rational functions

What is the significance of the degrees of the numerator and denominator in rational functions?

• They determine the vertical asymptote of the function.

• They determine the horizontal asymptote and the end behavior of the function.

• They have no effect on the function's graph.

• They indicate the number of x-intercepts the function can have.

What is the end behavior of the function @@f(x) = \frac{3x+2}{x^2+1}@@ as x approaches infinity?

• The graph approaches y = 0 as x gets larger.

• The graph approaches y = 3 as x gets larger.

• The graph approaches y = infinity as x gets larger.

• The graph approaches y = -3 as x gets larger.

What is a rational function?

• A function that can be expressed as the quotient of two polynomials, where the denominator is not zero.

• A function that has a constant value for all inputs.

• A function that can only take integer values.

• A function that is defined only for positive numbers.

How do you determine horizontal asymptotes for rational functions?

• By comparing the coefficients of the numerator and denominator polynomials.

• By evaluating the function at x = 0.

• By comparing the degrees of the numerator and denominator polynomials.

• By finding the limits of the function as x approaches infinity.

• The graph approaches y = 0 as x gets larger.

• The graph approaches y = 5 as x gets larger.

• The graph approaches y = 2 as x gets larger.

• The graph approaches y = infinity as x gets larger.

What happens to the graph of a rational function as x approaches a vertical asymptote?

• The function's value remains constant.

• The function's value increases or decreases without bound (approaches infinity or negative infinity).

• The graph becomes a straight line.

• The function's value approaches zero.

How can you identify the x-intercepts of a rational function?

• Set the denominator equal to zero and solve for x.

• Set the numerator equal to zero and solve for x.

• Find the maximum value of the function.

• Evaluate the function at x = 0.

What is the general form of a rational function?

• f(x) = \frac{P(x)}{Q(x)} where P(x) and Q(x) are polynomials.

• f(x) = P(x) + Q(x) where P(x) and Q(x) are polynomials.

• f(x) = P(x) - Q(x) where P(x) and Q(x) are polynomials.

• f(x) = P(x) \cdot Q(x) where P(x) and Q(x) are polynomials.

What is the importance of identifying asymptotes in graphing rational functions?

• Asymptotes help to understand the behavior of the graph and guide the sketching of the function.

• Asymptotes are used to find the x-intercepts of the function.

• Asymptotes indicate the maximum value of the function.

• Asymptotes are irrelevant in graphing rational functions.

What is the relationship between the x-intercepts and vertical asymptotes of a rational function?

• X-intercepts occur where the denominator is zero, while vertical asymptotes occur where the numerator is zero.

• X-intercepts occur where the numerator is zero, while vertical asymptotes occur where the denominator is zero.

• X-intercepts and vertical asymptotes are the same thing in a rational function.

• X-intercepts occur at the same points as vertical asymptotes in a rational function.