Write the transformations that occur from the parent function. $$ \begin{array}{l} ext { 1. } 4 f(3 x)-5\end{array} $$ 3. $$ y=\left(\frac{2}{3} x ight)^{2}+5 $$ 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 4. $$ y=-3 \sqrt{-6 x} $$ 6. A horizontal stretch of $$ 9 / 2 $$, shift up 13 , vertical compression of $$ 3 / 8 $$. 8. A reflection of the $$ y $$-axis, a vertical compression of $$ 1 / 8 $$, a horizontal compression of $$ 1 / 7 $$, and a shift up 24 . 7. A vertical stretch of $$ 5 / 2 $$, a horizontal compression of $$ 2 / 5 $$, and a shift down 33

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Write the transformations that occur from the parent function. $$ \begin{array}{l}	ext { 1. } 4 f(3 x)-5\end{array} $$ 3. $$ y=\left(\frac{2}{3} xight)^{2}+5 $$ 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 4. $$ y=-3 \sqrt{-6 x} $$ 6. A horizontal stretch of $$ 9 / 2 $$, shift up 13 , vertical compression of $$ 3 / 8 $$. 8. A reflection of the $$ y $$-axis, a vertical compression of $$ 1 / 8 $$, a horizontal compression of $$ 1 / 7 $$, and a shift up 24 . 7. A vertical stretch of $$ 5 / 2 $$, a horizontal compression of $$ 2 / 5 $$, and a shift down 33

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Let's analyze the transformations for each of the given functions step by step.

### 1. Transformations of $$ 4 f(3x) - 5 $$

- **Parent Function**: $$ f(x) $$
- **Transformations**:
  - **Horizontal Compression**: The factor of $$ 3 $$ inside the function $$ f(3x) $$ indicates a horizontal compression by a factor of $$ \frac{1}{3} $$.
  - **Vertical Stretch**: The factor of $$ 4 $$ outside the function indicates a vertical stretch by a factor of $$ 4 $$.
  - **Vertical Shift**: The $$ -5 $$ indicates a downward shift of $$ 5 $$ units.

### 3. Transformations of $$ y = \left(\frac{2}{3} x\right)^{2} + 5 $$

- **Parent Function**: $$ f(x) = x^2 $$
- **Transformations**:
  - **Horizontal Stretch**: The factor $$ \frac{2}{3} $$ indicates a horizontal stretch by a factor of $$ \frac{3}{2} $$.
  - **Vertical Shift**: The $$ +5 $$ indicates an upward shift of $$ 5 $$ units.

### 5. Equation using the square root function for the given transformations

- **Transformations**:
  - **Vertical Stretch**: $$ 28 $$ indicates a vertical stretch by a factor of $$ 28 $$.
  - **Shift Left**: $$ 19 $$ indicates a shift left by $$ 19 $$ units.
  - **Reflection**: Reflecting across the x-axis indicates a negative sign in front of the function.
  - **Shift Down**: $$ 22 $$ indicates a downward shift of $$ 22 $$ units.

The equation can be written as:
$$
y = -28 \sqrt{x + 19} - 22
$$

### 4. Transformations of $$ y = -3 \sqrt{-6x} $$

- **Parent Function**: $$ f(x) = \sqrt{x} $$
- **Transformations**:
  - **Horizontal Compression**: The factor of $$ -6 $$ indicates a horizontal compression by a factor of $$ \frac{1}{6} $$ and a reflection across the y-axis.
  - **Vertical Stretch**: The factor of $$ -3 $$ indicates a vertical stretch by a factor of $$ 3 $$ and a reflection across the x-axis.

### 6. Transformations

- **Transformations**:
  - **Horizontal Stretch**: $$ \frac{9}{2} $$ indicates a horizontal stretch by a factor of $$ \frac{2}{9} $$.
  - **Shift Up**: $$ 13 $$ indicates an upward shift of $$ 13 $$ units.
  - **Vertical Compression**: $$ \frac{3}{8} $$ indicates a vertical compression by a factor of $$ \frac{3}{8} $$.

The equation can be written as:
$$
y = \frac{3}{8} \sqrt{\frac{2}{9} x} + 13
$$

### 8. Transformations

- **Transformations**:
  - **Reflection**: Across the y-axis indicates a negative sign in front of the function.
  - **Vertical Compression**: $$ \frac{1}{8} $$ indicates a vertical compression by a factor of $$ \frac{1}{8} $$.
  - **Horizontal Compression**: $$ \frac{1}{7} $$ indicates a horizontal compression by a factor of $$ 7 $$.
  - **Shift Up**: $$ 24 $$ indicates an upward shift of $$ 24 $$ units.

The equation can be written as:
$$
y = -\frac{1}{8} \sqrt{7x} + 24
$$

### 7. Transformations

- **Transformations**:
  - **Vertical Stretch**: $$ \frac{5}{2} $$ indicates a vertical stretch by a factor of $$ \frac{5}{2} $$.
  - **Horizontal Compression**: $$ \frac{2}{5} $$ indicates a horizontal compression by a factor of $$ \frac{5}{2} $$.
  - **Shift Down**: $$ 33 $$ indicates a downward shift of $$ 33 $$ units.

The equation can be written as:
$$
y = \frac{5}{2} \sqrt{\frac{5}{2} x} - 33
$$

These equations represent the transformations based on the given conditions.
Here are the transformed equations based on the given conditions: 1. $$ y = 4f(3x) - 5 $$ 2. $$ y = \left(\frac{2}{3}x\right)^2 + 5 $$ 3. $$ y = -28\sqrt{x + 19} - 22 $$ 4. $$ y = -3\sqrt{-6x} $$ 5. $$ y = \frac{3}{8}\sqrt{\frac{2}{9}x} + 13 $$ 6. $$ y = -\frac{1}{8}\sqrt{7x} + 24 $$ 7. $$ y = \frac{5}{2}\sqrt{\frac{5}{2}x} - 33 $$ These equations incorporate the specified transformations for each function.

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