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4.6.4: Families of Lines

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  • 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.

    Figure 4.6.4.1

    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.

    Figure 4.6.4.2

    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.

    1. 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.

    1. 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

    1. What is a family of lines?
    2. Find the equation of the line parallel to 5x−2y=2 that passes through the point (3, –2).
    3. Find the equation of the line perpendicular to y=−25x−3 that passes through the point (2, 8).
    4. Find the equation of the line parallel to 7y+2x−10=0 that passes through the point (2, 2).
    5. Find the equation of the line perpendicular to y+5=3(x−2) that passes through the point (6, 2).
    6. Find the equation of the line through (2, –4) perpendicular to y=2/7x+3.
    7. 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.

    1. All lines pass through point (0, 4).
    2. All lines are perpendicular to 4x+3y−1=0.
    3. All lines are parallel to y−3=4x+2.
    4. All lines pass through point (0, –1).
    5. Write an equation for a line parallel to the equation graphed below.
    6. Write an equation for a line perpendicular to the equation graphed below and passing through the point (0, –1).
    Figure 4.6.4.3

    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

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