Exercises - Graphs and Transformations

  1. Graph the function $f(x)=3x^2 + 7x - 20$ and find the following: (a) the vertex; (b) the domain of $f$; (c) the range of $f$; (d) the axis of symmetry; (e) the $y$-intercept; and (f) the $x$-intercept(s), if they exist.   $\tooltip{\ians{}}{
    \begin{array}{l}
    \textrm{vertex: } (-7/6,-289/12); \textrm{ domain: } (-\infty,\infty); \textrm{ range: } [-289/12,\infty); \\
    \textrm{axis of symmetry: } x=-7/6;\textrm{ $y$-intercept: } (0,-20); \textrm{ $x$-intercepts: } (-4,0), (5/3,0)
    \end{array}}$

  2. Use $f(x) = |x|$ and transformations to sketch $f(x)=-2|x+3|-1$. Describe the transformation in words, and label three points.   $\ians{}$

  3. Suppose $y = 1 - \sqrt{x+4}$.   Find the following:

    1. the domain   $\ans{\displaystyle{[-4,\infty)}}$

    2. the range   $\ans{\displaystyle{(-\infty,1]}}$

    3. the $x$ and $y$ intercepts   $\ans{\displaystyle{(-3,0), (0,-1)}}$

    4. the graph of this function $\ians{}$

  4. Suppose $y = \left| x-2 \right| - 3$.   Find the following:

    1. the domain   $\ans{\displaystyle{\textrm{all reals, } \mathbb{R}}}$

    2. the range   $\ans{\displaystyle{[-3,\infty)}}$

    3. the $x$ and $y$ intercepts   $\ans{\displaystyle{(0,-1), (5,0), (-1,0)}}$

    4. the graph of this function $\ians{}$

  5. The graph of $y=g(x)$ is given below. Describe the transformations necessary to turn the graph of $g(x)$ into the graph of $y=f(x)$ for each such function given. Make sure to indicate the order in which these transformations should be applied. Finally, graph $y=f(x)$.

    1. $\displaystyle{f(x) = 2g(x-1)+1}$   $\ians{}$

    2. $\displaystyle{f(x) = \frac{1}{4}g(1-x)-2}$   $\ians{}$

    3. $\displaystyle{f(x) = -3g(x/2)}$   $\ians{}$

    4. $\displaystyle{f(x) = 2g(-3x)+1}$   $\ians{}$

    5. $\displaystyle{f(x) = g(-2(x+1))}$   $\ians{}$

  6. Sketch the graphs of the following functions. Label all important aspects. Also, give the domain and range.

    1. $\displaystyle{f(x)=x^2+2x+1}$   $\tooltip{\ians{}}{\textrm{domain: } (-\infty,\infty); \textrm{ range: } [0,\infty)}$

    2. $\displaystyle{f(x)=-|1-x|}$   $\tooltip{\ians{}}{\textrm{domain: } (-\infty,\infty); \textrm{ range: }(-\infty,0] }$

    3. $\displaystyle{f(x) =
      \begin{cases}
      -1, & x \lt 0\\
      0, & x = 0\\
      1, & x \gt 0
      \end{cases}}$   $\tooltip{\ians{}}{\textrm{domain: } (-\infty,\infty); \textrm{ range: }\{-1,0,1\} }$

    4. $\displaystyle{H(x)=5-\sqrt{1-x^2}}$   $\tooltip{\ians{}}{\textrm{domain: } [-1,1]; \textrm{ range: }[4,5] }$

    5. $\displaystyle{F(x)=-2+\sqrt{16-x^2}}$   $\tooltip{\ians{}}{\textrm{domain: } [-4,4]; \textrm{ range: }[-2,2] }$

    6. $\displaystyle{y=3-12x^2}$   $\tooltip{\ians{}}{\textrm{domain: } (-\infty,\infty); \textrm{ range: }(-\infty,3] }$

    7. $\displaystyle{f(x)=
      \begin{cases}
      3, & x \le -2\\
      x^2-1, & -2 \lt x \le 3\\
      -\sqrt{25-x^2}, & x \gt 3
      \end{cases}}$   $\tooltip{\ians{}}{\textrm{domain: } (-\infty,5]; \textrm{ range: } (-4,8]}$

    8. $\displaystyle{G(t)=
      \begin{cases}
      1-t, & t \lt 1\\
      t^2-2t+1, &1 \le t \le 3\\
      -2, & t \gt 3
      \end{cases}}$   $\tooltip{\ians{}}{\textrm{domain: } (-\infty,\infty); \textrm{ range: } \{-2\} \cup [0,\infty)}$

    9. $\displaystyle{y=\sqrt{x-2}+1}$   $\tooltip{\ians{}}{\textrm{domain: } [2,\infty); \textrm{ range: }[1,\infty) }$

    10. $\displaystyle{f(x)=\sqrt{x+2}-4}$   $\tooltip{\ians{}}{\textrm{domain: } [-2,\infty); \textrm{ range: } [-4,\infty)}$

    11. $\displaystyle{y=4-\sqrt{4-x^2}}$   $\tooltip{\ians{}}{\textrm{domain: } [-2,2]; \textrm{ range: } [2,4]}$

    12. $\displaystyle{H(x)=5-\sqrt{3-x}}$   $\tooltip{\ians{}}{\textrm{domain: } (-\infty,3]; \textrm{ range: } (-\infty,5]}$

    13. $\displaystyle{f(x)=
      \begin{cases}
      -4x, & x \le -1\\
      \sqrt{1-x^2}, & -1 \lt x \lt 1\\
      -2, & x \ge 1
      \end{cases}}$   $\tooltip{\ians{}}{\textrm{domain: } (-\infty,\infty); \textrm{ range: } \{-2\} \cup (0,1] \cup [4,\infty)}$

    14. $\displaystyle{y=\frac{x^2-9}{x-3}}$   $\tooltip{\ians{}}{\textrm{domain: } (-\infty,3) \cup (3,\infty); \textrm{ range: } (-\infty,6) \cup (6,\infty)}$

    15. $\displaystyle{y=|x-3|}$   $\tooltip{\ians{}}{\textrm{domain: } (-\infty,\infty); \textrm{ range: } [0,\infty)}$

    16. $\displaystyle{y=
      \begin{cases}
      4-x^2, & x \lt -1\\
      -2, & x = -1\\
      \sqrt{4-x^2}, & x \gt -1
      \end{cases}}$   $\tooltip{\ians{}}{\textrm{domain: } (-\infty,2]; \textrm{ range: } (-\infty,3]}$