Understanding and Graphing Rational Functions

This video tutorial covers the basics of rational functions, including their definition, how to determine their domain, and how to graph them. It explains the importance of avoiding division by zero and demonstrates how to identify vertical asymptotes and holes in the graph. The tutorial also introduces an interactive tool for exploring the effects of changing coefficients on the graph of a rational function. Several examples are provided to illustrate the process of finding domains and graphing rational functions, emphasizing the identification of common factors and their impact on the graph.

What is a rational function?

• A function that is always linear

• A function defined by the ratio of two polynomials

• A function with no restrictions on its domain

• A function that cannot be graphed

Why is the domain of a rational function restricted?

• Because the numerator cannot be zero

• Because the function must be linear

• Because division by zero is undefined

• Because the function must be quadratic

How do you find the domain of a rational function?

• By setting the numerator equal to zero

• By graphing the function

• By setting the denominator equal to zero and solving

• By finding the x-intercepts

What values must be excluded from the domain of a rational function?

• All positive values

• Values that make the numerator zero

• Values that make the denominator zero

• All negative values

What is a vertical asymptote in the context of rational functions?

• A point where the graph crosses the x-axis

• A line where the graph approaches but never touches

• A line that the graph intersects at multiple points

• A point where the graph crosses the y-axis

What causes a hole in the graph of a rational function?

• A zero in the numerator that is not in the denominator

• A zero in the denominator only

• A zero in the numerator only

• A zero in the denominator that is also a zero in the numerator

How can you explore the effects of changing parameters in a rational function?

• By manually calculating each point

• By using an interactive applet with sliders

• By using a graphing calculator

• By drawing the graph on paper

What happens to the graph of a rational function when you change the value of 'D' in the denominator?

• The horizontal asymptote shifts

• The graph becomes a parabola

• The vertical asymptote shifts

• The graph becomes a straight line

In the example of graphing a rational function, what is the significance of a common factor in the numerator and denominator?

• It indicates a point of intersection

• It indicates a horizontal asymptote

• It indicates a hole in the graph

• It indicates a vertical asymptote

What is the domain of a rational function with a hole at x = 3 and a vertical asymptote at x = -2?

• All real numbers except 3

• All real numbers

• All real numbers except -2

• All real numbers except 3 and -2