Section 7.2: Graph Simple Rational Functions

What is the significance of the degrees of the numerator and denominator in determining horizontal asymptotes?

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.

• If the degree of the numerator is greater than the degree of the denominator, the horizontal asymptote is y = infinity.

• If the degrees of the numerator and denominator are equal, the horizontal asymptote is y = leading coefficient of numerator / leading coefficient of denominator.

• The horizontal asymptote is always y = 1 regardless of the degrees.

How do you determine the x-intercepts of a rational function?

• By setting the denominator equal to zero and solving for x.

• By setting the numerator equal to zero and solving for x.

• By finding the vertex of the function.

• By evaluating the function at x = 0.

What is the general form of a rational function?

• 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)/Q(x), where P(x) and Q(x) are polynomials.

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

What is the horizontal asymptote of a rational function?

• A horizontal line that the graph approaches as x approaches infinity or negative infinity.

• A vertical line that the graph approaches as x approaches infinity.

• The point where the graph intersects the x-axis.

• The maximum value of the function as x approaches infinity.

What are the steps to graph a rational function?

• 1. Identify vertical and horizontal asymptotes, determine intercepts, analyze behavior around asymptotes, and plot additional points.

• 2. Find the roots of the function, calculate the derivative, and plot the function's critical points.

• 3. Simplify the function, find the domain, and plot the function on a graphing calculator.

• 4. Identify the function type, calculate the area under the curve, and sketch the graph.

What is a rational function?

• A function that can be expressed as the ratio of two polynomials.

• A function that is always increasing or decreasing.

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

• A function that can only be defined for integer values.

How do you find vertical asymptotes in a rational function?

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

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

• Find the limits of the function as x approaches infinity.

• Differentiate the function and find critical points.

What does it mean for a function to have no vertical asymptotes?

• A function has no vertical asymptotes if the denominator does not have any real roots, meaning it does not equal zero for any real value of x.

• A function has no vertical asymptotes if it is a linear function with a non-zero slope.

• A function has no vertical asymptotes if it approaches a horizontal line as x approaches infinity.

• A function has no vertical asymptotes if it is defined for all real numbers.

What is the difference between vertical and horizontal asymptotes?

• Vertical asymptotes are horizontal lines that the function approaches as x goes to infinity, while horizontal asymptotes are vertical lines where the function approaches infinity.

• Vertical asymptotes are vertical lines where the function approaches infinity, while horizontal asymptotes are horizontal lines that the function approaches as x goes to infinity.

• Vertical asymptotes are points where the function is undefined, while horizontal asymptotes indicate the end behavior of the function.

• Vertical asymptotes are lines that the function never touches, while horizontal asymptotes are lines that the function can cross.

What happens to the graph of a rational function at its vertical asymptotes?

• The graph approaches a finite value, indicating continuity.

• The graph approaches infinity or negative infinity, indicating a discontinuity.

• The graph becomes a straight line at the asymptote.

• The graph oscillates between positive and negative values.