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