Question

if the following function is of the form : $a \cdot f(x - h) + k$,
state the function as cubic or cube root,
give the coordinates of the inflection point
and the factor of vertical dilation (as a fraction when necessary)
$y = \frac{1}{4}\sqrt3{x+4}-6$

• function:
• point of inflection : $(h,k)$
• factor of vertical dilation a=
• increasing or decreasing:

Explanation:

Step1: Identify function type

The function contains $\sqrt[3]{x+4}$, so it is a cube root function, a transformation of $f(x)=\sqrt[3]{x}$.

Step2: Match to $a\cdot f(x-h)+k$

Rewrite given function: $y=\frac{1}{4}\sqrt[3]{x-(-4)} + (-6)$. Compare to $a\cdot f(x-h)+k$:

• $a=\frac{1}{4}$, $h=-4$, $k=-6$

Step3: Find inflection point

For $a\cdot f(x-h)+k$, inflection point is $(h,k)$ = $(-4, -6)$.

Step4: Determine vertical dilation

From the matched form, vertical dilation factor $a=\frac{1}{4}$.

Step5: Check monotonicity

The parent cube root function $f(x)=\sqrt[3]{x}$ is always increasing. Vertical dilation by positive $a$ preserves increasing behavior, so the function is increasing.

Answer:

• Function: Cube Root
• Point of inflection: $(-4, -6)$
• Factor of vertical dilation $a$: $\frac{1}{4}$
• Increasing or Decreasing: Increasing