Question

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

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

Explanation:

Step1: Determine the degree

The end - behavior of a polynomial: If the leading coefficient is positive, as \(x
ightarrow\pm\infty\), the function behaves like \(a_nx^n\) (where \(a_n\) is the leading coefficient and \(n\) is the degree). Here, as \(x
ightarrow+\infty\) and \(x
ightarrow-\infty\), the function \(f(x)\) goes to \(-\infty\). For a polynomial, if the degree \(n\) is even, and the leading coefficient \(a_n<0\), the end - behavior is \(x
ightarrow\pm\infty,f(x)
ightarrow-\infty\). Also, the number of turning points (local maxima and minima) of a polynomial \(y = f(x)\) of degree \(n\) is at most \(n - 1\). The graph has 3 turning points (2 local maxima and 1 local minimum), so \(n-1\geq3\), so \(n\geq4\). Since the end - behavior is consistent with an even degree (both ends go to the same direction) and the leading coefficient negative, and the number of turning points suggests a degree of 4 (a quartic polynomial, degree 4 has at most 3 turning points). So the degree is 4 (even).

Step2: Determine the leading coefficient

From the end - behavior, as \(x
ightarrow\pm\infty\), \(f(x)
ightarrow-\infty\). For a polynomial \(f(x)=a_nx^n + a_{n - 1}x^{n-1}+\cdots+a_0\), when \(n\) is even, if \(a_n<0\), then as \(x
ightarrow\pm\infty\), \(f(x)
ightarrow-\infty\). So the leading coefficient is negative.

Step3: Determine the number of real zeros

The real zeros of a polynomial are the \(x\) - intercepts. Looking at the graph, it intersects the \(x\) - axis at 2 distinct points. So there are 2 different real zeros.

Step4: Determine the number of relative maxima

A relative maximum is a point where the function changes from increasing to decreasing. Looking at the graph, there are 2 "peaks", so there are 2 relative maxima.

Answer:

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