Question

determine the parent function from which the graph of the function shown below can be obtained. next, identify each transformation that can be applied to the parent function in order to obtain the graph of the function shown below.

(f(x) = -\frac{3}{4}(-x)^2 + 8)

a) choose the correct parent function.

• (y = x^2)
• (y = x^3)
• (y = |x|)
• (y = \sqrt{x})
• (y = \sqrt3{x})

b) choose the correct transformation (reflections).

c) choose the correct transformation (stretches/compressions).

Explanation:

Step1: Identify the parent function

The function is \(f(x) = -\frac{3}{4}(-x)^2 + 8\), which contains a squared term, so the parent function is \(y = x^2\).

$$y = x^2$$

Step2: Identify the reflections

The negative sign outside, \(-f(x)\), reflects across the x-axis, and the negative sign inside, \(f(-x)\), reflects across the y-axis.

$$\text{Reflect across both the x-axis and the y-axis}$$

Step3: Identify the stretch or compression

The factor \(\frac{3}{4}\) multiplying the function vertically compresses the graph because \(0 < \frac{3}{4} < 1\).

$$\text{Vertically compress by a factor of } \frac{3}{4}$$

Answer:

a) \(y = x^2\)
b) Reflect across both the x-axis and the y-axis
c) Vertically compress by a factor of \(\frac{3}{4}\)