Rational Functions and Asymptotes

The video tutorial covers graphing rational functions, starting with an introduction to rational functions as ratios of polynomials. It explains the parent function f(x) = 1/x and its table of values, highlighting asymptotes. The tutorial outlines steps for graphing rational functions, including finding intercepts, holes, and asymptotes. Several examples are provided to illustrate these steps, showing how to graph different rational functions. The video concludes with an invitation for further learning and engagement.

What is a rational function?

• A function that is a sum of two polynomials

• A function that is a ratio of two polynomials

• A function that is a product of two polynomials

• A function that is a difference of two polynomials

What is the parent function of rational functions?

• f(x) = x - 1

• f(x) = 1/x

• f(x) = x + 1

• f(x) = x^2

What is an asymptote?

• A point where the graph crosses the x-axis

• A line that the graph approaches but never touches

• A line that the graph intersects

• A point where the graph crosses the y-axis

What is the first step in graphing a rational function?

• Find the horizontal asymptotes

• Find the vertical asymptotes

• Find the holes

• Find the x and y-intercepts

How do you find a vertical asymptote?

• Set the function equal to zero

• Set the denominator equal to zero

• Set the x-intercept equal to zero

• Set the numerator equal to zero

In the function f(x) = (3x + 6) / (x - 1), what is the vertical asymptote?

• x = 1

• x = -1

• x = 0

• x = 3

In the function f(x) = (x - 2) / (x - 4), what is the horizontal asymptote?

• y = -1

• y = 1

• y = 2

• y = 0

For the function f(x) = 1 / (3x^2 + 3x - 18), what is the y-intercept?

• (0, 1)

• (0, 0)

• (0, 1/18)

• (0, -1/18)

In the function f(x) = (x^3 - 9x) / (3x^2 - 6x - 9), what is the slant asymptote?

• y = 2x - 1

• y = -1/3x + 1

• y = 1/3x + 2/3

• y = x + 1

For the function f(x) = -4 / (x^2 - 3x), what are the vertical asymptotes?

• x = 0 and x = 3

• x = -3 and x = 3

• x = 0 and x = -3

• x = 3 and x = -1