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:

To solve this, we analyze the polynomial graph:

Step 1: Determine the Degree

The end - behavior of a polynomial is determined by the leading term \(a_nx^n\). For a polynomial, if the ends of the graph go in the same direction (both down here, as \(x\to\pm\infty\), \(f(x)\to-\infty\)), the degree \(n\) is even. Also, the number of turning points (local max/min) is related to the degree. The graph has 3 turning points (2 local maxima and 1 local minimum). The formula for the maximum number of turning points of a polynomial of degree \(n\) is \(n - 1\). If there are 3 turning points, then \(n-1 = 3\), so \(n=4\). So the degree is even (specifically 4, which is even).

Step 2: Determine the Leading Coefficient Sign

When the degree is even, if the ends of the graph go down (as \(x\to\pm\infty\), \(f(x)\to-\infty\)), the leading coefficient \(a_n\) is negative (because for \(y = a_nx^n\) with \(n\) even, if \(a_n<0\), as \(|x|\) gets large, \(y\) is negative).

Step 3: Count Real Zeros

A real zero of a polynomial is a point where the graph intersects the \(x\) - axis. Looking at the graph, we can see that it intersects the \(x\) - axis at 1 distinct point (since it crosses the \(x\) - axis once). Wait, no, wait. Wait, the graph: let's re - examine. Wait, the graph crosses the \(x\) - axis once? Wait, no, maybe I made a mistake. Wait, the graph: when we look at the \(x\) - intercepts, the number of times the graph crosses or touches the \(x\) - axis. Wait, the given graph: let's see, the left - most part comes from below, rises, has a local max, then a local min, then a local max, then falls. So it crosses the \(x\) - axis once? Wait, no, maybe the graph crosses the \(x\) - axis once? Wait, no, maybe I misread. Wait, the problem says "different real zeros". Wait, maybe the graph intersects the \(x\) - axis at 1 point? Wait, no, let's think again. Wait, the graph: when \(x\) is very negative, \(f(x)\) is negative, then it rises, has a local max, then a local min (still above or below? Wait, the local min: if the graph has a local min between the two local maxima. Wait, maybe the graph touches the \(x\) - axis once? Wait, no, maybe the graph crosses the \(x\) - axis once. So the number of different real zeros is 1? Wait, no, maybe I made a mistake. Wait, the graph: let's count the \(x\) - intercepts. The graph intersects the \(x\) - axis at 1 distinct point (so 1 different real zero).

Step 4: Count Relative Maxima

A relative maximum is a point where the function changes from increasing to decreasing. Looking at the graph, we can see that there are 2 relative maxima (the two "peaks" in the graph).

Final Answers

• The degree of \(f(x)\) is \(\boldsymbol{\text{even (4)}}\) and the leading coefficient is \(\boldsymbol{\text{negative}}\). There are \(\boldsymbol{1}\) different real zeros and \(\boldsymbol{2}\) relative maximums.

Wait, but let's correct the real zero count. Wait, maybe the graph crosses the \(x\) - axis once? Wait, no, maybe I misinterpreted the graph. Wait, the original graph: let's see, the user's graph: the left end goes down, then up to a peak, then down to a valley, then up to a peak, then down. So the graph intersects the \(x\) - axis at 1 point? So the number of different real zeros is 1. And the number of relative maxima: the two peaks, so 2.

Answer:

The degree of \(f(x)\) is \(\boldsymbol{\text{even (4)}}\) and the leading coefficient is \(\boldsymbol{\text{negative}}\). There are \(\boldsymbol{1}\) different real zeros and \(\boldsymbol{2}\) relative maximums.

(If we consider the degree as 4, which is even, leading coefficient negative, 1 real zero, 2 relative maxima)