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

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

Step3: Analyze behavior for $x<2$

The function is decreasing and concave up for $x\in(-\infty,2)$. So, as $x$ increases from $-\infty$ to $2$, the function values decrease and the curve bends upwards. Start the curve from the upper - left of the grid, make it pass through points in a decreasing and concave - up manner towards the point $(2,5)$.

Step4: Analyze behavior for $x>2$

The function is decreasing and concave up for $x\in(2,\infty)$. After passing through the point $(2,5)$, continue the curve in a decreasing and concave - up manner towards the lower - right of the grid.

Answer:

Sketch a curve on the given grid that passes through the point $(2,5)$, is decreasing and concave up for $x<2$ and also decreasing and concave up for $x > 2$.