Question

positive: $-3 < x < 3$
negative: $x < -3$ and $x > 3$
increasing: $x < 0$
decreasing: $x > 0$
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(b) determine the end behavior of the graph.
$x \to -\infty, f(x) \to$
$x \to \infty, f(x) \to$

Explanation:

Step1: Analyze the function's degree and leading coefficient

Assume the function is \(y = -x^{2}+9\) (since positive when \(-3 < x < 3\), negative when \(x < - 3\) or \(x>3\), increasing when \(x < 0\) and decreasing when \(x>0\), which is a quadratic function \(y=ax^{2}+bx + c\) with \(a=-1\), \(b = 0\), \(c = 9\)).
For a quadratic function \(y = ax^{2}+bx + c\), the degree \(n = 2\) (even) and the leading coefficient \(a=-1<0\).

Step2: Determine the end - behavior

When \(n\) is even and \(a<0\):
As \(x\to-\infty\), \(y = ax^{2}+bx + c\approx ax^{2}\). Substitute \(a=-1\) and \(x\to-\infty\), \(f(x)=-x^{2}+9\to-\infty\) (because \(x^{2}\to+\infty\) as \(x\to\pm\infty\) and \(a=-1\)).
As \(x\to+\infty\), \(y=-x^{2}+9\approx -x^{2}\), so \(f(x)\to-\infty\).

Answer:

When \(x\to-\infty\), \(f(x)\to-\infty\); when \(x\to+\infty\), \(f(x)\to-\infty\)