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. 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. state whether the graph has a turning point in this window. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the graph has a turning point in this window. the turning point(s) is/are. (type an ordered pair. use a comma to separate answers as needed.) b. there is no turning point.

Explanation:

Step1: Find y-values at x=-15

$y=(-15)^3 + 3(-15)^2 -45(-15) = -3375 + 675 + 675 = -2025$

Step2: Find y-values at x=15

$y=(15)^3 + 3(15)^2 -45(15) = 3375 + 675 - 675 = 3375$

Step3: Find critical points (turning points)

First, take derivative: $y'=3x^2+6x-45$
Set to 0: $3x^2+6x-45=0$
Simplify: $x^2+2x-15=0$
Factor: $(x+5)(x-3)=0$
Solutions: $x=-5, x=3$

Step4: Find y at critical points

For $x=-5$: $y=(-5)^3+3(-5)^2-45(-5)=-125+75+225=175$
For $x=3$: $y=(3)^3+3(3)^2-45(3)=27+27-135=-81$

Step5: Determine valid viewing window

The x-range is $[-15,15]$. The y-values range from -2025 to 3375, but the given options: none fully cover, but we check the graph shape. The cubic rises left? No: as $x\to-\infty$, $y\to-\infty$; $x\to+\infty$, $y\to+\infty$. It has a local max at $(-5,175)$ and local min at $(3,-81)$. The only window that can show both positive and negative y-values is none of C/D, but wait, the first part's options: the correct window that can contain the key points is none of the given, but the graph A matches the cubic shape (falls left, rises right, with a local max then min).

Step6: Confirm turning points

The critical points $(-5,175)$ and $(3,-81)$ are within $[-15,15]$, so they are in the window.

Answer:

1. Appropriate viewing window: (Note: The given options C/D are incomplete, but the only valid window that includes both positive and negative y-values for the key points would need to cover [-2025, 3375], but from the provided options, if forced to choose, none are fully correct. However, the graph selection:

A. [The graph with left end falling, right end rising, one local maximum and one local minimum]

1. State turning points:

A. The graph has a turning point in this window. The turning point(s) is/are $(-5, 175), (3, -81)$