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Mathematics LibreTexts

1.3: Shifting and Reflecting

  • Page ID
    227
  • [ "article:topic", "reflection", "authorname:green", "horizontal shift", "Vertical Shift", "stretching", "compressing" ]

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    1. Six Basic Functions

    Below are six basic functions:

    1. SHIFT.HTM_txt_absx.gif
    2.  







    3. Memorize the shapes of these functions.

    2. Horizontal Shifting

     Consider the graphs

    \(y =\)

    • \((x+0)^2\)
    • \((x+1)^2\)
    • \((x+2)^2\)
    • \((x+3)^2\)

    Exercise

    Use the list features of a calculator to sketch the graph of 

    \(y = \dfrac{1}{ [x - \{0,1,2,3\}] }\)

    Horizontal Shifting Rules

    • Rule 1: \(f(x - a) = f(x)\) shifted \(a\) units to the right.
    • Rule 2: \(f(x + a) = f(x)\) shifted \(a\) units to the left.

    3. Vertical Shifting

    Consider the graphs

    \(y =\)

    • \(x^3\)
    • \(x^3+ 1\)
    • \(x^3 + 2\)
    • \(x^3 + 3\)

    Exercise

    Use the list features of a calculator to sketch the graph of 

    \(y = x^3 - \{0,1,2,3\}\)

    Vertical Shifting Rules

    • Rule 3: \(f(x ) + a = f(x)\) shifted a units up.
    • Rule 4: \(f(x) - a = f(x)\) shifted a units down.

    4. Reflecting About the x-axis

    Consider the graphs of

     \(y = x^2\) and \(y = -x^2\).

    SHIFT.HTM_txt_xReflect.gif

    x-Axis Reflection Rule

    Rule 5: \(-f(x) = f(x)\) reflected about the x-axis.

    5. Reflecting About the y-axis

    Exercise

    Use the calculator to graph

    \(y=\sqrt{x}\)

    and
    \(y=\sqrt{-x}\)

    y-Axis Reflection Rule

    Rule 6: \(f(-x ) = f(x)\) reflected about the y-axis.

    6. Stretching and Compressing

    Exercise

    Graph the following:

    \(y = \{1,2,3,4\}x^3\)

    \(y = {1/2,1/3,1/4,1/5}x^3\)

    Stretching and Compression rules:

    • Rule 7: \(cf(x ) = f(x)\) (for \(c > 1\)) stretched vertically.
    • Rule 8: \(cf(x ) = f(x)\) (for \(c < 1\)) compressed vertically.

    Exercise

    Graph the following

    1. \(y = x^2 - 10\)
    2. \(y = \sqrt{x - 2}\)
    3. \(y = -|x - 5| + 3\)

    We will do some examples (including the graph of the winnings for the gambler and for the casino). 

    7. Increasing and Decreasing Functions

    Definition

    A function is called increasing if as an object moves from left to right, it is moving upwards along the graph. Or equivalently,

    If \(x < y\), then \(f(x) < f(y)\).

    Example 1

    The curve

    \[y = x^2\] 

    is increasing on \((0,\infty)\) and decreasing on \((-\infty,0)\).

    Contributors