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5.3: Exercises

  • Page ID
    48977
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    Exercise \(\PageIndex{1}\)

    Find a possible formula of the graph displayed below.

    1. clipboard_e8b4a7bb9bf98f2eca140c515f3c07a73.png
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    Answer
    1. \(y=|x|+1\)
    2. \(y=-\sqrt{x}\)
    3. \(y=(x-2)^{2}+1\)
    4. \(y=\dfrac{1}{x+2}+2\)
    5. \(y=-(x+2)^{3}\)
    6. \(y=-|x-3|+2\)

    Exercise \(\PageIndex{2}\)

    Sketch the graph of the function, based on the basic graphs in Section 5.1 and the transformations described above. Confirm your answer by graphing the function with the calculator.

    1. \(f(x)=|x|-3\)
    2. \(f(x)=\dfrac 1 {x+2}\)
    3. \(f(x)=-x^2\)
    4. \(f(x)=(x-1)^3\)
    5. \(f(x)=\sqrt{-x}\)
    6. \(f(x)=4\cdot |x-3|\)
    7. \(f(x)=-\sqrt{x}+1\)
    8. \(f(x)=(\dfrac{1}{2}\cdot x)^2+3\)
    Answer
    1. clipboard_ee265927e6582495145348c203441d81b.png
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    Exercise \(\PageIndex{3}\)

    Consider the graph of \(f(x)=x^2-7x+1\). Find the formula of the function that is given by performing the following transformations on the graph.

    1. Shift the graph of \(f\) down by \(4\).
    2. Shift the graph of \(f\) to the left by \(3\) units.
    3. Reflect the graph of \(f\) about the \(x\)-axis.
    4. Reflect the graph of \(f\) about the \(y\)-axis.
    5. Stretch the graph of \(f\) away from the \(y\)-axis by a factor \(3\).
    6. Compress the graph of \(f\) towards the \(y\)-axis by a factor \(2\).
    Answer
    1. \(y=x^{2}-7 x-3\)
    2. \(y=(x+3)^{2}-7 \cdot(x+3)+1=x^{2}-x-11\)
    3. \(y=-x^{2}+7 x-1\)
    4. \(y=x^{2}+7 x+1\)
    5. \(y=\dfrac{1}{9} x^{2}-\dfrac{7}{3} x+1\)
    6. \(y=4 x^{2}-14 x+1\)

    Exercise \(\PageIndex{4}\)

    How are the graphs of \(f\) and \(g\) related?

    1. \(f(x)=\sqrt{x}, \quad g(x)=\sqrt{x-5}\)
    2. \(f(x)=|x|, \quad g(x)=2 \cdot|x|\)
    3. \(f(x)=(x+1)^3 , \quad g(x)=(x-3)^3\)
    4. \(f(x)=x^2+3x+5, \quad g(x)=(2x)^2+3(2x)^2+5\)
    5. \(f(x)=\dfrac 1 {x+3}, \quad g(x)=-\dfrac 1 x\)
    6. \(f(x)= 2 \cdot |x|, \quad g(x)=|x+1|+1\)
    Answer
    1. shift to the right by \(5\)
    2. stretched away from the \(x\)-axis by a factor \(2\)
    3. shift to the right by \(4\)
    4. compressed towards the \(y\)-axis by a factor \(2\)
    5. shifted to the right by \(3\) (to get the graph of \(y=\dfrac{1}{x}\) ) and then reflected about the \(x\)-axis
    6. compressed towards the \(x\)-axis by a factor \(2\) (you get \(y=|x|\)) then shifted to the left by \(1\) and up by \(1\)

    Exercise \(\PageIndex{5}\)

    Determine, if the function is even, odd, or neither.

    1. \(y=2x^3\)
    2. \(y=5x^2\)
    3. \(y=3x^4-4x^2+5\)
    4. \(y=2x^3+5x^2\)
    5. \(y=|x|\)
    6. \(y=\dfrac{1}{x}\)
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    Answer
    1. odd
    2. even
    3. even
    4. neither
    5. even
    6. odd
    7. even
    8. neither
    9. odd

    Exercise \(\PageIndex{6}\)

    The graph of the function \(y=f(x)\) is displayed below.

    clipboard_e524d31e279dc102e65d09af47dad4860.png

    Sketch the graph of the following functions.

    1. \(y=f(x)+1\)
    2. \(y=f(x-3)\)
    3. \(y=-f(x)\)
    4. \(y=2f(x)\)
    5. \(y=f(2x)\)
    6. \(y=f\left (\dfrac 1 2 x\right )\)
    Answer
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    This page titled 5.3: Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Thomas Tradler and Holly Carley (New York City College of Technology at CUNY Academic Works) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.