Question

choose the end behavior of the graph of each polynomial function.

(a) choose the end behavior for \\(f(x) = 2x^5 + x^4 - 8x^2 - 7\\).

• the graph of \\(f\\) falls to the left and rises to the right.
• the graph of \\(f\\) rises to the left and falls to the right.
• the graph of \\(f\\) rises to the left and rises to the right.
• the graph of \\(f\\) falls to the left and falls to the right.

(b) choose the end behavior for \\(f(x) = -2x^4 + 7x^3 + 4x^2 - 3x\\).

• the graph of \\(f\\) falls to the left and rises to the right.
• the graph of \\(f\\) rises to the left and falls to the right.
• the graph of \\(f\\) rises to the left and rises to the right.
• the graph of \\(f\\) falls to the left and falls to the right.

(c) choose the end behavior for \\(f(x) = -4x(x - 4)(x + 2)\\).

• the graph of \\(f\\) falls to the left and rises to the right.
• the graph of \\(f\\) rises to the left and falls to the right.
• the graph of \\(f\\) rises to the left and rises to the right.
• the graph of \\(f\\) falls to the left and falls to the right.

Explanation:

Determine the end behavior of function (a)

The given function is:

$$f(x) = 2x^5 + x^4 - 8x^2 - 7$$

The leading term is \(2x^5\).

• The degree of the polynomial is \(n = 5\), which is odd.
• The leading coefficient is \(a_n = 2\), which is positive.

For an odd-degree polynomial with a positive leading coefficient:

• As \(x \to -\infty\), \(f(x) \to -\infty\) (falls to the left).
• As \(x \to \infty\), \(f(x) \to \infty\) (rises to the right).

Determine the end behavior of function (b)

The given function is:

$$f(x) = -2x^4 + 7x^3 + 4x^2 - 3x$$

The leading term is \(-2x^4\).

• The degree of the polynomial is \(n = 4\), which is even.
• The leading coefficient is \(a_n = -2\), which is negative.

For an even-degree polynomial with a negative leading coefficient:

• As \(x \to -\infty\), \(f(x) \to -\infty\) (falls to the left).
• As \(x \to \infty\), \(f(x) \to -\infty\) (falls to the right).

Determine the end behavior of function (c)

The given function is:

$$f(x) = -4x(x - 4)(x + 2)$$

To find the leading term, we multiply the leading terms of each factor:

$$\text{Leading term} = -4x \cdot x \cdot x = -4x^3$$

• The degree of the polynomial is \(n = 3\), which is odd.
• The leading coefficient is \(a_n = -4\), which is negative.

For an odd-degree polynomial with a negative leading coefficient:

• As \(x \to -\infty\), \(f(x) \to \infty\) (rises to the left).
• As \(x \to \infty\), \(f(x) \to -\infty\) (falls to the right).

Answer:

Question (a)

• The graph of \(f\) falls to the left and rises to the right. (Correct answer)
• The graph of \(f\) rises to the left and falls to the right.
• The graph of \(f\) rises to the left and rises to the right.
• The graph of \(f\) falls to the left and falls to the right.

Question (b)

• The graph of \(f\) falls to the left and rises to the right.
• The graph of \(f\) rises to the left and falls to the right.
• The graph of \(f\) rises to the left and rises to the right.
• The graph of \(f\) falls to the left and falls to the right. (Correct answer)

Question (c)

• The graph of \(f\) falls to the left and rises to the right.
• The graph of \(f\) rises to the left and falls to the right. (Correct answer)
• The graph of \(f\) rises to the left and rises to the right.
• The graph of \(f\) falls to the left and falls to the right.