Question

12.
deg:
coeff:
justify:
15.
deg:
coeff:
justify:

Explanation:

Problem 12 (Graph 1)

Step1: Determine Degree (deg)

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 the first graph, as \(x
ightarrow+\infty\), the graph goes down, and as \(x
ightarrow-\infty\), the graph goes up. The rule for end - behavior is: if the degree \(n\) is odd, and the leading coefficient \(a_n<0\), then as \(x
ightarrow+\infty\), \(f(x)
ightarrow-\infty\) and as \(x
ightarrow-\infty\), \(f(x)
ightarrow+\infty\). Also, the number of turning points of a polynomial is at most \(n - 1\). The graph has 2 turning points, so \(n-1\geq2\), so \(n\geq3\). Since the end - behavior is consistent with an odd degree, and the number of turning points suggests a degree of at least 3. A cubic polynomial (degree 3) has at most 2 turning points, which matches the graph. So the degree is odd. Let's assume the degree is 3 (a common case for this end - behavior and number of turning points).

Step2: Determine Leading Coefficient (coeff)

From the end - behavior, as \(x
ightarrow+\infty\), \(y
ightarrow-\infty\) and as \(x
ightarrow-\infty\), \(y
ightarrow+\infty\). For a polynomial \(y = a_nx^n+\cdots\), when \(n\) is odd, if \(a_n<0\), the end - behavior is \(x
ightarrow+\infty,y
ightarrow-\infty\) and \(x
ightarrow-\infty,y
ightarrow+\infty\). So the leading coefficient is negative.

Step1: Determine Degree (deg)

For the second graph, as \(x
ightarrow+\infty\) and \(x
ightarrow-\infty\), the graph goes down. The rule for end - behavior: if the degree \(n\) is even, and the leading coefficient \(a_n<0\), then as \(x
ightarrow\pm\infty\), \(f(x)
ightarrow-\infty\). The number of turning points of a polynomial is at most \(n - 1\). The graph has 3 turning points, so \(n-1\geq3\), so \(n\geq4\). A quartic polynomial (degree 4) has at most 3 turning points, which matches the graph. Since the end - behavior is consistent with an even degree, and the number of turning points suggests a degree of at least 4. A quartic polynomial (degree 4) is a likely candidate.

Step2: Determine Leading Coefficient (coeff)

Since as \(x
ightarrow\pm\infty\), the graph goes down, for a polynomial \(y=a_nx^n+\cdots\) with even degree \(n\), if \(a_n < 0\), the end - behavior is \(x
ightarrow\pm\infty,y
ightarrow-\infty\). So the leading coefficient is negative.

Answer:

deg: 3 (odd, at least 3, cubic is likely)
coeff: negative

Problem 15 (Graph 2)