Question

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(f(x) = \frac{1}{5}(x + 3)^2 + 9)

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).
no reflections

c) choose the correct transformation (stretches/compressions).
vertical compression by a factor of 1/5

d) choose the correct transformation (vertical shifts).
select an answer

e) choose the correct transformation (horizontal shifts).
select an answer

Explanation:

Step1: Identify the parent function

The function \(f(x) = \frac{1}{5}(x + 3)^2 + 9\) is a quadratic function, so its parent function is \(y = x^2\).

Step2: Determine the reflection transformation

There is no negative sign in front of the function or the \(x\) term, meaning there are no reflections.

Step3: Determine the stretch or compression

The coefficient \(\frac{1}{5}\) is outside the squared term and \(0 < \frac{1}{5} < 1\), representing a vertical compression by a factor of \(\frac{1}{5}\).

Step4: Determine the vertical shift

The constant term \(+9\) outside the squared term represents a vertical shift up by \(9\) units.

Step5: Determine the horizontal shift

The term \((x + 3)\) inside the squared term represents a horizontal shift left by \(3\) units.

Answer:

a) \(y = x^2\)
b) No reflections
c) Vertical compression by a factor of 1/5
d) Shift up 9 units
e) Shift left 3 units