Question

level 4: open ended questions 4. sketch the graph of a single function with a domain (-∞,∞) that has all of the following characteristics: a. a cubic function b. zeros at -1, 0, and 1 c. inflection point at (0,0) d. concave up on the interval (-∞, 0) e. concave down on the interval (0,∞) f. end behavior x→∞, f(x)→ -∞ and x→ -∞, f(x)→∞

Explanation:

Step1: Write the general form of cubic

A cubic function with zeros at $x = - 1,0,1$ can be written as $y=a(x + 1)x(x - 1)=a(x^{3}-x)$ where $a
eq0$.

Step2: Determine the second - derivative

First derivative $y'=a(3x^{2}-1)$. Second derivative $y'' = 6ax$.

Step3: Use concavity and inflection - point information

The inflection point is at $(0,0)$. For $x\in(-\infty,0)$, $y''>0$ when $a<0$ (concave up), and for $x\in(0,\infty)$, $y''<0$ when $a < 0$ (concave down). Let $a=-1$. So the function is $y=-x^{3}+x$.

Step4: Sketch the graph

• Mark the zeros at $x=-1,0,1$.
• Since it is a cubic with $a=-1$, as $x\to-\infty,y\to\infty$ and as $x\to\infty,y\to-\infty$.
• The function is concave up on $(-\infty,0)$ and concave down on $(0,\infty)$ with an inflection point at $(0,0)$. Draw a smooth curve passing through the zeros and following the concavity and end - behavior.

Answer:

Sketch a smooth curve for $y=-x^{3}+x$ passing through $(-1,0),(0,0),(1,0)$, concave up on $(-\infty,0)$ and concave down on $(0,\infty)$ with end - behavior $x\to-\infty,y\to\infty$ and $x\to\infty,y\to-\infty$.