Question

find an appropriate viewing window for the function, using the given x - values. then graph the function.
y = x³ + 3x² - 45x, for x - values between - 15 and 15.
which of the following is the viewing window for the function, using the given x - values?
a. -15,15 by -10,10
b. -15,15 by -90,190
c. -15,15 by -90,0
d. -15,15 by 0,190
which of the following is the graph of y = x³ + 3x² - 45x with the above viewing window?
a.
b.
c.
d.

Explanation:

Step1: Find y at x=-15

Substitute $x=-15$ into $y=x^3+3x^2-45x$:

$$\begin{align*} y&=(-15)^3 + 3(-15)^2 - 45(-15)\\ &=-3375 + 675 + 675\\ &=-2025 \end{align*}$$

Correction: Recalculate accurately

$$\begin{align*} y&=(-15)^3 + 3(-15)^2 - 45(-15)\\ &=-3375 + 3(225) + 675\\ &=-3375 + 675 + 675\\ &=-3375 + 1350\\ &=-2025 \end{align*}$$

Wait, correction again: The given option B has y-range [-90,190], so we need to check critical points and endpoints within relevant range.

Step2: Find critical points

Take derivative $y'=3x^2+6x-45$, set to 0:
$$3x^2+6x-45=0 \implies x^2+2x-15=0$$
Factor: $(x+5)(x-3)=0$, so $x=-5, x=3$.

Step3: Calculate y at critical points

For $x=-5$:

$$\begin{align*} y&=(-5)^3 + 3(-5)^2 - 45(-5)\\ &=-125 + 75 + 225\\ &=175 \end{align*}$$

For $x=3$:

$$\begin{align*} y&=(3)^3 + 3(3)^2 - 45(3)\\ &=27 + 27 - 135\\ &=-81 \end{align*}$$

Step4: Calculate y at x=15

$$\begin{align*} y&=(15)^3 + 3(15)^2 - 45(15)\\ &=3375 + 675 - 675\\ &=3375 \end{align*}$$

The relevant y-values for the curve's key features (local min/max) are -81 and 175, so the window must include these. Option B: $[-15,15]$ by $[-90,190]$ covers these.

Step5: Match the graph

The cubic function $y=x^3+3x^2-45x$ has a local maximum at $x=-5$ (y=175) and local minimum at $x=3$ (y=-81). As $x\to+\infty$, $y\to+\infty$; as $x\to-\infty$, $y\to-\infty$. This matches graph D: rises to the right, falls to the left, has a peak at negative x, valley at positive x.

Answer:

First question: B. $[-15,15]$ by $[-90,190]$
Second question: D. (The graph that rises to the right, falls to the left, with a local maximum on the negative x-axis and local minimum on the positive x-axis)