Question

for problems 14-15: write the square root function given the transformations from the parent function. 14. vertical compression by a factor \\(\frac{2}{7}\\), translate right 1 and up 3 15. reflect over the x-axis, vertical stretch by a factor of \\(\frac{4}{3}\\)

Explanation:

Problem 14:

Step 1: Recall the parent function

The parent square root function is \( y = \sqrt{x} \).

Step 2: Apply vertical compression

A vertical compression by a factor of \( \frac{2}{7} \) means we multiply the function by \( \frac{2}{7} \), so we get \( y = \frac{2}{7}\sqrt{x} \).

Step 3: Apply horizontal translation (right 1)

To translate a function right by \( h \) units, we replace \( x \) with \( x - h \). Here, \( h = 1 \), so we have \( y = \frac{2}{7}\sqrt{x - 1} \).

Step 4: Apply vertical translation (up 3)

To translate a function up by \( k \) units, we add \( k \) to the function. Here, \( k = 3 \), so the final function is \( y = \frac{2}{7}\sqrt{x - 1} + 3 \).

Step 1: Recall the parent function

The parent square root function is \( y = \sqrt{x} \).

Step 2: Apply reflection over the x - axis

A reflection over the x - axis means we multiply the function by \( - 1 \), so we get \( y=-\sqrt{x} \).

Step 3: Apply vertical stretch

A vertical stretch by a factor of \( \frac{4}{3} \) means we multiply the function (after reflection) by \( \frac{4}{3} \). So the function becomes \( y =-\frac{4}{3}\sqrt{x} \).

Answer:

\( y = \frac{2}{7}\sqrt{x - 1} + 3 \)

Problem 15: