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Mathematics LibreTexts

11.3E: Exercises

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    Practice Makes Perfect

    Exercise \(\PageIndex{17}\) Graph Vertical Parabolas

    In the following exercises, graph each equation by using properties.

    1. \(y=-x^{2}+4 x-3\)
    2. \(y=-x^{2}+8 x-15\)
    3. \(y=6 x^{2}+2 x-1\)
    4. \(y=8 x^{2}-10 x+3\)
    Answer

    1.

    This graph shows a parabola opening downward with vertex (2, 1) and x intercepts (1, 0) and (3, 0).
    Figure 11.2.83

    3.

    This graph shows a parabola opening upward. The vertex is (negative 0.167, negative 1.167), the x intercepts are (negative 0.608) and (negative 0.274, 0), and the y-intercept is (0, negative 1).
    Figure 11.2.84
    Exercise \(\PageIndex{18}\) Graph Vertical Parabolas

    In the following exercises,

    1. Write the equation in standard form and
    2. Use properties of the standard form to graph the equation.
    1. \(y=-x^{2}+2 x-4\)
    2. \(y=2 x^{2}+4 x+6\)
    3. \(y=-2 x^{2}-4 x-5\)
    4. \(y=3 x^{2}-12 x+7\)
    Answer

    1.

    1. \(y=-(x-1)^{2}-3\)
    This graph shows a parabola opening downward with vertex (1, negative 3) and y intercept (0, 4).
    Figure 11.2.85

    3.

    1. \(y=-2(x+1)^{2}-3\)
    This graph shows a parabola opening downward with vertex (negative 1, negative 3) and x intercepts (negative 5, 0).
    Figure 11.2.86
    Exercise \(\PageIndex{19}\) Graph Horizontal Parabolas

    In the following exercises, graph each equation by using properties.

    1. \(x=-2 y^{2}\)
    2. \(x=3 y^{2}\)
    3. \(x=4 y^{2}\)
    4. \(x=-4 y^{2}\)
    5. \(x=-y^{2}-2 y+3\)
    6. \(x=-y^{2}-4 y+5\)
    7. \(x=y^{2}+6 y+8\)
    8. \(x=y^{2}-4 y-12\)
    9. \(x=(y-2)^{2}+3\)
    10. \(x=(y-1)^{2}+4\)
    11. \(x=-(y-1)^{2}+2\)
    12. \(x=-(y-4)^{2}+3\)
    13. \(x=(y+2)^{2}+1\)
    14. \(x=(y+1)^{2}+2\)
    15. \(x=-(y+3)^{2}+2\)
    16. \(x=-(y+4)^{2}+3\)
    17. \(x=-3(y-2)^{2}+3\)
    18. \(x=-2(y-1)^{2}+2\)
    19. \(x=4(y+1)^{2}-4\)
    20. \(x=2(y+4)^{2}-2\)
    Answer

    1.

    This graph shows a parabola opening to the left with vertex (0, 0). Two points on it are (negative 2, 1) and (negative 2, negative 1).
    Figure 11.2.87

    3.

    This graph shows a parabola opening to the right with vertex (0, 0). Two points on it are (4, 1) and (4, negative 1).
    Figure 11.2.88

    5.

    This graph shows a parabola opening to the left with vertex (4, negative 1) and y intercepts (0, 1) and (0, negative 3).
    Figure 11.2.89

    7.

    This graph shows a parabola opening to the right with vertex (negative 1, negative 3) and y intercepts (0, negative 2) and (0, negative 4).
    Figure 11.2.90

    9.

    This graph shows a parabola opening to the right with vertex (3, 2) and x intercept (7, 0).
    Figure 11.2.91

    11.

    This graph shows a parabola opening to the left with vertex (2, 1) and x intercept (1, 0).
    Figure 11.2.92

    13.

    This graph shows a parabola opening to the right with vertex (1, negative 2) and x intercept (5, 0).
    Figure 11.2.93

    15.

    This graph shows a parabola opening to the left with vertex (2, negative 3). Two points on it are (negative 2, negative 1) and (negative 2, 5).
    Figure 11.2.94

    17.

    This graph shows a parabola opening to the left with vertex (3, 2) and y intercepts (0, 1) and (0, 3).
    Figure 11.2.95

    19.

