Rational Functions and Polynomial Zeros

Consider the function f(x) = 7x^4 and identify all key characteristics. Fill in the following: Domain, Range, x-intercept(s), y-intercept, Increasing Interval(s), Decreasing Interval(s), End Behavior.

• Domain: All real numbers, Range: [0, ∞), x-intercept(s): None, y-intercept: (0,0), Increasing Interval(s): (0, ∞), Decreasing Interval(s): (-∞, 0), End Behavior: As x → ±∞, f(x) → ∞

• Domain: All real numbers, Range: [0, ∞), x-intercept(s): (0,0), y-intercept: (0,0), Increasing Interval(s): (0, ∞), Decreasing Interval(s): None, End Behavior: As x → ±∞, f(x) → ∞

• Domain: All real numbers, Range: [0, ∞), x-intercept(s): None, y-intercept: (0,0), Increasing Interval(s): None, Decreasing Interval(s): (-∞, 0), End Behavior: As x → ±∞, f(x) → -∞

• Domain: All real numbers, Range: [0, ∞), x-intercept(s): (0,0), y-intercept: (0,0), Increasing Interval(s): (-∞, 0), Decreasing Interval(s): (0, ∞), End Behavior: As x → ±∞, f(x) → ∞

Consider the function f(x) = -2x^3 and identify all key characteristics. Fill in the following: Domain, Range, x-intercept(s), y-intercept, Increasing Interval(s), Decreasing Interval(s), End Behavior.

• Domain: All real numbers, Range: All real numbers, x-intercept: (0,0), y-intercept: (0,0), Increasing Interval: None, Decreasing Interval: (-∞, ∞), End Behavior: As x → ±∞, f(x) → ∓∞

• Domain: All real numbers, Range: y ≤ 0, x-intercept: (0,0), y-intercept: (0,0), Increasing Interval: (-∞, 0), Decreasing Interval: (0, ∞), End Behavior: As x → ±∞, f(x) → ±∞

• Domain: All real numbers, Range: y ≥ 0, x-intercept: (0,0), y-intercept: (0,0), Increasing Interval: (0, ∞), Decreasing Interval: (-∞, 0), End Behavior: As x → ±∞, f(x) → ∓∞

• Domain: All real numbers, Range: All real numbers, x-intercept: (0,0), y-intercept: (0,0), Increasing Interval: (0, ∞), Decreasing Interval: (-∞, 0), End Behavior: As x → ±∞, f(x) → ±∞

Consider the function f(x) = x^3 + 11x^2 + 35x + 27 and identify all key characteristics. Fill in the following: Domain, Range, x-intercept(s), y-intercept, Increasing Interval(s), Decreasing Interval(s), End Behavior.

• Domain: All real numbers, Range: All real numbers, x-intercept(s): -1, -3, -9, y-intercept: 27, Increasing Interval(s): (-∞, -3) ∪ (0, ∞), Decreasing Interval(s): (-3, 0), End Behavior: As x → ±∞, f(x) → ±∞

• Domain: All real numbers, Range: All real numbers, x-intercept(s): -1, -3, -9, y-intercept: 27, Increasing Interval(s): (-∞, -1) ∪ (1, ∞), Decreasing Interval(s): (-1, 1), End Behavior: As x → ±∞, f(x) → ±∞

• Domain: All real numbers, Range: All real numbers, x-intercept(s): -1, -3, -9, y-intercept: 27, Increasing Interval(s): (-∞, -9) ∪ (-3, ∞), Decreasing Interval(s): (-9, -3), End Behavior: As x → ±∞, f(x) → ±∞

• Domain: All real numbers, Range: All real numbers, x-intercept(s): -1, -3, -9, y-intercept: 27, Increasing Interval(s): (-∞, -9) ∪ (0, ∞), Decreasing Interval(s): (-9, 0), End Behavior: As x → ±∞, f(x) → ±∞

Which of the following is the correct domain of the function f(x) = x^4 - 4x^2 - 1?

• All real numbers

• x ≥ 0

• x ≤ 0

• x ≠ 0

• The function has a vertical asymptote at x = 2.

• The function has a horizontal asymptote at y = 1/3.

• The function has a hole at x = -2.

• The function is increasing on the interval (-∞, -1).

For the function f(x) = (x^2 - 3x - 4)/(x^2 + 3x), which of the following is true about its vertical asymptote?

• x = -3 and x = 0

• x = 3 and x = 0

• x = 4 and x = -3

• x = -4 and x = 3

For the function f(x) = \frac{2x^3 + 8x^2 - 10x}{2x^2 + 2x} , which of the following is a Vertical Asymptote (VA)?

• x = -1

• x = 0

• x = 1

• x = 2

For the function f(x) = (3x^2 - 27)/(x^3 - 6x^2 - 9x + 54), which of the following is a Vertical Asymptote (VA)?

• x = 3

• x = -3

• x = 6

• x = -6

Given the graph of each polynomial function below, determine the sign of the leading coefficient and whether the function has an even or odd degree. (Question 9)

• The leading coefficient is positive and the degree is even.

• The leading coefficient is positive and the degree is odd.

• The leading coefficient is negative and the degree is even.

• The leading coefficient is negative and the degree is odd.

Given the graph of each polynomial function below, determine the sign of the leading coefficient and whether the function has an even or odd degree. (Question 10)

• The leading coefficient is positive and the degree is even.

• The leading coefficient is positive and the degree is odd.

• The leading coefficient is negative and the degree is even.

• The leading coefficient is negative and the degree is odd.