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11.5: Graphing with Intercepts (Part 1)

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    21784
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    Learning Objectives
    • Identify the intercepts on a graph
    • Find the intercepts from an equation of a line
    • Graph a line using the intercepts
    • Choose the most convenient method to graph a line
    be prepared!

    Before you get started, take this readiness quiz.

    1. Solve: 3x + 4y = −12 for x when y = 0. If you missed this problem, review Example 9.11.6.
    2. Is the point (0, −5) on the x-axis or y-axis? If you missed this problem, review Example 11.1.5.
    3. Which ordered pairs are solutions to the equation 2x − y = 6? (a) (6, 0) (b) (0, −6) (c) (4, −2). If you missed this problem, review Example 11.2.8.

    Identify the Intercepts on a Graph

    Every linear equation has a unique line that represents all the solutions of the equation. When graphing a line by plotting points, each person who graphs the line can choose any three points, so two people graphing the line might use different sets of points.

    At first glance, their two lines might appear different since they would have different points labeled. But if all the work was done correctly, the lines will be exactly the same line. One way to recognize that they are indeed the same line is to focus on where the line crosses the axes. Each of these points is called an intercept of the line.

    Definition: Intercepts of a Line

    Each of the points at which a line crosses the x-axis and the y-axis is called an intercept of the line.

    Let’s look at the graph of the lines shown in Figure \(\PageIndex{1}\).

    The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through two labeled points, “ordered pair 0, 6” and ordered pair 3, 0”.

    Figure \(\PageIndex{1}\)

    First, notice where each of these lines crosses the x- axis:

    Table \(\PageIndex{1}\)
    Figure: The line crosses the x-axis at: Ordered pair of this point
    Figure \(\PageIndex{1a}\) 3 (3,0)
    Figure \(\PageIndex{1b}\) 4 (4,0)
    Figure \(\PageIndex{1c}\) 5 (5,0)
    Figure \(\PageIndex{1d}\) 0 (0,0)

    Do you see a pattern?

    For each row, the y-coordinate of the point where the line crosses the x-axis is zero. The point where the line crosses the x-axis has the form (a, 0); and is called the x-intercept of the line. The x-intercept occurs when y is zero.

    Now, let's look at the points where these lines cross the y-axis.

    Table \(\PageIndex{2}\)
    Figure: The line crosses the x-axis at: Ordered pair of this point
    Figure \(\PageIndex{1a}\) 6 (0, 6)
    Figure \(\PageIndex{1b}\) -3 (0, -3)
    Figure \(\PageIndex{1c}\) -5 (0, -5)
    Figure \(\PageIndex{1d}\) 0 (0, 0)
    Definition: x-intercept and y-intercept of a line

    The x-intercept is the point, (a, 0), where the graph crosses the x-axis.

    The x-intercept occurs when y is zero.

    The y-intercept is the point, (0, b), where the graph crosses the y-axis.

    The y-intercept occurs when x is zero.

    Example \(\PageIndex{1}\)

    Find the x- and y-intercepts of each line:

    (a) x + 2y = 4

    The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through the points “ordered pair 0, 2” and “ordered pair 4, 0”.

    (b) 3x - y = 6

    The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through the points “ordered pair 0, -6” and “ordered pair 2, 0”.

    (c) x + y = -5

    The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through the points “ordered pair 0, -5” and “ordered pair -5, 0”.

    Solution

    (a)

    The graph crosses the x-axis at the point (4, 0). The x-intercept is (4, 0).
    The graph crosses the y-axis at the point (0, 2). The x-intercept is (0, 2).

    (b)

    The graph crosses the x-axis at the point (2, 0). The x-intercept is (2, 0).
    The graph crosses the y-axis at the point (0, -6). The x-intercept is (0, -6).

    (c)

    The graph crosses the x-axis at the point (-5, 0). The x-intercept is (-5, 0).
    The graph crosses the y-axis at the point (0, -5). The x-intercept is (0, -5).
    Exercise \(\PageIndex{1A}\)

    Find the x- and y-intercepts of the graph: x − y = 2.

    The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through the points “ordered pair 0, 2” and “ordered pair 2, 0”.

    Answer

    x-intercept (2,0); y-intercept (0,-2)

    Exercise \(\PageIndex{1B}\)

    Find the x- and y-intercepts of the graph: 2x + 3y = 6.

    The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through the points “ordered pair 0, 2” and “ordered pair 3, 0”.

