Question

compared with the graph of the parent function, which equation shows a vertical stretch by a factor of 6, a shift of 7 units right, and a reflection over the x-axis?
$y = -6\sqrt{x - 7}$
$y = -\frac{1}{6}\sqrt{x - 7}$
$y = 6\sqrt{x} + 7$
$y = 6\sqrt{x + 7}$

Explanation:

Step1: Recall transformation rules

For a function \( y = f(x) \), vertical stretch by factor \( a \) ( \( a>1 \)): \( y = a f(x) \); reflection over \( x \)-axis: \( y=-f(x) \); shift right by \( h \) units: \( y = f(x - h) \). The parent function here is \( y=\sqrt{x} \).

Step2: Apply vertical stretch

Vertical stretch by factor 6: \( y = 6\sqrt{x} \).

Step3: Apply reflection over x - axis

Reflection over x - axis: \( y=-6\sqrt{x} \).

Step4: Apply horizontal shift

Shift 7 units right: Replace \( x \) with \( x - 7 \), so \( y=-6\sqrt{x - 7} \).

Answer:

\( y = -6\sqrt{x - 7} \) (the first option among the given choices)