Question

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 $\square$ different real zeros and $\square$ relative maximums.
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Explanation:

Step1: Determine the degree (even/odd)

The end - behavior of a polynomial function is determined by the degree (even or odd) and the leading coefficient. For a polynomial \(y = a_nx^n+\cdots+a_1x + a_0\), if the degree \(n\) is odd:

• If the leading coefficient \(a_n>0\), as \(x

ightarrow+\infty\), \(y
ightarrow+\infty\) and as \(x
ightarrow-\infty\), \(y
ightarrow-\infty\).

• If the leading coefficient \(a_n < 0\), as \(x

ightarrow+\infty\), \(y
ightarrow-\infty\) and as \(x
ightarrow-\infty\), \(y
ightarrow+\infty\).

If the degree \(n\) is even:

• If the leading coefficient \(a_n>0\), as \(x

ightarrow\pm\infty\), \(y
ightarrow+\infty\).

• If the leading coefficient \(a_n < 0\), as \(x

ightarrow\pm\infty\), \(y
ightarrow-\infty\).

Looking at the graph, as \(x
ightarrow+\infty\), \(y
ightarrow-\infty\) and as \(x
ightarrow-\infty\), \(y
ightarrow+\infty\). So the degree of the polynomial is odd.

Step2: Determine the leading coefficient sign

Since the end - behavior is \(x
ightarrow+\infty,y
ightarrow-\infty\) and \(x
ightarrow-\infty,y
ightarrow+\infty\) (which is the behavior of an odd - degree polynomial with a negative leading coefficient), the leading coefficient is negative.

Step3: Count the number of real zeros

A real zero of a polynomial function \(y = f(x)\) is a value of \(x\) for which \(f(x)=0\), i.e., the \(x\) - intercepts of the graph. From the graph, we can see that the graph intersects the \(x\) - axis at 3 distinct points. So there are 3 different real zeros.

Step4: Count the number of relative maximums

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

Answer:

The degree of \(f(x)\) is \(\text{odd}\) and the leading coefficient is \(\text{negative}\). There are \(\boldsymbol{3}\) different real zeros and \(\boldsymbol{2}\) relative maximums.