    Answer

    x-intercept (3,0); y-intercept (0,2)

    Find the Intercepts from an Equation of a Line

    Recognizing that the x-intercept occurs when y is zero and that the y-intercept occurs when x is zero gives us a method to find the intercepts of a line from its equation. To find the x-intercept, let y = 0 and solve for x. To find the y-intercept, let x = 0 and solve for y.

    Definition: Find the x and y from the Equation of a Line

    Use the equation to find:

    • the x-intercept of the line, let y = 0 and solve for x.
    • the y-intercept of the line, let x = 0 and solve for y
    Table \(\PageIndex{3}\)
    x y
      0
    0  
    Example \(\PageIndex{2}\)

    Find the intercepts of 2x + y = 6

    Solution

    We'll fill in Figure \(\PageIndex{2}\).

    ...

    Figure \(\PageIndex{2}\)

    To find the x- intercept, let y = 0:

    Substitute 0 for y. \(2x + \textcolor{red}{0} = 6\)
    Add. 2x = 6
    Divide by 2. x = 3

    The x-intercept is (3, 0).

    To find the y- intercept, let x = 0:

    Substitute 0 for x. \(2 \cdot \textcolor{red}{0} + y = 6\)
    Multiply. 0 + y = 6
    Add. y = 6

    The y-intercept is (0, 6).

    2x + y = 6
    x y
    3 0
    0 6

    Figure \(\PageIndex{3}\)

    The intercepts are the points (3, 0) and (0, 6).

    Exercise \(\PageIndex{2A}\)

    Find the intercepts: 3x + y = 12.

    Answer

    x-intercept (4,0); y-intercept (0,12)

    Exercise \(\PageIndex{2B}\)

    Find the intercepts: x + 4y = 8.

    Answer

    x-intercept (8,0); y-intercept (0,2)

    Example \(\PageIndex{3}\)

    Find the intercepts of 4x−3y = 12.

    Solution

    To find the x-intercept, let y = 0.

    Substitute 0 for y. 4x − 3 • 0 = 12
    Multiply. 4x − 0 = 12
    Subtract. 4x = 12
    Divide by 4. x = 3

    The y-intercept is (0, −4). The intercepts are the points (−3, 0) and (0, −4).

    4x - 3y = 12
    x y
    3 0
    0 -4
    Exercise \(\PageIndex{3A}\)

    Find the intercepts of the line: 3x−4y = 12.

    Answer

    x-intercept (4,0); y-intercept (0,-3)

    Exercise \(\PageIndex{3B}\)

    Find the intercepts of the line: 2x−4y = 8.

    Answer

    x-intercept (4,0); y-intercept (0,-2)

    Graph a Line Using the Intercepts

    To graph a linear equation by plotting points, you can use the intercepts as two of your three points. Find the two intercepts, and then a third point to ensure accuracy, and draw the line. This method is often the quickest way to graph a line.

    Example \(\PageIndex{4}\)

    Graph −x + 2y = 6 using intercepts.

    Solution

    First, find the x-intercept. Let y = 0,

    \[\begin{split} -x + 2y &= 6 \\ -x + 2(0) &= 6 \\ -x &= 6 \\ x &= -6 \end{split}\]

    The x-intercept is (–6, 0).

    Now find the y-intercept. Let x = 0.

    \[\begin{split} -x + 2y &= 6 \\ -0 + 2y &= 6 \\ 2y &= 6 \\ y &= 3 \end{split}\]

    The y-intercept is (0, 3).

    Find a third point. We’ll use x = 2,

    \[\begin{split} -x + 2y &= 6 \\ -2 + 2y &= 6 \\ 2y &= 8 \\ y &= 4 \end{split}\]

    A third solution to the equation is (2, 4).

    Summarize the three points in a table and then plot them on a graph.

    -x + 2y = 6
    x y (x,y)
    -6 0 (−6, 0)
    0 3 (0, 3)
    2 4 (2, 4)

    The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10. Three labeled points are shown at “ordered pair -6, 0”, “ordered pair 0, 3” and “ordered pair 2, 4”.

    Do the points line up? Yes, so draw line through the points.

    The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10. Three labeled points are shown at “ordered pair -6, 0”, “ordered pair 0, 3” and “ordered pair 2, 4”.  A line passes through the three labeled points.

    Exercise \(\PageIndex{4A}\)

    Graph the line using the intercepts: x−2y = 4.

