# 4.3E: Exercises - Understanding Transformations of Functions

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Exercise $$\PageIndex{1}$$

Match the graph to the function definition.

1. $$f(x) = \sqrt{x + 4}$$
2. $$f(x) = |x − 2| − 2$$
3. $$f(x) = \sqrt{x + 1} -1$$
4. $$f(x) = |x − 2| + 1$$
5. $$f(x) = \sqrt{x + 4} + 1$$
6. $$f(x) = |x + 2| − 2$$

1. e

3. d

5. f

Exercise $$\PageIndex{2}$$

Graph the given function. Identify the basic function and translations used to sketch the graph. Then state the domain and range.

1. $$g(x) = −4$$
2. $$g(x) = 2$$
3. $$f(x) = x + 3$$
4. $$f(x) = x − 2$$
5. $$g(x) = x^{2} + 1$$
6. $$g(x) = x^{2} − 4$$
7. $$g(x) = (x − 5)^{2} + 2$$
8. $$g(x) = (x + 2)^{2} − 5$$
9. $$h(x) = |x − 1| − 3$$
10. $$h(x) = |x + 2| − 5$$
11. $$g(x) = \sqrt{x − 2} + 1$$
12. $$g(x) = \sqrt{x + 2} + 3$$
13. $$h(x) = (x − 1)^{3} − 4$$
14. $$h(x) = (x + 1)^{3} + 3$$
15. $$f(x) = \frac{1}{x+1} − 2$$
16. $$f(x) = \frac{1}{x−3} + 3$$
17. $$f ( x ) = \sqrt [ 3 ] { x - 2 } + 6$$
18. $$f ( x ) = \sqrt [ 3 ] { x + 8 } - 4$$

1. Basic graph $$y = −4$$; domain: $$(- \infty , \infty)$$; range: $$\{−4\}$$

3. $$y = x$$; Shift up $$3$$ units; domain: $$(- \infty , \infty)$$; range: $$(- \infty , \infty)$$

5. $$y = x^{2}$$; Shift up $$1$$ unit; domain: $$(- \infty , \infty)$$; range: $$[1, ∞)$$

7. $$y = x^{2}$$; Shift right $$5$$ units and up $$2$$ units; domain: $$(- \infty , \infty)$$; range: $$[2, ∞)$$

9. $$y = |x|$$; Shift right $$1$$ unit and down $$3$$ units; domain: $$(- \infty , \infty)$$; range: $$[−3, ∞)$$

11. $$y = \sqrt{x}$$; Shift right $$2$$ units and up $$1$$ unit; domain: $$[2, ∞)$$; range: $$[1, ∞)$$

13. $$y = x^{3}$$; Shift right $$1$$ unit and down $$4$$ units; domain: $$(- \infty , \infty)$$; range: $$(- \infty , \infty)$$

15. $$y = \frac{1}{x}$$; Shift left $$1$$ unit and down $$2$$ units; domain: $$(−∞, −1) ∪ (−1, ∞)$$; range: $$(−∞, −2) ∪ (−2, ∞)$$

17. $$y = \sqrt [ 3 ] { x }$$; Shift up $$6$$ units and right $$2$$ units; domain: $$(- \infty , \infty)$$; range: $$(- \infty , \infty)$$

Exercise $$\PageIndex{3}$$

Graph the piecewise functions.

1. $$h ( x ) = \left\{ \begin{array} { l l } { x ^ { 2 } + 2 } & { \text { if } x < 0 } \\ { x + 2 } & { \text { if } x \geq 0 } \end{array} \right.$$
2. $$h ( x ) = \left\{ \begin{array} { l l } { x ^ { 2 } - 3 \text { if } x < 0 } \\ { \sqrt { x } - 3 \text { if } x \geq 0 } \end{array} \right.$$
3. $$h ( x ) = \left\{ \begin{array} { l l } { x ^ { 3 } - 1 } & { \text { if } x < 0 } \\ { | x - 3 | - 4 } & { \text { if } x \geq 0 } \end{array} \right.$$
4. $$h ( x ) = \left\{ \begin{array} { c c } { x ^ { 3 } } & { \text { if } x < 0 } \\ { ( x - 1 ) ^ { 2 } - 1 } & { \text { if } x \geq 0 } \end{array} \right.$$

1.

3.

Exercise $$\PageIndex{4}$$

Write an equation that represents the function whose graph is given.

1.

2.

3.

4.

5.

6.

7.

8.

1. $$f ( x ) = \sqrt { x - 5 }$$

3. $$f ( x ) = ( x - 15 ) ^ { 2 } - 10$$

5. $$f ( x ) = \frac { 1 } { x + 8 } + 4$$

7. $$f ( x ) = \sqrt { x + 16 } - 4$$

Exercise $$\PageIndex{5}$$

Match the graph to the given function definition.

1. $$f ( x ) = - 3 | x |$$
2. $$f ( x ) = - ( x + 3 ) ^ { 2 } - 1$$
3. $$f ( x ) = - | x + 1 | + 2$$
4. $$f ( x ) = - x ^ { 2 } + 1$$
5. $$f ( x ) = - \frac { 1 } { 3 } | x |$$
6. $$f ( x ) = - ( x - 2 ) ^ { 2 } + 2$$

1. b

3. d

5. f

Exercise $$\PageIndex{6}$$

Use the transformations to graph the following functions.

1. $$f ( x ) = - x + 5$$
2. $$f ( x ) = - | x | - 3$$
3. $$g ( x ) = - | x - 1 |$$
4. $$f ( x ) = - ( x + 2 ) ^ { 2 }$$
5. $$h ( x ) = \sqrt { - x } + 2$$
6. $$h ( x ) = - \sqrt { x - 2 } + 1$$
7. $$g ( x ) = - x ^ { 3 } + 4$$
8. $$f ( x ) = - x ^ { 2 } + 6$$