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 $\boldsymbol{\vee}$ and the leading coefficient is $\boldsymbol{\vee}$. there are $\boldsymbol{\square}$ different real zeros and $\boldsymbol{\square}$ relative maximums.

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 up (as in this case, as \(x
ightarrow+\infty\) and \(x
ightarrow - \infty\), \(y
ightarrow+\infty\)), the degree \(n\) is even. Also, the number of turning points (local maxima and minima) of a polynomial is at most \(n - 1\). Looking at the graph, we can see that there are 3 turning points (2 minima and 1 maximum). So, if the number of turning points is \(n - 1=3\), then \(n = 4\)? Wait, no. Wait, the number of turning points: let's count again. Wait, the graph has a shape with two "valleys" and one "hill" in between? Wait, no, looking at the graph, the number of turning points (where the graph changes from increasing to decreasing or vice - versa) is 3? Wait, no, actually, for a polynomial of degree \(n\), the maximum number of turning points is \(n - 1\). Also, the end - behavior: both ends going up means the leading coefficient is positive and the degree is even. Now, let's look at the number of real zeros: the graph crosses the \(x\) - axis at 2 points, so there are 2 different real zeros. Now, the number of relative maxima: a relative maximum is a point where the graph changes from increasing to decreasing. Looking at the graph, there is 1 relative maximum (the "hill" part between the two valleys). Now, for the degree: since the end - behavior is both ends up (even degree) and the number of turning points: let's see, if we have a polynomial of degree 4, the maximum number of turning points is 3. Let's check the graph: it has 3 turning points (2 minima and 1 maximum). So the degree is 4 (even), leading coefficient is positive (because the ends go up), number of real zeros is 2, number of relative maxima is 1.

Wait, let's re - examine:

1. Degree: The end behavior: as \(x

ightarrow\pm\infty\), \(y
ightarrow+\infty\). So the degree \(n\) is even. The number of turning points (local extrema) is 3. For a polynomial, the number of turning points \(T\) satisfies \(T\leq n - 1\). If \(T = 3\), then \(n-1\geq3\), so \(n\geq4\). Since the end behavior is even - degree, and the graph has 3 turning points, the degree is 4 (a quartic polynomial).

1. Leading coefficient: Since the end behavior is \(y

ightarrow+\infty\) as \(x
ightarrow\pm\infty\) and the degree is even, the leading coefficient is positive.

1. Number of real zeros: The graph intersects the \(x\) - axis at 2 distinct points, so there are 2 different real zeros.

1. Number of relative maxima: A relative maximum is a point where the function changes from increasing to decreasing. Looking at the graph, there is 1 such point (the peak of the "hill" between the two valleys).

Step2: Analyze each part

• Degree: Even, with 3 turning points, so degree is 4 (since \(n-1 = 3\) implies \(n = 4\)).
• Leading coefficient: Positive (because both ends of the graph go up, for an even - degree polynomial, if leading coefficient is positive, ends go up).
• Number of different real zeros: The graph crosses the \(x\) - axis at 2 distinct points, so 2.
• Number of relative maxima: 1 (the point where the graph changes from increasing to decreasing).

Answer:

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