Question

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

answer attempt 1 out of 2
the degree of $f(x)$ is and the leading coefficient is . there are different real zeros and relative minimums.

Explanation:

Step1: Determine the degree of the polynomial

The end - behavior of a polynomial function is determined by the degree (even or odd) and the leading coefficient (positive or negative). For a polynomial function \(y = a_nx^n+\cdots+a_1x + a_0\), if the ends of the graph go in the same direction (both up or both down), the degree \(n\) is even. In the given graph, both ends of the polynomial graph point downwards. So the degree of the polynomial is even. Also, the number of turning points (relative maxima and minima) of a polynomial of degree \(n\) is at most \(n - 1\). Looking at the graph, we can see that there are 3 turning points (1 relative maximum on the left, 1 relative minimum, and 1 relative maximum on the right? Wait, no, let's count again. Wait, the graph has a "W" - like shape? Wait, no, the left end goes down, then up, then down, then up, then down? Wait, no, the graph as drawn: starts from the bottom left (down), goes up to a peak, then down to a valley, then up to a peak, then down to the bottom right. Wait, the number of turning points (where the slope changes from increasing to decreasing or vice - versa) is 3? Wait, no, the number of relative minima and maxima: relative minima are the low points, relative maxima are the high points. Wait, the degree of a polynomial is at least the number of turning points + 1. Let's see the end - behavior: both ends down, so degree is even. The number of real zeros: the graph crosses or touches the x - axis. Let's see the x - intercepts: the graph crosses the x - axis at two points? Wait, no, wait, the graph: first, it comes from the bottom left, goes up, crosses the x - axis, then goes up to a peak, then down, crosses the y - axis (below the x - axis), then down to a valley (relative minimum), then up, touches the x - axis (a repeated zero), then up to a peak, then down to the bottom right. Wait, so the number of real zeros: the graph crosses the x - axis at one point and touches it at another, so total of 2 distinct real zeros? Wait, no, maybe I misread. Wait, the problem says "different real zeros". Let's start with the degree:

End - behavior: as \(x ightarrow+\infty\) and \(x ightarrow-\infty\), \(f(x) ightarrow-\infty\). For a polynomial \(f(x)=a_nx^n+\cdots\), if \(n\) is even and \(a_n<0\), then as \(x ightarrow\pm\infty\), \(f(x) ightarrow-\infty\) (since \(x^n\) is positive for large \(|x|\) when \(n\) is even, and multiplying by a negative leading coefficient makes it negative). The number of turning points: the graph has 3 turning points (2 relative minima? Wait, no, let's look at the graph again. The graph has a "W" - shape? No, it's more like a "M" but inverted? Wait, no, the left end is down, goes up to a maximum, then down to a minimum, then up to a maximum, then down. So the number of turning points (where the derivative is zero) is 3. The degree of a polynomial is at least the number of turning points + 1. So if there are 3 turning points, the degree is at least 4. And since the end - behavior is even (both ends same direction), the degree is 4 (even, and 4 - 1 = 3 turning points, which matches).

Step2: Determine the leading coefficient

Since as \(x ightarrow\pm\infty\), \(f(x) ightarrow-\infty\), and for an even - degree polynomial \(y = a_nx^n\), if \(a_n<0\), then \(y ightarrow-\infty\) as \(|x| ightarrow\infty\). So the leading coefficient is negative.

Step3: Determine the number of different real zeros

The graph intersects the x - axis at two distinct points? Wait, no, wait: the graph crosses the x - axis at one point and touches it at another? Wait, no, looking at the graph, there are two distinct x - intercepts? Wait, no, the graph: first, it comes from the bottom left, goes up, crosses the x - axis, then goes up to a peak, then down, crosses the y - axis (below x - axis), then down to a valley (relative minimum), then up, touches the x - axis (a repeated root), then up to a peak, then down. Wait, but the problem says "different real zeros". So the number of distinct real zeros: how many times does the graph cross or touch the x - axis at distinct x - values? Let's see: the graph crosses the x - axis at one point (left of the y - axis) and touches it at another point (right of the y - axis). So that's 2 different real zeros? Wait, no, maybe I made a mistake. Wait, the graph: let's count the x - intercepts (where \(f(x) = 0\)). The graph intersects the x - axis at two distinct points? Wait, no, the left peak is above the x - axis, then it goes down, crosses the x - axis, then down to a valley (below x - axis), then up, touches the x - axis (so that's a zero with even multiplicity), then up to a peak (above x - axis), then down. Wait, no, the initial graph: the left end is down, goes up to a peak (above x - axis), then down, crosses the x - axis, then down to a valley (below x - axis), then up, touches the x - axis (so that's a zero), then up to a peak (above x - axis), then down. Wait, so the number of distinct real zeros: the graph crosses the x - axis at one point and touches it at another, so 2 different real zeros? Wait, no, maybe the graph has three x - intercepts? Wait, no, the way it's drawn: the first intersection (crossing) on the left, then a touch on the right. Wait, maybe the number of different real zeros is 2? Wait, no, let's look at the graph again. The graph: starts from bottom left, goes up, crosses x - axis, goes up to a maximum, then down, crosses y - axis (below x - axis), then down to a minimum, then up, touches x - axis, then up to a maximum, then down. So the x - intercepts: one crossing (left) and one touching (right), so 2 different real zeros? Wait, no, maybe the graph has three x - intercepts? Wait, I think I messed up. Wait, the degree is 4 (since 3 turning points, degree at least 4, even). For a degree 4 polynomial, the number of real zeros can be 2 or 4 (by Fundamental Theorem of Algebra, complex zeros come in pairs). Looking at the graph, the number of times it crosses or touches the x - axis: let's see, the graph crosses the x - axis at one point (left) and touches it at another (right), and maybe another? Wait, no, the left peak is above x - axis, then it goes down, crosses x - axis, then down to a valley (below x - axis), then up, touches x - axis (so that's a zero), then up to a peak (above x - axis), then down. So that's two x - intercepts (one crossing, one touching), so 2 different real zeros? Wait, no, the touching is a repeated zero, but it's still one distinct real zero? Wait, no, the problem says "different real zeros". So if the graph touches the x - axis at \(x = a\) (multiplicity even) and crosses at \(x = b\) (multiplicity odd), then the number of different real zeros is 2.

Step4: Determine the number of relative minima

Relative minima are the low points of the graph. Looking at the graph, there are 2 relative minima: one is the valley between the two peaks on the left - right, and wait, no: the graph has a "W" - like shape? No, it's inverted. Wait, the graph: from left, goes up (to a max), then down (to a min), then up (to a max), then down. So the number of relative minima is 2? Wait, no, the first minimum (after the first max) and the second minimum? Wait, no, let's count the number of times the graph changes from decreasing to increasing (which is a relative minimum). The graph: starts decreasing (from left end, going up? Wait, no, the left end is down, so as \(x\) increases from \(-\infty\), the function first increases (to the first max), then decreases (to the first min), then increases (to the second max), then decreases (to the end). So the relative minima are the points where the function changes from decreasing to increasing: there are 2 such points? Wait, no, the first min (after the first max) and the second min? Wait, no, the graph has two relative minima? Wait, no, looking at the graph, the number of relative minima is 2? Wait, no, the graph has a "W" - shape with two valleys (relative minima) and two peaks (relative maxima). Wait, yes, for a degree 4 polynomial with leading coefficient negative, the graph is "open down" (both ends down), so it has two relative minima and two relative maxima. Wait, so the number of relative minima is 2.

Answer:

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

So filling in the blanks:

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 minimums.