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)
$y = -\frac{1}{2}\sqrt3{x - 2} + 3$

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

Explanation:

Step1: Identify function type

The given function contains a cube root $\sqrt[3]{x-2}$, so it is a cube root function.

Step2: Find inflection point

For $a\cdot f(x-h)+k$, inflection point is $(h,k)$. Here $h=2$, $k=3$, so $(2,3)$.

Step3: Identify vertical dilation factor

The coefficient of the cube root term is $a=-\frac{1}{2}$.

Step4: Determine increasing/decreasing

Since $a=-\frac{1}{2}<0$, the parent cube root function (increasing) is reflected vertically, making it decreasing.

Answer:

• Function: Cube Root
• Point of Inflection: $(2, 3)$
• Factor of vertical dilation $a$: $-\frac{1}{2}$
• Increasing or Decreasing: Decreasing