Virginia SOL Math / Algebra II / Functions

A2.F.2

Characteristics of Advanced Functions

What you'll learn

  • Identify domain, range, and intercepts
  • Find vertical and horizontal asymptotes
  • Describe end behavior of a function

Words to know

Asymptote —
a line the graph approaches but never touches
Domain —
all allowed input values
Range —
all possible output values
End behavior —
what the graph does as x grows large
Rational function —
a ratio of two polynomials

Why it matters Asymptotes and end behavior show the limits of a model, such as a value a quantity approaches but can never reach.

Worked examples

Model ItI do

Find the asymptotes of y = 1 / (x - 2).

The denominator is 0 when x = 2, so there is a vertical asymptote at x = 2. As x grows large in either direction, the value gets closer to 0, so there is a horizontal asymptote at y = 0. The domain is all x not equal to 2.

Try It TogetherWe do

What is the domain of y = 1 / x?

The denominator cannot be 0, so x cannot be 0. The domain is all real numbers except x = 0.

Now YouYou do

Find the vertical asymptote of y = 1 / (x + 3).

Set the denominator equal to 0 and solve. Check your answer in the back of this chapter.

Practice

Mastered 0 of 11

Warm-up

Ease in with the basics.

  1. 1.Find the vertical asymptote of y = 1 / (x - 8).

  2. 2.What is the domain of y = 1 / (x + 2)?

  3. 3.Find the horizontal asymptote of y = 3 / x.

On Level

The core Quick Check from the book.

  1. 1.Find the vertical asymptote of y = 1 / (x - 5).

  2. 2.What is the domain of y = 1 / (x - 1)?

  3. 3.Find the horizontal asymptote of y = 1 / x.

  4. 4.Can a graph cross a vertical asymptote?

  5. 5.Find the vertical asymptote of y = 2 / x.

Challenge

Stretch your thinking.

  1. 1.Find the horizontal asymptote of y = (2x + 1) / (x - 3).

  2. 2.Find the vertical asymptotes of y = (x + 1) / (x^2 - 9).

  3. 3.Describe the end behavior of y = x^2 (as x approaches positive and negative infinity).