Question: For each of the functions below, sketch the graph. Clearly mark the starting point (or vertices) and state the Domain (D_f) and Range (R_f).
1. f(x) = \sqrt{x - 1} + 2
2. f(x) = \sqrt{x + 3} - 4
3. f(x) = -\sqrt{x - 2} + 1
4. f(x) = \sqrt{4 - x}
5. f(x) = -\sqrt{5 - x} - 2
6. f(x) = \sqrt{x + 2}
7. f(x) = -\sqrt{x + 4} - 3
8. f(x) = \sqrt{1 - x} + 5
9. f(x) = -\sqrt{x} + 6
10. f(x) = \sqrt{x^2 - 9}
1. **Problem:** Sketch the graph of $$f(x) = \sqrt{x - 1} + 2$$.
- **Domain:** $$x - 1 \geq 0 \Rightarrow x \geq 1$$.
- **Range:** Since $$\sqrt{x - 1} \geq 0$$, $$f(x) \geq 2$$, so $$[2, \infty)$$.
- **Starting point:** At $$x=1$$, $$f(1) = \sqrt{0} + 2 = 2$$.
2. **Problem:** Sketch the graph of $$f(x) = \sqrt{x + 3} - 4$$.
- **Domain:** $$x + 3 \geq 0 \Rightarrow x \geq -3$$.
- **Range:** $$f(x) \geq -4$$.
- **Starting point:** At $$x=-3$$, $$f(-3) = 0 - 4 = -4$$.
3. **Problem:** Sketch the graph of $$f(x) = -\sqrt{x - 2} + 1$$.
- **Domain:** $$x - 2 \geq 0 \Rightarrow x \geq 2$$.
- **Range:** Since $$-\sqrt{x - 2} \leq 0$$, $$f(x) \leq 1$$, so $$(-\infty, 1]$$.
- **Starting point:** At $$x=2$$, $$f(2) = -0 + 1 = 1$$.
4. **Problem:** Sketch the graph of $$f(x) = \sqrt{4 - x}$$.
- **Domain:** $$4 - x \geq 0 \Rightarrow x \leq 4$$.
- **Range:** $$f(x) \geq 0$$.
- **Starting point:** At $$x=4$$, $$f(4) = 0$$.
5. **Problem:** Sketch the graph of $$f(x) = -\sqrt{5 - x} - 2$$.
- **Domain:** $$5 - x \geq 0 \Rightarrow x \leq 5$$.
- **Range:** Since $$-\sqrt{5 - x} \leq 0$$, $$f(x) \leq -2$$.
- **Starting point:** At $$x=5$$, $$f(5) = -0 - 2 = -2$$.
6. **Problem:** Sketch the graph of $$f(x) = \sqrt{x + 2}$$.
- **Domain:** $$x + 2 \geq 0 \Rightarrow x \geq -2$$.
- **Range:** $$f(x) \geq 0$$.
- **Starting point:** At $$x=-2$$, $$f(-2) = 0$$.
7. **Problem:** Sketch the graph of $$f(x) = -\sqrt{x + 4} - 3$$.
- **Domain:** $$x + 4 \geq 0 \Rightarrow x \geq -4$$.
- **Range:** $$f(x) \leq -3$$.
- **Starting point:** At $$x=-4$$, $$f(-4) = -0 - 3 = -3$$.
8. **Problem:** Sketch the graph of $$f(x) = \sqrt{1 - x} + 5$$.
- **Domain:** $$1 - x \geq 0 \Rightarrow x \leq 1$$.
- **Range:** $$f(x) \geq 5$$.
- **Starting point:** At $$x=1$$, $$f(1) = 0 + 5 = 5$$.
9. **Problem:** Sketch the graph of $$f(x) = -\sqrt{x} + 6$$.
- **Domain:** $$x \geq 0$$.
- **Range:** $$f(x) \leq 6$$.
- **Starting point:** At $$x=0$$, $$f(0) = 6$$.
10. **Problem:** Sketch the graph of $$f(x) = \sqrt{x^2 - 9}$$.
- **Domain:** $$x^2 - 9 \geq 0 \Rightarrow |x| \geq 3$$.
- **Range:** $$f(x) \geq 0$$.
- **Vertices:** At $$x=\pm 3$$, $$f(\pm 3) = 0$$.
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**Summary of domains and ranges:**
| Function | Domain | Range | Starting Point |
|---|---|---|---|
| 1 | $$[1, \infty)$$ | $$[2, \infty)$$ | (1, 2) |
| 2 | $$[-3, \infty)$$ | $$[-4, \infty)$$ | (-3, -4) |
| 3 | $$[2, \infty)$$ | $$(-\infty, 1]$$ | (2, 1) |
| 4 | $$(-\infty, 4]$$ | $$[0, \infty)$$ | (4, 0) |
| 5 | $$(-\infty, 5]$$ | $$(-\infty, -2]$$ | (5, -2) |
| 6 | $$[-2, \infty)$$ | $$[0, \infty)$$ | (-2, 0) |
| 7 | $$[-4, \infty)$$ | $$(-\infty, -3]$$ | (-4, -3) |
| 8 | $$(-\infty, 1]$$ | $$[5, \infty)$$ | (1, 5) |
| 9 | $$[0, \infty)$$ | $$(-\infty, 6]$$ | (0, 6) |
| 10 | $$(-\infty, -3] \cup [3, \infty)$$ | $$[0, \infty)$$ | (-3, 0), (3, 0) |