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24.11: E1.10- Section 6 Part 2

  • Page ID
    51742
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    Start exploring. How does changing h change the graph?

    Start by changing h to -2.

    That makes the spreadsheet look like the illustration below.

      A B C D E F G H
    1 x y
    2 -6 36 2 a
    3 -5 22 -2 h
    4 -4 12 4 k
    5 -3 6
    6 -2 4
    7 -1 6
    8 0 12
    9 1 22
    10 2 36
    11 3 54
    12 4 76
    13 5 102
    14 6 132
    15
    16
    122.png

    Now we can notice that, when h=3, the lowest point on the graph is at x=3, and when h=-2, then the lowest point on the graph is at x=-2.

    This suggests that maybe the value that is subtracted from x in the original formula is the one that determines where the lowest y-value is – that is, where the lowest point on the graph is.

     

    Try h=0, h=4, and h=-3.

     

    h=0

    (leaving a=2 and k=4)

    h=4

    (leaving a=2 and k=4)

    h=-3

    (leaving a=2 and k=4)

     211.png 35.png  43.png

     

    Do these results support the conjecture we made in the previous sentence?   Answer: Yes.

     

    Example 21.   Using the same formula and spreadsheet as in Example 18, use the values a=1, h=0, and explore the effect of changing k.

    k=4

    (leaving a=1 and h=0)

    k=0

    (leaving a=1 and h=0)

    k=-7

    (leaving a=1 and h=0)

     53.png 36.png  44.png

    We find that changing k alone changes how far up or down the lowest point on the graph is. It appears that the y-value of that lowest point is k.

    Example 22.   Using the same formula and spreadsheet as in Example 17, use h=0 and k=0, and explore the effect of changing a.

    a=1

    (with h=0 and k=0)

    a=3

    (withh=0 and k=0)

    a=-3

    (with h=0 and k=0)

     82.png 91.png  101.png

    We find that changing a from a positive to a negative number makes the graph change from opening upward to opening downward. Making a larger (from 1 to 3) changes how large the y-values are, so that the y-values for a=3 are three times as large as those when a=1.

     

     

    CC licensed content, Shared previously
    • Mathematics for Modeling. Authored by: Mary Parker and Hunter Ellinger. License: CC BY: Attribution

    24.11: E1.10- Section 6 Part 2 is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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