Question

the graph of g(x) is the image of $f\left(x\
ight)=x^{2}$ after the given transformations.

• a reflection over the x- axis
• a vertical translation of 4 units up
• a horizontal translation of 3 units to the right

which equation represents g(x)?

• $g(x) = (x - 3)^{2} + 4$
• $g(x) = - (x + 3)^{2} - 4$
• $g(x) = - (x - 3)^{2} + 4$
• $g(x) = (x + 3)^{2} - 4$

Explanation:

Step1: Apply reflection over x - axis

For a function \(y = f(x)\), a reflection over the \(x\) - axis is given by \(y=-f(x)\). Given \(f(x)=x^{2}\), after reflection over the \(x\) - axis, the function becomes \(y = -x^{2}\).

Step2: Apply horizontal translation 3 units right

For a function \(y = f(x)\), a horizontal translation of \(h\) units to the right is given by \(y = f(x - h)\). Here, \(h = 3\) and the function after reflection is \(y=-x^{2}\), so after horizontal translation, the function becomes \(y=-(x - 3)^{2}\).

Step3: Apply vertical translation 4 units up

For a function \(y = f(x)\), a vertical translation of \(k\) units up is given by \(y=f(x)+k\). Here, \(k = 4\) and the function after horizontal translation is \(y=-(x - 3)^{2}\), so after vertical translation, the function becomes \(y=-(x - 3)^{2}+4\). So \(g(x)=-(x - 3)^{2}+4\).

Answer:

\(g(x)=-(x - 3)^{2}+4\) (the third option: \(g(x)=-(x - 3)^{2}+4\))