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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The Deep Dive
To transform a square root function based on your instructions, we start with the basic square root function, \( f(x) = \sqrt{x} \), and apply the given transformations. 1. For the transformation of a vertical stretch of 28, shift left 19, reflect the x-axis, and shift down 22, the equation becomes: \[ y = -28\sqrt{(x + 19)} - 22 \] 2. To create the equation for a horizontal stretch of \( \frac{9}{2} \), a shift up 13, and a vertical compression of \( \frac{3}{8} \), we have: \[ y = \frac{3}{8} \sqrt{\left(\frac{2}{9} x\right)} + 13 \] 3. For a vertical stretch of \( \frac{5}{2} \), a horizontal compression of \( \frac{2}{5} \), and a shift down 33, the equation would be: \[ y = \frac{5}{2} \sqrt{\left(5x\right)} - 33 \] 4. With 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, the resulting equation is: \[ y = \frac{1}{8} \sqrt{(-7x)} + 24 \]
