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 leading term \(a_nx^n\), where \(n\) is the degree and \(a_n\) is the leading coefficient. For a polynomial, if the ends of the graph both go down (as \(x
ightarrow+\infty\) and \(x
ightarrow - \infty\), \(f(x)
ightarrow-\infty\)), the degree \(n\) is even and the leading coefficient \(a_n\) is negative. 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 (2 relative maxima and 1 relative minimum). So, if the number of turning points \(T=n - 1\), and \(T = 3\), then \(n=T + 1=4\)? Wait, no, wait. Wait, the graph has 3 turning points? Wait, let's count again. The graph has two "hills" (relative maxima) and one "valley" (relative minimum)? Wait, no, looking at the graph: from left to right, it comes from the bottom, goes up to a peak (relative maximum), then down to a valley (relative minimum), then up to another peak (relative maximum), then down. So the number of turning points (changes in direction) is 3. The formula for the maximum number of turning points of a polynomial of degree \(n\) is \(n - 1\). So if \(n-1 = 3\), then \(n = 4\)? Wait, no, wait, the degree: since the ends are both going down, the degree is even. Let's check the number of real zeros: the graph crosses the \(x\) - axis at 2 points (since it intersects the \(x\) - axis at two distinct points). Now, the relative minima: we can see that there is 1 relative minimum (the valley at \(x = 0\)) and 2 relative maxima (the two peaks).

For the degree: since the end - behavior is that as \(x
ightarrow\pm\infty\), \(f(x)
ightarrow-\infty\), the leading coefficient is negative and the degree is even. Also, the number of turning points: if we have a polynomial of degree \(n\), the number of turning points is at most \(n - 1\). Let's assume the degree is 4 (even). Let's verify:

• Degree: Even (since both ends go down), and the number of turning points is 3. For a degree \(n\) polynomial, the maximum number of turning points is \(n - 1\). So if \(n-1=3\), then \(n = 4\). So the degree of \(f(x)\) is 4 (even).

Step2: Determine the leading coefficient

Since as \(x
ightarrow+\infty\) and \(x
ightarrow-\infty\), \(f(x)
ightarrow-\infty\), for a polynomial \(y=a_nx^n+\cdots+a_0\), when \(n\) is even, if \(a_n<0\), the end - behavior is down on both ends. So the leading coefficient is negative.

Step3: Determine the number of real zeros

The real zeros of a polynomial are the \(x\) - intercepts (where the graph crosses the \(x\) - axis). From the graph, we can see that the graph intersects the \(x\) - axis at 2 distinct points. So there are 2 different real zeros.

Step4: Determine the number of relative minima

A relative minimum is a point where the function changes from decreasing to increasing. Looking at the graph, there is 1 such point (the valley at \(x = 0\)).

Answer:

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