Question

if the following function is of the form : $acdot f(x - h) + k$, 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) $y = -(x - 3)^3 + 2$

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

Explanation:

Step1: Identify function type

The given function $y = -(x - 3)^3 + 2$ has the variable raised to the power of 3, so it is a cubic function.

Step2: Find inflection point

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

Step3: Find vertical dilation factor

In the form $a\cdot f(x-h)+k$, $a$ is the vertical dilation factor. Comparing to $y = -(x - 3)^3 + 2$, we get $a=-1$.

Step4: Determine increasing/decreasing

Take the derivative: $y' = -3(x - 3)^2$. Since $(x - 3)^2\geq0$ for all real $x$, $y'\leq0$ (and only equals 0 at $x=3$), so the function is decreasing.

Answer:

• Function: Cubic
• Point of inflection: $(3, 2)$
• Factor of vertical dilation $a$: $-1$
• Increasing or Decreasing: Decreasing