Question

what is the end behavior of the polynomial function shown in the graph? select one ① left up, right down ② left down, right up ③ both up ④ both down

Explanation:

To determine the end - behavior of a polynomial function from its graph, we analyze the direction the graph is heading as \(x\to-\infty\) (left - hand side) and as \(x\to+\infty\) (right - hand side).

Step 1: Recall the rules for end - behavior of polynomials

The end - behavior of a polynomial function \(y = a_nx^n+a_{n - 1}x^{n - 1}+\cdots+a_1x + a_0\) is determined by the leading term \(a_nx^n\), where \(n\) is the degree of the polynomial (the highest power of \(x\)) and \(a_n\) is the leading coefficient.

• If the degree \(n\) is even:
• If \(a_n>0\), as \(x\to-\infty\) and \(x\to+\infty\), \(y\to+\infty\) (both ends up).
• If \(a_n < 0\), as \(x\to-\infty\) and \(x\to+\infty\), \(y\to-\infty\) (both ends down).
• If the degree \(n\) is odd:
• If \(a_n>0\), as \(x\to-\infty\), \(y\to-\infty\) (left down) and as \(x\to+\infty\), \(y\to+\infty\) (right up).
• If \(a_n < 0\), as \(x\to-\infty\), \(y\to+\infty\) (left up) and as \(x\to+\infty\), \(y\to-\infty\) (right down).

Step 2: Analyze the options

We assume we can visually inspect the graph (even though the graph is not fully shown, we can use the general rules and the options).

• Option 1: "Left up, right down" occurs when the degree is odd and the leading coefficient is negative.
• Option 2: "Left down, right up" occurs when the degree is odd and the leading coefficient is positive.
• Option 3: "Both up" occurs when the degree is even and the leading coefficient is positive.
• Option 4: "Both down" occurs when the degree is even and the leading coefficient is negative.

Since the most common case for a polynomial graph (especially for a cubic - like behavior which is odd - degree) with a positive leading coefficient has the end - behavior of left down and right up.

Answer:

1. Left down, right up