Question

level 4: open ended questions 2. sketch the graph of a single function with a domain (- ∞, ∞) that has all of the following characteristics: a. a cubic function b. inflection point at (2,5) c. decreasing and concave up on the interval (- ∞,2) d. decreasing and concave up on the interval (2,∞)

Explanation:

Step1: Recall cubic - function properties

A general cubic function is of the form $y = ax^{3}+bx^{2}+cx + d$. The second - derivative $y''$ determines concavity and the first - derivative $y'$ determines increasing/decreasing behavior.

Step2: Mark the inflection point

Plot the point $(2,5)$ on the graph. Since it is an inflection point, the concavity changes at $x = 2$.

Step3: Sketch for $x<2$

For $x\in(-\infty,2)$, the function is decreasing ($y'<0$) and concave up ($y''>0$). So, draw a curve that is going downwards and curving upwards as it approaches $x = 2$ from the left.

Step4: Sketch for $x>2$

For $x\in(2,\infty)$, the function is decreasing ($y'<0$) and concave up ($y''>0$). So, draw a curve that is going downwards and curving upwards as it moves away from $x = 2$ to the right.

Answer:

A hand - sketched graph on the provided grid with an inflection point at $(2,5)$, decreasing and concave up on $(-\infty,2)$ and $(2,\infty)$.