Graphing Rational Functions: Key Concepts and Techniques

The video tutorial covers graphing rational functions, focusing on asymptotes and holes. It explains horizontal and vertical asymptotes, how to identify holes, and graphing techniques for rational functions with and without tails. The tutorial also discusses domain, range, and end behavior, providing examples and encouraging practice.

What is an asymptote in the context of graphing rational functions?

• A line the graph touches at the origin

• The minimum value of the function

• The maximum value of the function

• A line the graph approaches but never touches

How is a horizontal asymptote determined for a rational function without a tail?

• By comparing the degrees of the numerator and denominator

• By subtracting the degree of the numerator from the denominator

• By adding the degrees of the numerator and denominator

• By comparing the coefficients of the highest degree terms

How can you identify a horizontal asymptote when the degrees of the numerator and denominator are the same?

• By setting the numerator equal to the denominator

• By subtracting the highest degree term from the function

• By adding the degrees of the numerator and denominator

• By comparing the leading coefficients

What is the significance of the degree of the polynomial in the numerator compared to the denominator for horizontal asymptotes?

• It indicates the number of x-intercepts

• It helps in determining the equation of the horizontal asymptote

• It has no significance

• It determines the number of horizontal asymptotes

What indicates the presence of a hole in a rational function's graph?

• A common factor in the numerator and denominator

• A higher degree in the numerator than the denominator

• A vertical asymptote

• A horizontal asymptote

How do you find the x-coordinate of a hole in a rational function?

• By setting the denominator equal to zero

• By setting the numerator equal to zero

• By setting the common factor equal to zero

• By dividing the numerator by the denominator

What is the equation for a vertical asymptote found in rational functions?

• x = value that makes the denominator 0

• x = 0

• y = value that makes the numerator 0

• y = 0

When graphing a rational function, what does the domain represent?

• All possible y-values

• All possible x-values except where vertical asymptotes occur

• Only positive x-values

• Only negative x-values

What determines the range of a rational function?

• Only the values of the vertical asymptotes

• All real numbers except values at horizontal asymptotes and holes

• The y-intercepts of the function

• The x-intercepts of the function

In the context of rational functions, what does end behavior describe?

• The behavior of the graph as it approaches the y-axis

• The behavior of the graph as x approaches positive or negative infinity

• The behavior of the graph near its intercepts

• The behavior of the graph as it crosses the x-axis