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Mathematics LibreTexts

9.8E: Exercises

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    Practice Makes Perfect

    Exercise \(\PageIndex{23}\) Graph Quadratic Functions of the Form \(f(x)=x^{2}=k\)

    In the following exercises,

    1. Graph the quadratic functions on the same rectangular coordinate system
    2. Describe what effect adding a constant, \(k\), to the function has on the basic parabola.
      1. \(f(x)=x^{2}, g(x)=x^{2}+4, \text { and } h(x)=x^{2}-4\)
      2. \(f(x)=x^{2}, g(x)=x^{2}+7, \text { and } h(x)=x^{2}-7\)
    Answer

    1.


    1. This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The top curve has been moved up 4 units, and the bottom has been moved down 4 units.
      Figure 9.7.71
    2. The graph of \(g(x)=x^{2}+4\) is the same as the graph of \(f(x)=x^{2}\) but shifted up \(4\) units. The graph of \(h(x)=x^{2}-4\) is the same as the graph of \(f(x)=x^{2}\) but shift down \(4\) units.
    Exercise \(\PageIndex{24}\) Graph Quadratic Functions of the Form \(f(x)=x^{2}=k\)

    In the following exercises, graph each function using a vertical shift.

    1. \(f(x)=x^{2}+3\)
    2. \(f(x)=x^{2}-7\)
    3. \(g(x)=x^{2}+2\)
    4. \(g(x)=x^{2}+5\)
    5. \(h(x)=x^{2}-4\)
    6. \(h(x)=x^{2}-5\)
    Answer

    1.

    This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 3) and other points (7, 2) and (7, negative 2).
    Figure 9.7.72

    3.

    This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 2) and other points (negative 2, 6) and (2, 6).
    Figure 9.7.73

    5.

    This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, negative 4) and other points (negative 2, 0) and (2, 0).
    Figure 9.7.74
    Exercise \(\PageIndex{25}\) Graph Quadratic Functions of the Form \(f(x)=(x-h)^{2}\)

    In the following exercises,

    1. Graph the quadratic functions on the same rectangular coordinate system
    2. Describe what effect adding a constant, \(h\), inside the parentheses has
      1. \(f(x)=x^{2}, g(x)=(x-3)^{2}, \text { and } h(x)=(x+3)^{2}\)
      2. \(f(x)=x^{2}, g(x)=(x+4)^{2}, \text { and } h(x)=(x-4)^{2}\)
    Answer

    1.


    1. This figure shows 3 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The graph to the right is shifted 3 units to the right to produce g of x equals the quantity of x minus 3 squared. The graph the left is shifted 3 units to the left to produce h of x equals the quantity of x plus 3 squared.
      Figure 9.7.75
    2. The graph of \(g(x)=(x−3)^{2}\) is the same as the graph of \(f(x)=x^{2}\) but shifted right \(3\) units. The graph of \(h(x)=(x+3)^{2}\) is the same as the graph of \(f(x)=x^{2}\) but shifted left \(3\) units.
    Exercise \(\PageIndex{26}\) Graph Quadratic Functions of the Form \(f(x)=(x-h)^{2}\)

    In the following exercises, graph each function using a horizontal shift.

    1. \(f(x)=(x-2)^{2}\)
    2. \(f(x)=(x-1)^{2}\)
    3. \(f(x)=(x+5)^{2}\)
    4. \(f(x)=(x+3)^{2}\)
    5. \(f(x)=(x-5)^{2}\)
    6. \(f(x)=(x+2)^{2}\)
    Answer

    1.

    This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (2, 0) and other points (0, 4) and (4, 4).
    Figure 9.7.76

    3.

    This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 5, 0) and other points (negative 7, 4) and (negative 3, 4).
    Figure 9.7.77

    5.

    This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (5, 0) and other points (3, 4) and (7, 4).
    Figure 9.7.78
    Exercise \(\PageIndex{27}\) Graph Quadratic Functions of the Form \(f(x)=(x-h)^{2}\)

    In the following exercises, graph each function using transformations.

    1. \(f(x)=(x+2)^{2}+1\)
    2. \(f(x)=(x+4)^{2}+2\)
    3. \(f(x)=(x-1)^{2}+5\)
    4. \(f(x)=(x-3)^{2}+4\)
    5. \(f(x)=(x+3)^{2}-1\)
    6. \(f(x)=(x+5)^{2}-2\)
    7. \(f(x)=(x-4)^{2}-3\)
    8. \(f(x)=(x-6)^{2}-2\)
    Answer

    1.

