Question

level 4: open ended questions

1. 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 coordinate grid. This is where the concavity of the cubic function changes.

Step3: Sketch for $x<2$

Since the function is decreasing and concave up on $(-\infty,2)$, the slope of the tangent line is negative and the curve bends upwards. As $x\to-\infty$, the function value decreases. Draw a smooth curve approaching the point $(2,5)$ from the left with a negative - sloped tangent and upward concavity.

Step4: Sketch for $x > 2$

Since the function is decreasing and concave up on $(2,\infty)$, the slope of the tangent line is negative and the curve bends upwards. As $x\to\infty$, the function value decreases. Draw a smooth curve moving away from the point $(2,5)$ to the right with a negative - sloped tangent and upward concavity.

Answer:

A hand - sketched curve on the provided grid that meets the described criteria: a cubic function with an inflection point at $(2,5)$, decreasing and concave up on $(-\infty,2)$ and $(2,\infty)$.