Questions: Use the given function to answer the questions that follow. f(x) = x^4 - 4x^2 a) Use the Leading Coefficient Test to determine the graph's end behavior. A. The graph of f(x) rises left and falls right. B. The graph of f(x) falls left and falls right. C. The graph of f(x) falls left and rises right. D. The graph of f(x) rises left and rises right. b) Find the x-intercepts. x = (Use a comma to separate answers as needed.)

Solution Steps

Using the Leading Coefficient Test, we analyze the function \( f(x) = x^4 - 4x^2 \). The leading term is \( x^4 \), which has a positive coefficient (1) and an even degree (4). Therefore, the end behavior of the graph is:

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

Next, we factor the polynomial \( f(x) = x^4 - 4x^2 \). The factorization yields:

\[ f(x) = x^2 (x - 2)(x + 2) \]

To find the x-intercepts, we set the factored expression equal to zero:

\[ x^2 = 0 \quad \Rightarrow \quad x = 0 \] \[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \] \[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \]

Thus, the x-intercepts are \( x = 0, 2, -2 \).

Final Answer