Practice Problems

Answers

  1. The expression  k\cdot f(x) when | k| >1 is a vertical stretch: The y-value of every point on the graph of y=f(x) is multiplied by  k, so the points get farther away from the x-axis.

    The graph of g is a stretched version of the graph of f, so g(x)= k\cdot f(x) for some value of  k. Let's find that value.

    The graph of f passes through the point (1, 2) and the graph of g passes through the point (1, 5), so  2\cdot k= 5.

    q1-answer

    We found that  {k=\dfrac{5}2}. Now let's find the equation of
    g.

    \begin{aligned}g(x)&= {\dfrac{5}2}f(x)\\\\&= {\dfrac{5}2}\left( x^3+x^2 \right)\\\\&=\dfrac{5}2x^3+\dfrac{5}2x^2\end{aligned}

    In conclusion, g(x)=\dfrac{5}2x^3+\dfrac{5}2x^2.


  2. First, we note that g(x)= {3}f(x).

    The expression  k\cdot f(x) when | k| >1 is a vertical stretch: The y-value of every point on the graph of y=f(x) is multiplied by  k, so the points get farther away from the x-axis.

    Let's use this information to determine how the graph of g should look.

    The graph of f passes through the points (-6,0),(-3,-3), and (0,0).

    So the graph of g should pass through the following points:

    • (-6,0\cdot {3})=(-6,0)
    • (-3,-3\cdot {3})=(-3,-9)
    • (0,0\cdot {3})=(0,0)

    So the correct answer is B.

    q2-answer-b

    Notice how the graph of g looks as if we took the graph of f and stretched it from both sides of the x-axis.


  3. The expression  k\cdot f(x) when | k| is a vertical squash (or compression): The y-value of every point on the graph of y=f(x) is multiplied by  k, so the points get closer to the x-axis.

    The graph of g is a squashed version of the graph of f, so g(x)= k\cdot f(x) for some value of  k. Let's find that value and then the expression for g.

    The graph of f passes through the point (4, {16}) and the graph of g passes through the point (4, 4), so  {16}\cdot k= 4.

    q3-answer

    We found that  {k=\dfrac{1}4}.

    In conclusion, g(x)=\dfrac{1}4f(x).


  4. First, we note that g(x)= {\dfrac{1}2}f(x)

    The expression  k\cdot f(x) when | k| is a vertical squash (or compression): The y-value of every point on the graph of y=f(x) is multiplied by  k, so the points get closer to the x-axis.

    Let's use this information to determine how the graph of g should look.

    The graph of f passes through the points (1,0), (2,1), and (4,2).

    q4

    So the graph of g should pass through the following points:

    • \left(1,0\cdot {\dfrac{1}2}\right)=(1,0)
    • \left(2,1\cdot {\dfrac{1}2}\right)=\left(2,\dfrac{1}2\right)
    • \left(4,2\cdot {\dfrac{1}2}\right)=(4,1)

    So the correct answer is A.

    q4-answer-a

    Notice how the graph of g looks as if we took the graph of f and squashed it towards both sides of the x-axis.