Pg. 222 Homework• Pg. 223 #31 – 43 odd• Pg. 224 #48• Pg. 234 #1

• #1 (-∞,-1)U(-1, 2)U(2, ∞) #3 (-∞,-3)U(-3, 1)U(1, ∞) • #5 (-∞,-1)U(-1, 1)U(1, ∞)• #7 (-∞, 2 – √5)U(2 – √5, 2 + √5)U(2 + √5, ∞)• #9 (-∞,-1)U(-1, 0)U(0, 1)U(1, ∞) • #11 y = 0 #13 y = -4 #15 y = -12• #17 y = 0, x = -1 #19 y = 4, x = -1 #21 y = 0, x = 4• #23 right 3; y = 0, x = 3• #25 stretch 2, reflect x – axis, left 5; y = 0, x = -5• #27 left 1, down 3; y = -3, x = -1• #29 stretch 5, reflect x – axis, right 1; y = 0, x = 1

4.1/4.2 Rational Functions, Asymptotes and Graphs

Basic Information• Domain is determined by existence

of the denominator.• Horizontal Asymptotes tell you the

potential end behavior of a function. They can be crossed “in the middle.”

• Vertical Asymptotes occur when the function heads to ±∞ from either side of a value. They can not be crossed.

• Vertical Asymptotes are found when the denominator is set equal to zero and solved.

• Horizontal Asymptotes are determined by the degrees in the numerator and denominator.– Top Heavy means there is no

Horizontal Asymptote, but there is a Hidden Asymptote of sorts.

– Bottom Heavy means the Horizontal Asymptote is y = 0

– Equal Degrees means the Horizontal Asymptote is where a and b are coefficients of the degree terms in the numerator and denominator

ayb

4.1/4.2 Rational Functions, Asymptotes and Graphs

Steps to Graphing Rational Functions

• Determine both the x and y intercepts. (How?)

• Find the Vertical Asymptotes.• Find the End Behavior, and

any Horizontal Asymptotes.• Find values of f(x) on either

side of the vertical asymptote(s) and use the calculator to help finish the graph!

Examples• Graph the following rational

functions using the given steps provided to the left.

2

1

4 3

xf x

x x

2

2

3 12

1

xg x

x

4.1/4.2 Rational Functions, Asymptotes and Graphs

Hidden Asymptotes• Sometimes there will be an

asymptote that is not perfectly horizontal or vertical. This occurs when the degrees are “Top Heavy”

• To find the “hidden” asymptote when the functions are “Top Heavy” you must divide the functions out.

Examples• Graph the following rational

function using the given steps provided.

3 2

2

2 7 4

2 3

x xf x

x x

4.1/4.2 Rational Functions, Asymptotes and Graphs

• Use the information you just learned to algebraically determine why

has only two horizontal asymptotes and no vertical asymptotes.

• What can you do to a function like

to see all the asymptotes without basing you decision off of shifts?

1

2

xf x

x

31

5 2g x

x