    Answer
    The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12.  A line passes through the points “ordered pair 0, -2” and “ordered pair 4, 0”.
    Exercise \(\PageIndex{4B}\)

    Graph the line using the intercepts: −x + 3y = 6.

    Answer

    The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12.  A line passes through the points “ordered pair 0, 2” and “ordered pair -6, 0”.

    HOW TO: GRAPH A LINE USING THE INTERCEPTS

    Step 1. Find the x - and y-intercepts of the line.

    • Let y = 0 and solve for x.
    • Let x = 0 and solve for y.

    Step 2. Find a third solution to the equation.

    Step 3. Plot the three points and then check that they line up.

    Step 4. Draw the line.

    Example \(\PageIndex{5}\)

    Graph 4x−3y = 12 using intercepts.

    Solution

    Find the intercepts and a third point.

    $$\begin{split} x-intercept,\; &let\; y = 0 \\ 4x - 3y &= 12 \\ 4x - 3(\textcolor{red}{0}) &= 12 \\ 4x &= 12 \\ x &= 3 \end{split}$$ $$\begin{split} y-intercept,\; &let\; x = 0 \\ 4x - 3y &= 12 \\ 4(\textcolor{red}{0}) - 3y &= 12 \\ 4x - 3(\textcolor{red}{4}) &= 12 \\ -3y &= 12 \\ y &= -4 \end{split}$$ $$\begin{split} third\; point,\; &let\; y = 4 \\ 4x - 3y &= 12 \\ 4x - 12 &= 12 \\ 4x &= 24 \\ x &= 6 \end{split}$$

    We list the points and show the graph.

    4x - 3y = 12
    x y (x. y)
    3 0 (3, 0)
    0 -4 (0, −4)
    6 4 (6, 4)

    The graph shows the x y-coordinate plane. Both axes run from -7 to 7. Three unlabeled points are drawn at  “ordered pair 0, -4”, “ordered pair 3, 0” and “ordered pair  6, 4”.  A line passes through the points.

    Exercise \(\PageIndex{5A}\)

    Graph the line using the intercepts: 5x−2y = 10.

    Answer

    The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7.  A line passes through the points “ordered pair 0, -5” and “ordered pair 2, 0”.

    Exercise \(\PageIndex{5B}\)

    Graph the line using the intercepts: 3x−4y = 12.

    Answer

    The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7.  A line passes through the points “ordered pair 0, -3” and “ordered pair 4, 0”.

    Example \(\PageIndex{6}\)

    Graph \(y = 5x\) using the intercepts.

    Solution

    $$\begin{split} x-intercept;\; &Let\; y = 0 \ldotp \\ y &= 5x\\ \textcolor{red}{0} &= 5x \\ 0 &= x \\ x &= 0 \\ The\; x-intercept\; &is\; (0, 0) \ldotp \end{split}$$ $$\begin{split} y-intercept;\; &Let\; x = 0 \ldotp \\ y &= 5x \\ y &= 5(\textcolor{red}{0}) \\ y &= 0 \\ The\; y-intercept\; &is\; (0, 0) \ldotp \end{split}$$

    This line has only one intercept! It is the point (0, 0).

    To ensure accuracy, we need to plot three points. Since the intercepts are the same point, we need two more points to graph the line. As always, we can choose any values for x, so we’ll let x be 1 and −1.

    $$\begin{split} x &= 1 \\ y &= 5x \\ y &= 5(\textcolor{red}{1}) \\ y &= 5 \\ (1, &-5) \end{split}$$ $$\begin{split} x &= -1 \\ y &= 5x \\ y &= 5(\textcolor{red}{-1}) \\ y &= -5 \\ (-1, &-5) \end{split}$$

    Organize the points in a table.

    y = 5x
    x y (x, y)
    0 0 (0, 0)
    1 5 (1, 5)
    -1 -5 (−1, −5)

    Plot the three points, check that they line up, and draw the line.

    The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10.  A line passes through three labeled points, “ordered pair -1, -5”, “ordered pair 0, 0”, and ordered pair 1, 5”.

    Exercise \(\PageIndex{6A}\)

    Graph using the intercepts: \(y = 3x\).

    Answer

    The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12.  A line passes through the points “ordered pair 0, 0” and “ordered pair 1, 3”.

    Exercise \(\PageIndex{6B}\)

    Graph using the intercepts: \(y = − x\).

    Answer

    The graph shows the x y-coordinate plane. The x and y-axis each run from -12 to 12.  A line passes through the points “ordered pair 0, 0” and “ordered pair 1, -1”.

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