Question

1. the graph of

$y = \sqrt{x}$
is reflected over the x - axis, right 4 units, and up 5 units.
which is the equation of the translated graph?
\\( y = -\sqrt{x - 5}-4 \\)
\\( y = -\sqrt{x + 4}-5 \\)
\\( y = -\sqrt{x + 4}+5 \\)
\\( y = \sqrt{x - 5}-4 \\)
\\( y = -\sqrt{x - 4}+5 \\)

Explanation:

Step1: Reflect over x - axis

To reflect a function \(y = f(x)\) over the \(x\) - axis, we replace \(y\) with \(-y\). So for \(y=\sqrt{x}\), after reflection over the \(x\) - axis, the function becomes \(y =-\sqrt{x}\).

Step2: Shift right 4 units

To shift a function \(y = f(x)\) to the right by \(h\) units, we replace \(x\) with \(x - h\). Here, \(h = 4\), so we replace \(x\) with \(x-4\) in the function \(y =-\sqrt{x}\). The function now becomes \(y=-\sqrt{x - 4}\).

Step3: Shift up 5 units

To shift a function \(y = f(x)\) up by \(k\) units, we add \(k\) to the function. Here, \(k = 5\), so we add 5 to the function \(y=-\sqrt{x - 4}\). The final function is \(y=-\sqrt{x - 4}+5\).

Answer:

\(y =-\sqrt{x - 4}+5\) (the last option: \(y =-\sqrt{x - 4}+5\))