Skip to main content
Mathematics LibreTexts

4.6E: Exercises

  1. Last updated
  2. Save as PDF
  • Page ID
    108361

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    Practice Makes Perfect

    Exercises 1 - 4: Recognize the Graph of a Quadratic Function

    In the following exercises, graph the functions by plotting points.

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

    1.

    clipboard_eb78a0f78325e7c8a9cceea709788ca1d.png

    2.

    clipboard_ef318ed788d73edacb2b69f9a778e9ce9.png

    Exercises 5 - 8: Recognize the Graph of a Quadratic Function

    For each of the following exercises, determine if the parabola opens up or down.

    3. a. \(f(x)=-2 x^{2}-6 x-7\) b. \(f(x)=6 x^{2}+2 x+3\)

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

    Answer

    3. a. down b. up

    4. a. down b. up

    Exercises 9 - 12: Find the Axis of Symmetry and Vertex of a Parabola

    In the following functions, find

    1. The equation of the axis of symmetry
    2. The vertex of its graph
    1. \(f(x)=x^{2}+8 x-1\)
    2. \(f(x)=x^{2}+10 x+25\)
    3. \(f(x)=-x^{2}+2 x+5\)
    4. \(f(x)=-2 x^{2}-8 x-3\)
    Answer
    1. a. Axis of symmetry: \(x=-4\) b. Vertex: \((-4,-17)\)
    2. a. Axis of symmetry: \(x=-5\) b. Vertex: \((-5,0)\)
    3. a. Axis of symmetry: \(x=1\) b. Vertex: \((1,6)\)
    4. a. Axis of symmetry: \(x=-2\) b. Vertex: \((-2,5)\)
    Find the Intercepts of a Parabola

    In the following exercises, find the intercepts of the parabola whose function is given.

    1. \(f(x)=x^{2}+7 x+6\)
    2. \(f(x)=x^{2}+8 x+12\)
    3. \(f(x)=-x^{2}+8 x-19\)
    4. \(f(x)=x^{2}+6 x+13\)
    5. \(f(x)=4 x^{2}-20 x+25\)
    6. \(f(x)=-x^{2}-6 x-9\)
    Answer
    1. \(y\)-intercept: \((0,6)\); \(x\)-intercept(s): \((-1,0), (-6,0)\)
    2. \(y\)-intercept: \((0,12)\); \(x\)-intercept(s): \((-2,0), (-6,0)\)
    3. \(y\)-intercept: \((0,-19)\); \(x\)-intercept(s): none
    4. \(y\)-intercept: \((0,13)\); \(x\)-intercept(s): none
    5. \(y\)-intercept: \((0,25)\); \(x\)-intercept(s): \((\frac{5}{2},0)\)
    6. \(y\)-intercept: \((0,-9)\); \(x\)-intercept(s): \((-3,0)\)
    Graph Quadratic Functions Using Properties

    In the following exercises, graph the function by using its properties.  That is, find its \(x\)-intercepts, \(y\)-intercept, vertex, and determine its end behavior.  Then use this information to graph the function.

    1. \(f(x)=x^{2}+6 x+5\)
    2. \(f(x)=x^{2}+4 x+3\)
    3. \(f(x)=9 x^{2}+12 x+4\)
    4. \(f(x)=-x^{2}+2 x-7\)
    5. \(f(x)=2 x^{2}-4 x+1\)
    6. \(f(x)=2 x^{2}-4 x+2\)
    7. \(f(x)=-x^{2}-4 x+2\)
    8. \(f(x)=5 x^{2}-10 x+8\)
    9. \(f(x)=3 x^{2}+18 x+20\)
    Answer

    15.

    This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 3, negative 4). The y-intercept, point (0, 5), is plotted as are the x-intercepts, (negative 5, 0) and (negative 1, 0).
    Figure 9.6.136

    16.

    This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 2, negative 1). The y-intercept, point (0, 3), is plotted as are the x-intercepts, (negative 3, 0) and (negative 1, 0).
    Figure 9.6.137

    17.

    This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis of the plane runs from negative 4 to 4. The parabola has a vertex at (negative 2 thirds, 0). The y-intercept, point (0, 4), is plotted. The axis of symmetry, x equals negative 2 thirds, is plotted as a dashed vertical line.
    Figure 9.6.138

    18.

    This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 15 to 10. The parabola has a vertex at (1, negative 6). The y-intercept, point (0, negative 7), is plotted. The axis of symmetry, x equals 1, is plotted as a dashed vertical line.
    Figure 9.6.139

    19.

    This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (1, negative 1). The y-intercept, point (0, 1), is plotted as are the x-intercepts, approximately (0.3, 0) and (1.7, 0). The axis of symmetry is the vertical line x equals 1, plotted as a dashed line.
    Figure 9.6.140

    20.

    This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (1, 0). This point is the only x-intercept. The y-intercept, point (0, 2), is plotted. The axis of symmetry is the vertical line x equals 1, plotted as a dashed line.
    Figure 9.6.141

    21.

    This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 2, 6). The y-intercept, point (0, 2), is plotted as are the x-intercepts, approximately (negative 4.4, 0) and (0.4, 0). The axis of symmetry is the vertical line x equals 2, plotted as a dashed line.
    Figure 9.6.142

    22.

    This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (1, 3). The y-intercept, point (0, 8), is plotted; there are no x-intercepts. The axis of symmetry is the vertical line x equals 1, plotted as a dashed line.
    Figure 9.6.143

    23.

    This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 3, negative 7). The x-intercepts are plotted at the approximate points (negative 4.5, 0) and (negative 1.5, 0). The axis of symmetry is the vertical line x equals negative 3, plotted as a dashed line.
    Figure 9.6.144

    This page titled 4.6E: Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Stanislav A. Trunov and Elizabeth J. Hale via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.