Question

use the given function to complete parts a) through e) below.
$f(x) = -x^4 + 4x^2$

a) use the leading coefficient test to determine the graph’s end behavior.
\\(\bigcirc\\) a. the graph of \\(f(x)\\) rises left and rises right.
\\(\bigcirc\\) b. the graph of \\(f(x)\\) falls left and rises right.
\\(\bigcirc\\) c. the graph of \\(f(x)\\) rises left and falls right.
\\(\bigcirc\\) d. the graph of \\(f(x)\\) falls left and falls right.

b) find the \\(x\\)-intercepts.
\\(x = \square\\)
(type an integer or a decimal. use a comma to separate answers as needed.)
at which zeros does the graph of the function cross the \\(x\\)-axis? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
\\(\bigcirc\\) a. \\(x = \square\\) (type an integer or a decimal. use a comma to separate answers as needed.)
\\(\bigcirc\\) b. there are no \\(x\\)-intercepts at which the graph crosses the \\(x\\)-axis.
at which zeros does the graph of the function touch the \\(x\\)-axis and turn around? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
\\(\bigcirc\\) a. \\(x = \square\\) (type an integer or a decimal. use a comma to separate answers as needed.)
\\(\bigcirc\\) b. there are no \\(x\\)-intercepts at which the graph touches the \\(x\\)-axis and turns around.

c) find the \\(y\\)-intercept by computing \\(f(0)\\).
\\(f(0) = \square\\)

Explanation:

Part a)

Step1: Identify Degree and Leading Coefficient

The function is \( f(x) = -x^4 + 4x^2 \). The degree (highest power of \( x \)) is 4 (even), and the leading coefficient (coefficient of the highest power term) is -1 (negative).

Step2: Apply Leading Coefficient Test

For a polynomial with even degree:

• If leading coefficient is positive, the graph rises left and rises right.
• If leading coefficient is negative, the graph falls left and falls right.

Since degree is 4 (even) and leading coefficient is -1 (negative), the graph falls left and falls right.

Step1: Find x-intercepts (solve \( f(x) = 0 \))

Set \( -x^4 + 4x^2 = 0 \). Factor out \( -x^2 \):
\( -x^2(x^2 - 4) = 0 \).
Factor \( x^2 - 4 \) (difference of squares): \( x^2 - 4 = (x - 2)(x + 2) \).
So, \( -x^2(x - 2)(x + 2) = 0 \).

Step2: Solve for \( x \)

Set each factor to zero:
\( -x^2 = 0 \implies x = 0 \) (multiplicity 2),
\( x - 2 = 0 \implies x = 2 \) (multiplicity 1),
\( x + 2 = 0 \implies x = -2 \) (multiplicity 1).

Step3: Determine Cross/Touch x-axis

• A zero with odd multiplicity: graph crosses the x-axis.
• A zero with even multiplicity: graph touches and turns around.

For \( x = -2 \) and \( x = 2 \) (multiplicity 1, odd): graph crosses x-axis.
For \( x = 0 \) (multiplicity 2, even): graph touches and turns around.

Step1: Compute \( f(0) \)

Substitute \( x = 0 \) into \( f(x) = -x^4 + 4x^2 \):
\( f(0) = -(0)^4 + 4(0)^2 = 0 + 0 = 0 \).

Answer:

D. The graph of \( f(x) \) falls left and falls right.

Part b)