3=4x+2.
- All lines pass through point (0, –1).
- Write an equation for a line parallel to the equation graphed below.
- Write an equation for a line perpendicular to the equation graphed below and passing through the point (0, –1).
Quick Quiz
1. Write an equation for a line with a slope of 4/3 and a y−intercept of (0, 8).
2. Write an equation for a line containing (6, 1) and (7, –3).
3. A plumber charges $75 for a 2.5-hour job and $168.75 for a 5-hour job.
Assuming the situation is linear, write an equation to represent the plumber’s charge and use it to predict the cost of a 1-hour job.
4. Rewrite in standard form: y=6/5x+11.
5. Sasha took tickets for the softball game. Student tickets were $3.00 and adult tickets were $3.75. She collected a total of $337.50 and sold 75 student tickets. How many adult tickets were sold?
Review (Answers)
To see the Review answers, open this PDF file and look for section 5.9.
Vocabulary
Term | Definition |
---|---|
vertical shifts | When all lines look the same but they are shifted up and down the y−axis, this behavior is called a vertical shift. As b gets larger, the line rises on the y−axis, and as b gets smaller the line goes lower on the y−axis. |
Additional Resources
PLIX: Play, Learn, Interact, eXplore: Families of Lines: Room in Perspective
Video:
Activity: Families of Lines Discussion Questions
Practice: Families of Lines
Real World Application: Parallel & Perpendicular Lines