Question

for the polynomial function f(x) = -2x⁴ - 6x², answer the parts a through e.

a. the x-intercept(s) at which the graph touches the x-axis and turns around is/are
(type an integer or a decimal. use a comma to separate answers as needed. type each answer only once.) -3,0.

b. there are no x-intercepts at which the graph touches the x-axis and turns around.

c. find the y-intercept.
the y-intercept is
(simplify your answer. type an integer or a decimal.) 0

d. determine whether the graph has y-axis symmetry, origin symmetry, or neither. choose the correct answer below.
a. the graph of f is symmetric about the y-axis.
b. the graph of f is symmetric about the origin.
c. the graph of f is neither symmetric about the y-axis nor symmetric about the origin.

e. if necessary, find a few additional points and graph the function. use the maximum number of turning points to check whether it is drawn correctly. choose the correct graph below.

a. image of graph

b. image of graph

c. image of graph

d. image of graph

Explanation:

Step1: Analyze the polynomial function

The given polynomial function is \( f(x) = -2x^4 - 6x^2 \). Let's check for symmetry. A function \( f(x) \) is symmetric about the y - axis if \( f(-x)=f(x) \), and symmetric about the origin if \( f(-x)= - f(x) \).

Calculate \( f(-x) \):
\( f(-x)=-2(-x)^4 - 6(-x)^2=-2x^4 - 6x^2 \) (since \( (-x)^n=x^n \) when \( n \) is even)
And \( f(x)=-2x^4 - 6x^2 \), so \( f(-x) = f(x) \). This means the function is even, and the graph of an even function is symmetric about the y - axis.

Step2: Analyze the x - intercepts

To find the x - intercepts, set \( f(x) = 0 \):
\( -2x^4-6x^2 = 0 \)
Factor out \( -2x^2 \): \( -2x^2(x^2 + 3)=0 \)
Set each factor equal to zero:

• \( -2x^2=0\Rightarrow x = 0 \) (with multiplicity 2, so the graph touches the x - axis at \( x = 0 \))
• \( x^2+3=0\Rightarrow x^2=- 3 \), which has no real solutions. So the x - intercept is \( x = 0 \) (or the point \( (0,0) \))

Step3: Analyze the y - intercept

The y - intercept is found by setting \( x = 0 \) in the function. \( f(0)=-2(0)^4-6(0)^2 = 0 \), so the y - intercept is \( (0,0) \)

Step4: Analyze the end - behavior and turning points

The degree of the polynomial is \( n = 4 \) (even) and the leading coefficient \( a=-2\) (negative). So as \( x
ightarrow\pm\infty \), \( f(x)
ightarrow-\infty \) (since for even degree, the ends of the graph go in the same direction, and for negative leading coefficient, they go down). The maximum number of turning points of a polynomial of degree \( n \) is \( n - 1 \). For \( n = 4 \), the maximum number of turning points is \( 4-1 = 3 \)

Now, let's analyze the graph options:

• Option A: The graph seems to have a different end - behavior or symmetry.
• Option B: Since the function is symmetric about the y - axis, has x - intercept at \( x = 0 \), and end - behavior going to \( -\infty \) as \( x

ightarrow\pm\infty \), and the shape with turning points consistent with a 4th - degree polynomial with negative leading coefficient and y - axis symmetry.

• Option C: The graph does not show the correct symmetry or end - behavior.
• Option D: The graph does not show the correct symmetry or end - behavior.

Answer:

Part a:

The x - intercept(s) at which the graph touches the x - axis and turns around is \( 0 \) (the point \( (0,0) \))

Part b:

There are no x - intercepts at which the graph crosses the x - axis (since the only real x - intercept is a touch - point)

Part c:

The y - intercept is \( 0 \) (the point \( (0,0) \))

Part d:

The graph of \( f \) is symmetric about the y - axis (Option A)

Part e:

The correct graph is Option B (assuming the visual analysis based on symmetry, intercepts, and end - behavior)