Question

the polynomial function $f(x)$ is graphed below. fill in the form below regarding the features of this graph.

answer
the degree of $f(x)$ is and the leading coefficient is ; there are $\square$ different real zeros and $\square$ relative maximums.

Explanation:

Step1: Determine the degree (end behavior)

The ends of the polynomial graph both go down (as \( x \to \pm\infty \), \( f(x) \to -\infty \)). For even - degree polynomials, the ends have the same behavior. So the degree is even. Also, the number of turning points (relative max and min) helps. The graph has 3 turning points (2 minima and 1 maximum? Wait, no: looking at the graph, it has a "w" - like shape? Wait, the graph has two "humps" above the x - axis? Wait, no, the graph: let's count the turning points. The graph comes from the bottom left, rises to a peak, then falls to a valley (on the y - axis), then rises to another peak, then falls to the bottom right. So the number of turning points (local max and min) is 3. The formula for the maximum number of turning points of a polynomial of degree \( n \) is \( n - 1 \). So if there are 3 turning points, \( n-1 = 3\) implies \( n = 4 \). So the degree is 4 (even). And since the ends go down, for a polynomial \( f(x)=a_nx^n+\cdots+a_0 \), when \( n \) is even and \( a_n<0 \), the ends go down. So the leading coefficient \( a_n \) is negative.

Step2: Determine the number of real zeros

The graph intersects the x - axis at how many points? Looking at the graph, it intersects the x - axis at 2 points (since it crosses the x - axis twice? Wait, no, maybe it touches? Wait, the graph: if it's a 4th - degree polynomial, and it has two real zeros (since it crosses the x - axis twice? Wait, no, maybe the graph is symmetric about the y - axis? Wait, the graph is symmetric about the y - axis (since it's a function with a graph that is symmetric with respect to the y - axis, so it's an even function). So if it crosses the x - axis at two points (left and right of the y - axis), but maybe it's tangent? Wait, no, the graph: let's see, the ends go down, and it has two "peaks" above the x - axis? Wait, no, the y - axis is the vertical axis. The graph: when \( x = 0 \), it's at a minimum? Wait, no, the graph: comes from \( -\infty \), rises to a maximum, then falls to a minimum (on the y - axis), then rises to a maximum, then falls to \( -\infty \). So the number of times it crosses the x - axis: let's say it crosses the x - axis at two points (left and right of the y - axis), so 2 different real zeros.

Step3: Determine the number of relative maximums

A relative maximum is a point where the function changes from increasing to decreasing. Looking at the graph, there are 2 relative maximums? Wait, no: first, it rises to a peak (relative max), then falls to a valley (relative min), then rises to another peak (relative max), then falls. Wait, no, that's 2 relative maximums? Wait, no, the first peak and the second peak: so 2 relative maximums? Wait, no, let's count again. The graph: starts at bottom left (decreasing? No, it comes from bottom left, so as \( x \) increases from \( -\infty \), it first increases (so the first turning point is a relative maximum), then decreases (to a relative minimum), then increases (to a relative maximum), then decreases. So there are 2 relative maximums.

Step4: Leading coefficient sign

Since the degree is 4 (even) and the ends of the graph go down (as \( x\to\pm\infty \), \( f(x)\to-\infty \)), for a polynomial \( f(x)=a_nx^n+\cdots \), when \( n \) is even, if \( a_n<0 \), the ends go down. So the leading coefficient is negative.

Answer:

The degree of \( f(x) \) is \( \boldsymbol{4} \) and the leading coefficient is \( \boldsymbol{\text{negative}} \). There are \( \boldsymbol{2} \) different real zeros and \( \boldsymbol{2} \) relative maximums.