Question

level 4: open ended questions

1. sketch the graph of a single function with a domain $(-\infty,\infty)$ 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 $(-\infty,0)$
e. concave down on the interval $(0,\infty)$
f. end behavior $x \to \infty,f(x) \to -\infty$ and $x \to -\infty,f(x) \to \infty$

Explanation:

Step1: Determine the general form of the cubic function

Since the zeros are at \(x = - 1\), \(x = 0\), and \(x = 1\), the cubic function can be written in factored form as \(y=a(x + 1)x(x - 1)=a(x^{3}-x)\), where \(a\) is a non - zero constant.

Step2: Use the end - behavior to find the value of \(a\)

The end - behavior is \(x
ightarrow\infty,f(x)
ightarrow-\infty\) and \(x
ightarrow-\infty,f(x)
ightarrow\infty\). For a cubic function \(y = ax^{3}+bx^{2}+cx + d\), the leading term \(ax^{3}\) determines the end - behavior. When \(x
ightarrow\infty\), if \(y
ightarrow-\infty\) and \(x
ightarrow-\infty\), \(y
ightarrow\infty\), then \(a<0\). Let \(a=-1\), so the function is \(y=-x^{3}+x\).

Step3: Check the inflection point and concavity

First, find the second derivative. The first derivative \(y^\prime=-3x^{2}+1\), and the second derivative \(y^{\prime\prime}=-6x\).
Set \(y^{\prime\prime}=0\), then \(-6x = 0\) gives \(x = 0\). When \(x = 0\), \(y=0\), so the inflection point is \((0,0)\).
For the concavity:

• When \(x\in(-\infty,0)\), \(y^{\prime\prime}=-6x>0\), so the function is concave up on \((-\infty,0)\).
• When \(x\in(0,\infty)\), \(y^{\prime\prime}=-6x<0\), so the function is concave down on \((0,\infty)\).

Step4: Sketch the graph

• Plot the zeros at \(x=-1\), \(x = 0\), and \(x = 1\).
• Since the function is \(y=-x^{3}+x\), when \(x=-2\), \(y=-(-2)^{3}+(-2)=8 - 2=6\); when \(x = 2\), \(y=-2^{3}+2=-8 + 2=-6\).
• Mark the inflection point \((0,0)\).
• Use the end - behavior (\(x

ightarrow\infty,y
ightarrow-\infty\) and \(x
ightarrow-\infty,y
ightarrow\infty\)) and the concavity information to draw a smooth curve passing through the plotted points.

Answer:

Sketch the graph of the function \(y=-x^{3}+x\) with zeros at \(x=-1\), \(x = 0\), \(x = 1\), inflection point at \((0,0)\), concave up on \((-\infty,0)\), concave down on \((0,\infty)\) and the given end - behavior.