Question

select all the correct answers. graph the following function. \\( f(x) = x^4 - 2x^2 + 1 \\) which of the following statements correctly describe the graph of the function above? when \\( x = -1 \\), \\( f(x) = 0 \\). as \\( x \\) approaches positive infinity, \\( f(x) \\) approaches positive infinity. when \\( x = -2 \\), \\( f(x) = 0 \\). as \\( x \\) approaches positive infinity, \\( f(x) \\) approaches negative infinity. as \\( x \\) approaches negative infinity, \\( f(x) \\) approaches negative infinity. as \\( x \\) approaches negative infinity, \\( f(x) \\) approaches positive infinity.

Explanation:

Step1: Check x=-1, compute f(-1)

$f(-1) = (-1)^4 - 2(-1)^2 + 1 = 1 - 2 + 1 = 0$

Step2: Check x=-2, compute f(-2)

$f(-2) = (-2)^4 - 2(-2)^2 + 1 = 16 - 8 + 1 = 9$

Step3: Analyze end behavior (x→+∞)

For $f(x)=x^4-2x^2+1$, the leading term is $x^4$. As $x\to+\infty$, $x^4\to+\infty$, so $f(x)\to+\infty$.

Step4: Analyze end behavior (x→-∞)

For $x\to-\infty$, $x^4=(-|x|)^4=|x|^4\to+\infty$, so $f(x)\to+\infty$.

Answer:

• When x = -1, f(x) = 0.
• As x approaches positive infinity, f(x) approaches positive infinity.
• As x approaches negative infinity, f(x) approaches positive infinity.