    This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 2, 1) and other points (negative 4, 5) and (0, 5).
    Figure 9.7.79

    3.

    This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (1, 5) and other points (negative 1, 9) and (3, 9).
    Figure 9.7.80

    5.

    This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 3, 1) and other points (negative 4, 0) and (negative 2, 0).
    Figure 9.7.81

    7.

    This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (4, negative 2) and other points (3, negative 2) and (5, negative 2).
    Figure 9.7.82
    Exercise \(\PageIndex{28}\) Graph Quadratic Functions of the Form \(f(x)=ax^{2}\)

    In the following exercises, graph each function.

    1. \(f(x)=-2 x^{2}\)
    2. \(f(x)=4 x^{2}\)
    3. \(f(x)=-4 x^{2}\)
    4. \(f(x)=-x^{2}\)
    5. \(f(x)=\frac{1}{2} x^{2}\)
    6. \(f(x)=\frac{1}{3} x^{2}\)
    7. \(f(x)=\frac{1}{4} x^{2}\)
    8. \(f(x)=-\frac{1}{2} x^{2}\)
    Answer

    1.

    This figure shows a downward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 0) and other points (negative 1, negative 2) and (1, negative 2).
    Figure 9.7.83

    3.

    This figure shows a downward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 0) and other points (negative 1, negative 4) and (1, negative 4).
    Figure 9.7.84

    5.

    This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 0) and other points (negative 2, 2) and (2, 2).
    Figure 9.7.85

    7.

    This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 0) and other points (2, 1) and (negative 2, 1).
    Figure 9.7.86
    Exercise \(\PageIndex{29}\) Graph Quadratic Functions Using Transformations

    In the following exercises, rewrite each function in the \(f(x)=a(x−h)^{2}+k\) form by completing the square.

    1. \(f(x)=-3 x^{2}-12 x-5\)
    2. \(f(x)=2 x^{2}-12 x+7\)
    3. \(f(x)=3 x^{2}+6 x-1\)
    4. \(f(x)=-4 x^{2}-16 x-9\)
    Answer

    1. \(f(x)=-3(x+2)^{2}+7\)

    3. \(f(x)=3(x+1)^{2}-4\)

    Exercise \(\PageIndex{30}\) Graph Quadratic Functions Using Transformations

    In the following exercises,

    1. Rewrite each function in \(f(x)=a(x−h)^{2}+k\) form
    2. Graph it by using transformations
      1. \(f(x)=x^{2}+6 x+5\)
      2. \((x)=x^{2}+4 x-12\)
      3. \(f(x)=x^{2}+4 x-12\)
      4. \(f(x)=x^{2}-6 x+8\)
      5. \(f(x)=x^{2}-6 x+15\)
      6. \(f(x)=x^{2}+8 x+10\)
      7. \(f(x)=-x^{2}+8 x-16\)
      8. \(f(x)=-x^{2}+2 x-7\)
      9. \(f(x)=-x^{2}-4 x+2\)
      10. \(f(x)=-x^{2}+4 x-5\)
      11. \(f(x)=5 x^{2}-10 x+8\)
      12. \(f(x)=3 x^{2}+18 x+20\)
      13. \(f(x)=2 x^{2}-4 x+1\)
      14. \(f(x)=3 x^{2}-6 x-1\)
      15. \(f(x)=-2 x^{2}+8 x-10\)
      16. \(f(x)=-3 x^{2}+6 x+1\)
    Answer

    1.

    1. f(x)=(x+3)^{2}-4
    This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 3, 3), y-intercept of (0, 5), and axis of symmetry shown at x equals negative 3.
    Figure 9.7.87

    3.

    1. \(f(x)=(x+2)^{2}-1\)
    This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 2, negative 1), y-intercept of (0, 3), and axis of symmetry shown at x equals negative 2.
    Figure 9.7.88

    5.

    1. \(f(x)=(x-3)^{2}+6\)
    This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (3, 6), y-intercept of (0, 10), and axis of symmetry shown at x equals 3.
    Figure 9.7.89

    7.

    1. \(f(x)=-(x-4)^{2}+0\)
    This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (4, 0), y-intercept of (0, negative 16), and axis of symmetry shown at x equals 4.
    Figure 9.7.90

    9.

    1. \(f(x)=-(x+2)^{2}+6\)
    This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 2, 6), y-intercept of (0, 2), and axis of symmetry shown at x equals negative 2.
    Figure 9.7.91

    11.

    1. \(f(x)=5(x-1)^{2}+3\)
    This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, 3), y-intercept of (0, 8), and axis of symmetry shown at x equals 1.
    Figure 9.7.92

    13.

