Write the transformations that occur from the parent function. \( \begin{array}{l}\text { 1. } 4 f(3 x)-5\end{array} \) 3. \( y=\left(\frac{2}{3} x\right)^{2}+5 \) 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

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.