Algebra 91 - Rational Functions and Vertical Asymptotes

This lecture by Professor Von Schmoohawk introduces rational functions, which are ratios of polynomial functions. It covers graphing these functions, focusing on vertical asymptotes and their impact on the domain. The lecture explains how to identify vertical asymptotes by setting the denominator to zero and discusses cases with multiple asymptotes. It also explores conditions where rational functions have no vertical asymptotes, such as when the denominator is constant or has no real zeros. Finally, it examines how the numerator affects graph intersections with the X-axis.

What is a rational function?

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

• A function that is undefined.

• A function that is always constant.

• A function that has no variables.

Why can't the denominator of a rational function be zero?

• Because division by zero is undefined.

• Because it would make the function quadratic.

• Because it would make the function constant.

• Because it would make the function linear.

What happens to the graph of 1/X as X approaches zero from the right?

• The graph approaches zero.

• The graph becomes negative.

• The graph grows positively without bound.

• The graph intersects the X-axis.

What is a vertical asymptote?

• A line that the graph approaches but never touches.

• A line that the graph crosses.

• A point where the graph intersects the X-axis.

• A point where the graph intersects the Y-axis.

How can you determine the location of vertical asymptotes?

• By finding where the numerator is zero.

• By finding where the function is constant.

• By finding where the denominator is zero.

• By finding where both numerator and denominator are zero.

What is the result of factoring the denominator X^2 - X - 6?

• X + 3 and X - 2

• X - 2 and X + 3

• X + 2 and X - 3

• X - 3 and X + 2

What condition allows a rational function to have no vertical asymptotes?

• The numerator and denominator are equal.

• The numerator is a constant.

• The denominator is a constant.

• The function is linear.

What happens when the numerator of a rational function is zero?

• The graph becomes undefined.

• The graph intersects the Y-axis.

• The graph has a vertical asymptote.

• The graph intersects the X-axis.

What is the domain of the function X^2 + 3 over 2?

• All positive numbers.

• All real numbers.

• All negative numbers.

• All real numbers except zero.

What is the result of setting the denominator X^2 + 1 to zero?

• X equals plus or minus 2.

• X equals zero.

• X equals plus or minus 1.

• X equals plus or minus i.