    1. \(f(x)=2(x-1)^{2}-1\)
    This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 1), y-intercept of (0, 1), and axis of symmetry shown at x equals 1.
    Figure 9.7.93

    15.

    1. \(f(x)=-2(x-2)^{2}-2\)
    This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (2, negative 2), y-intercept of (0, negative 10), and axis of symmetry shown at x equals 2.
    Figure 9.7.94
    Exercise \(\PageIndex{31}\) Graph Quadratic Functions Using Transformations

    In the following exercises,

    1. Rewrite each function in \(f(x)=a(x−h)^{2}+k\) form
    2. Graph it using properties
      1. \(f(x)=2 x^{2}+4 x+6\)
      2. \(f(x)=3 x^{2}-12 x+7\)
      3. \(f(x)=-x^{2}+2 x-4\)
      4. \(f(x)=-2 x^{2}-4 x-5\)
    Answer

    1.

    1. \(f(x)=2(x+1)^{2}+4\)
    This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 1, 4), y-intercept of (0, 6), and axis of symmetry shown at x equals negative 1.
    Figure 9.7.95

    3.

    1. \(f(x)=-(x-1)^{2}-3\)
    This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 3), y-intercept of (0, negative 4), and axis of symmetry shown at x equals 1.
    Figure 9.7.96
    Exercise \(\PageIndex{32}\) Matching

    In the following exercises, match the graphs to one of the following functions:

    1. \(f(x)=x^{2}+4\)
    2. \(f(x)=x^{2}-4\)
    3. \(f(x)=(x+4)^{2}\)
    4. \(f(x)=(x-4)^{2}\)
    5. \(f(x)=(x+4)^{2}-4\)
    6. \(f(x)=(x+4)^{2}+4\)
    7. \(f(x)=(x-4)^{2}-4\)
    8. \(f(x)=(x-4)^{2}+4\)

      1. This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 4, 0) and other points (negative 4, 4) and (negative 2, 4).
        Figure 9.7.97

      2. This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (0, negative 4) and other points (negative 2, 0) and (2, 0).
        Figure 9.7.98

      3. This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 4, negative 4) and other points (negative 4, 0) and (negative 2, 0).
        Figure 9.7.99

      4. This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 4, 4) and other points (negative 6, 8) and (negative 2, 8).
        Figure 9.7.100

      5. This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (4, 0) and other points (2, 4) and (2, 4).
        Figure 9.7.101

      6. This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (0, 4) and other points (negative 2, 8) and (2, 8).
        Figure 9.7.102

      7. This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (4, negative 4) and other points (2,0) and (6,0).
        Figure 9.7.103

      8. This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (4, 4) and other points (2,8) and (6,8).
        Figure 9.7.104
    Answer

    1. c

    3. e

    5. d

    7. g

    Exercise \(\PageIndex{33}\) Find a Quadratic Function from its Graph

    In the following exercises, write the quadratic function in \(f(x)=a(x−h)^{2}+k\) form whose graph is shown.


    1. This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 1, negative 5) and y-intercept (0, negative 4).
      Figure 9.7.105

    2. This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (2,4) and y-intercept (0, 8).
      Figure 9.7.106

    3. This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 3) and y-intercept (0, negative 1).
      Figure 9.7.107

    4. This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 1, negative 5) and y-intercept (0, negative 3).
      Figure 9.7.108
    Answer

    1. \(f(x)=(x+1)^{2}-5\)

    3. \(f(x)=2(x-1)^{2}-3\)

    Exercise \(\PageIndex{34}\) Writing Exercise
    1. Graph the quadratic function \(f(x)=x^{2}+4x+5\) first using the properties as we did in the last section and then graph it using transformations. Which method do you prefer? Why?
    2. Graph the quadratic function \(f(x)=2x^{2}−4x−3\) first using the properties as we did in the last section and then graph it using transformations. Which method do you prefer? Why?
    Answer

    1. Answers may vary.

    Self Check

    a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    This figure is a list to assess your understanding of the concepts presented in this section. It has 4 columns labeled I can…, Confidently, With some help, and No-I don’t get it! Below I can…, there is graph Quadratic Functions of the form f of x equals x squared plus k; graph Quadratic Functions of the form f of x equals the quantity x minus h squared; graph Quadratic functions of the form f of x equals a times x squared; graph Quadratic Functions Using Transformations; find a Quadratic Function from its Graph. The other columns are left blank for you to check you understanding.
    Figure 9.7.109

    b. After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?


    9.8E: Exercises is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.