1. $$ 4 f(3 x)-5 $$ 2. $$ -\frac{1}{5} f(x-6)+4 $$ 3. $$ y=\left(\frac{2}{3} x\right)^{2}+5 $$ 4. $$ y=-3 \sqrt{-6 x} $$ Write an equation using the square root function for the given transformations. 5. Vertical stretch 28 , shift left 19 , reflect the x-axis, shift down 22 6. A horizontal stretch of $$ 9 / 2 $$, shift up 13, vertical compression of $$ 3 / 8 $$. 7. A vertical stretch of $$ 5 / 2 $$, a horizontal compression of $$ 2 / 5 $$, and a shift down 33 8. A reflection of the $$ y $$-axis, a vertical compression of $$ 1 / 8 $$, a horizontal compression of $$ 1 / 7 $$, and a shift up 24.

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To write equations using the square root function for the given transformations, we will start with the basic square root function $$ f(x) = \sqrt{x} $$ and apply the transformations step by step.

### 5. Vertical stretch 28, shift left 19, reflect the x-axis, shift down 22

1. **Start with**: $$ f(x) = \sqrt{x} $$
2. **Reflect the x-axis**: $$ f(x) = -\sqrt{x} $$
3. **Vertical stretch by 28**: $$ f(x) = -28\sqrt{x} $$
4. **Shift left 19**: Replace $$ x $$ with $$ x + 19 $$: $$ f(x) = -28\sqrt{x + 19} $$
5. **Shift down 22**: Subtract 22: $$ f(x) = -28\sqrt{x + 19} - 22 $$

**Final equation**: 
$$
y = -28\sqrt{x + 19} - 22
$$

### 6. A horizontal stretch of $$ \frac{9}{2} $$, shift up 13, vertical compression of $$ \frac{3}{8} $$

1. **Start with**: $$ f(x) = \sqrt{x} $$
2. **Horizontal stretch by $$ \frac{9}{2} $$**: Replace $$ x $$ with $$ \frac{2}{9}x $$: $$ f(x) = \sqrt{\frac{2}{9}x} $$
3. **Vertical compression by $$ \frac{3}{8} $$**: Multiply by $$ \frac{3}{8} $$: $$ f(x) = \frac{3}{8}\sqrt{\frac{2}{9}x} $$
4. **Shift up 13**: Add 13: $$ f(x) = \frac{3}{8}\sqrt{\frac{2}{9}x} + 13 $$

**Final equation**: 
$$
y = \frac{3}{8}\sqrt{\frac{2}{9}x} + 13
$$

### 7. A vertical stretch of $$ \frac{5}{2} $$, a horizontal compression of $$ \frac{2}{5} $$, and a shift down 33

1. **Start with**: $$ f(x) = \sqrt{x} $$
2. **Horizontal compression by $$ \frac{2}{5} $$**: Replace $$ x $$ with $$ 5x $$: $$ f(x) = \sqrt{5x} $$
3. **Vertical stretch by $$ \frac{5}{2} $$**: Multiply by $$ \frac{5}{2} $$: $$ f(x) = \frac{5}{2}\sqrt{5x} $$
4. **Shift down 33**: Subtract 33: $$ f(x) = \frac{5}{2}\sqrt{5x} - 33 $$

**Final equation**: 
$$
y = \frac{5}{2}\sqrt{5x} - 33
$$

### 8. A reflection of the $$ y $$-axis, a vertical compression of $$ \frac{1}{8} $$, a horizontal compression of $$ \frac{1}{7} $$, and a shift up 24

1. **Start with**: $$ f(x) = \sqrt{x} $$
2. **Reflect the $$ y $$-axis**: Replace $$ x $$ with $$ -x $$: $$ f(x) = \sqrt{-x} $$
3. **Horizontal compression by $$ \frac{1}{7} $$**: Replace $$ x $$ with $$ 7x $$: $$ f(x) = \sqrt{-7x} $$
4. **Vertical compression by $$ \frac{1}{8} $$**: Multiply by $$ \frac{1}{8} $$: $$ f(x) = \frac{1}{8}\sqrt{-7x} $$
5. **Shift up 24**: Add 24: $$ f(x) = \frac{1}{8}\sqrt{-7x} + 24 $$

**Final equation**: 
$$
y = \frac{1}{8}\sqrt{-7x} + 24
$$

These equations represent the transformations applied to the basic square root function.
Here are the transformed square root equations: 1. $$ y = -28\sqrt{x + 19} - 22 $$ 2. $$ y = \frac{3}{8}\sqrt{\frac{2}{9}x} + 13 $$ 3. $$ y = \frac{5}{2}\sqrt{5x} - 33 $$ 4. $$ y = \frac{1}{8}\sqrt{-7x} + 24 $$

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