Question

name: period:
practice worksheet:
graphing square root functions
graph each square root function and describe its characteristics.

1. $y = 2\sqrt{x - 2} - 4$

endpoint:
guide point using $a$:
y - intercept:
extra guide point: (6, ___)
graph
domain: range:

1. $y = -3\sqrt{x + 6}$

endpoint:
guide point using $a$:
y - intercept:
extra guide point: (3, ___)
graph
domain: range:

1. $y = \sqrt{-(x - 4)} + 4$

endpoint:
guide point using $a$:
y - intercept:
extra guide point: (-5, ___)
graph
domain: range:

1. $y = -6\sqrt{-x + 4} + 3$

endpoint:
guide point using $a$:
y - intercept:
extra guide point: (1, ___)
graph
domain: range:

1. $y = \frac{1}{2}\sqrt{-x} - 1$

endpoint:
guide point using $a$:
y - intercept:
extra guide point: (-9, ___)
graph
domain: range:

1. $y = 1.5\sqrt{x + 2} + 0.5$

endpoint:
guide point using $a$:
y - intercept:
extra guide point: (2, ___)
graph
domain: range:

Explanation:

Step1: Identify endpoint for 1)

Set radicand ≥0: $x-2\geq0 \implies x\geq2$. At $x=2$, $y=2\sqrt{2-2}-4=-4$. Endpoint: $(2, -4)$

Step2: Guide point for 1)

For parent $\sqrt{x}$, guide point $(1,1)$. Scale $y$ by 2: $(2+1, -4+2)=(3,-2)$

Step3: y-intercept for 1)

Set $x=0$: $y=2\sqrt{0-2}-4$ (undefined, no y-intercept)

Step4: Extra point for 1)

Substitute $x=6$: $y=2\sqrt{6-2}-4=2*2-4=0$

Step5: Domain/Range for 1)

Domain: $x\geq2$; Range: $y\geq-4$

---

Step6: Identify endpoint for 2)

Set radicand ≥0: $x+6\geq0 \implies x\geq-6$. At $x=-6$, $y=-3\sqrt{-6+6}=0$. Endpoint: $(-6, 0)$

Step7: Guide point for 2)

Parent guide $(1,1)$: $(-6+1, 0+(-3*1))=(-5,-3)$

Step8: y-intercept for 2)

Set $x=0$: $y=-3\sqrt{0+6}=-3\sqrt{6}\approx-7.35$

Step9: Extra point for 2)

Substitute $x=3$: $y=-3\sqrt{3+6}=-3*3=-9$

Step10: Domain/Range for 2)

Domain: $x\geq-6$; Range: $y\leq0$

---

Step11: Identify endpoint for 3)

Set radicand ≥0: $-(x-4)\geq0 \implies x\leq4$. At $x=4$, $y=\sqrt{-(4-4)}+4=4$. Endpoint: $(4, 4)$

Step12: Guide point for 3)

Parent guide $(1,1)$: $(4-1, 4+1)=(3,5)$

Step13: y-intercept for 3)

Set $x=0$: $y=\sqrt{-(0-4)}+4=\sqrt{4}+4=2+4=6$

Step14: Extra point for 3)

Substitute $x=-5$: $y=\sqrt{-(-5-4)}+4=\sqrt{9}+4=3+4=7$

Step15: Domain/Range for 3)

Domain: $x\leq4$; Range: $y\geq4$

---

Step16: Identify endpoint for 4)

Set radicand ≥0: $-x+4\geq0 \implies x\leq4$. At $x=4$, $y=-6\sqrt{-4+4}+3=3$. Endpoint: $(4, 3)$

Step17: Guide point for 4)

Parent guide $(1,1)$: $(4-1, 3+(-6*1))=(3,-3)$

Step18: y-intercept for 4)

Set $x=0$: $y=-6\sqrt{-0+4}+3=-6*2+3=-9$

Step19: Extra point for 4)

Substitute $x=1$: $y=-6\sqrt{-1+4}+3=-6\sqrt{3}+3\approx-10.39+3=-7.39$

Step20: Domain/Range for 4)