    This graph shows a parabola opening to the right with vertex (negative 4, negative 1) and y intercepts (0, 0) and (0, negative 2).
    Figure 11.2.96
    Exercise \(\PageIndex{20}\) Graph Horizontal Parabolas

    In the following exercises,

    1. Write the equation in standard form and
    2. Use properties of the standard form to graph the equation.
    1. \(x=y^{2}+4 y-5\)
    2. \(x=y^{2}+2 y-3\)
    3. \(x=-2 y^{2}-12 y-16\)
    4. \(x=-3 y^{2}-6 y-5\)
    Answer

    1.

    1. \(x=(y+2)^{2}-9\)
    This graph shows a parabola opening to the right with vertex (negative 9, negative 2) and y intercepts (0, 1) and (0, negative 5).
    Figure 11.2.97

    3.

    1. \(x=-2(y+3)^{2}+2\)
    This graph shows a parabola opening to the left with vertex (2, negative 3) and y intercepts (0, negative 2) and (0, negative 4).
    Figure 11.2.98
    Exercise \(\PageIndex{21}\) Mixed Practice

    In the following exercises, match each graph to one of the following equations:

    1. \(x^{2}+y^{2}=64\)
    2. \(x^{2}+y^{2}=49\)
    3. \((x+5)^{2}+(y+2)^{2}=4\)
    4. \((x-2)^{2}+(y-3)^{2}=9\)
    5. \(y=-x^{2}+8 x-15\)
    6. \(y=6 x^{2}+2 x-1\)

    1.

    This graph shows circle with center (0, 0) and radius 8 units.
    Figure 11.2.99

    2.

    This graph shows a parabola opening upwards. Its vertex has an x value of slightly less than 0 and a y value of slightly less than negative 1. A point on it is close to (negative 1, 3).
    Figure 11.2.100

    3.

    This graph shows circle with center (0, 0) and radius 7 units.
    Figure 11.2.101

    4.

    This graph shows a parabola opening downwards with vertex (4, 1) and x intercepts (3, 0) and (5, 0).
    Figure 11.2.102

    5.

    This graph shows circle with center (2, 3) and radius 3 units.
    Figure 11.2.103

    6.

    This graph shows circle with center (negative 5, negative 2) and radius 2 units.
    Figure 11.2.104
    Answer

    1. a

    3. b

    5. d

    Exercise \(\PageIndex{22}\) Solve Applications with Parabolas

    Write the equation in standard form of the parabolic arch formed in the foundation of the bridge shown. Use the lower left side of the bridge as the origin \((0, 0)\).

    1.

    This graph shows circle with center (negative 5, negative 2) and radius 2 units.
    Figure 11.2.105

    2.

    This figure shows a parabolic arch formed in the foundation of a bridge. It is 50 feet high and 100 feet wide at the base.
    Figure 11.2.106

    3.

    This figure shows a parabolic arch formed in the foundation of a bridge. It is 90 feet high and 60 feet wide at the base.
    Figure 11.2.107

    4.

    This figure shows a parabolic arch formed in the foundation of a bridge. It is 45 feet high and 30 feet wide at the base.
    Figure 11.2.108
    Answer

    1. \(y=-\frac{1}{15}(x-15)^{2}+15\)

    3. \(y=-\frac{1}{10}(x-30)^{2}+90\)

    Exercise \(\PageIndex{23}\) Writing Exercises
    1. In your own words, define a parabola.
    2. Is the parabola \(y=x^{2}\) a function? Is the parabola \(x=y^{2}\) a function? Explain why or why not.
    3. Write the equation of a parabola that opens up or down in standard form and the equation of a parabola that opens left or right in standard form. Provide a sketch of the parabola for each one, label the vertex and axis of symmetry.
    4. Explain in your own words, how you can tell from its equation whether a parabola opens up, down, left or right.
    Answer

    1. Answers may vary

    3. Answers may vary

    Self Check

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

    This table has four columns, 3 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don’t get it. The first column has the following statements: graph vertical parabolas, graph horizontal parabolas, solve applications with parabolas. The remaining columns are blank.
    Figure 11.2.109

    b. After reviewing this checklist, what will you do to become confident for all objectives?


    11.3E: 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.

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