Domain: $x\leq4$; Range: $y\leq3$

---

Step21: Identify endpoint for 5)

Set radicand ≥0: $-x\geq0 \implies x\leq0$. At $x=0$, $y=\frac{1}{2}\sqrt{0}-1=-1$. Endpoint: $(0, -1)$

Step22: Guide point for 5)

Parent guide $(1,1)$: $(0-1, -1+\frac{1}{2})=(-1,-0.5)$

Step23: y-intercept for 5)

Set $x=0$: $y=-1$ (same as endpoint)

Step24: Extra point for 5)

Substitute $x=-9$: $y=\frac{1}{2}\sqrt{-(-9)}-1=\frac{1}{2}*3-1=1.5-1=0.5$

Step25: Domain/Range for 5)

Domain: $x\leq0$; Range: $y\geq-1$

---

Step26: Identify endpoint for 6)

Set radicand ≥0: $x+2\geq0 \implies x\geq-2$. At $x=-2$, $y=1.5\sqrt{-2+2}+0.5=0.5$. Endpoint: $(-2, 0.5)$

Step27: Guide point for 6)

Parent guide $(1,1)$: $(-2+1, 0.5+(1.5*1))=(-1,2)$

Step28: y-intercept for 6)

Set $x=0$: $y=1.5\sqrt{0+2}+0.5=1.5\sqrt{2}+0.5\approx2.12+0.5=2.62$

Step29: Extra point for 6)

Substitute $x=2$: $y=1.5\sqrt{2+2}+0.5=1.5*2+0.5=3+0.5=3.5$

Step30: Domain/Range for 6)

Domain: $x\geq-2$; Range: $y\geq0.5$

Answer:

1) $y=2\sqrt{x-2}-4$

Endpoint: $\boldsymbol{(2, -4)}$
Guide point: $\boldsymbol{(3, -2)}$
y-intercept: $\boldsymbol{\text{None}}$
Extra guide point: $\boldsymbol{0}$
Domain: $\boldsymbol{x\geq2}$
Range: $\boldsymbol{y\geq-4}$

2) $y=-3\sqrt{x+6}$

Endpoint: $\boldsymbol{(-6, 0)}$
Guide point: $\boldsymbol{(-5, -3)}$
y-intercept: $\boldsymbol{-3\sqrt{6}\approx-7.35}$
Extra guide point: $\boldsymbol{-9}$
Domain: $\boldsymbol{x\geq-6}$
Range: $\boldsymbol{y\leq0}$

3) $y=\sqrt{-(x-4)}+4$

Endpoint: $\boldsymbol{(4, 4)}$
Guide point: $\boldsymbol{(3, 5)}$
y-intercept: $\boldsymbol{6}$
Extra guide point: $\boldsymbol{7}$
Domain: $\boldsymbol{x\leq4}$
Range: $\boldsymbol{y\geq4}$

4) $y=-6\sqrt{-x+4}+3$

Endpoint: $\boldsymbol{(4, 3)}$
Guide point: $\boldsymbol{(3, -3)}$
y-intercept: $\boldsymbol{-9}$
Extra guide point: $\boldsymbol{-6\sqrt{3}+3\approx-7.39}$
Domain: $\boldsymbol{x\leq4}$
Range: $\boldsymbol{y\leq3}$

5) $y=\frac{1}{2}\sqrt{-x}-1$

Endpoint: $\boldsymbol{(0, -1)}$
Guide point: $\boldsymbol{(-1, -0.5)}$
y-intercept: $\boldsymbol{-1}$
Extra guide point: $\boldsymbol{0.5}$
Domain: $\boldsymbol{x\leq0}$
Range: $\boldsymbol{y\geq-1}$

6) $y=1.5\sqrt{x+2}+0.5$

Endpoint: $\boldsymbol{(-2, 0.5)}$
Guide point: $\boldsymbol{(-1, 2)}$
y-intercept: $\boldsymbol{1.5\sqrt{2}+0.5\approx2.62}$
Extra guide point: $\boldsymbol{3.5}$
Domain: $\boldsymbol{x\geq-2}$
Range: $\boldsymbol{y\geq0.5}$