1. Unit 4: Graphing Rational Equations
Graph each rational equation in your calculator and Sketch the graph.
[1]
[2]
[3]
[4]
[5]
[6]
Math 3 Hon: Unit 4

2. Observations based on the graphs #1 – 6:
Investigation: Graphing Rational Equations
Observations based on the graphs #1 – 6:
(1) What is different about these graphs from previous functions that you have drawn?
(2) Is there any relationship you see between the numerator and/or denominator with the behavior of the graph?

3. Graphing Rational Equations By Hand
Basic Steps
1. Factor Numerator and Denominator
2. Determine ZERO(S) of
Denominator and Numerator
3. Determine the types of Asymptotes and Discontinuity (Use zeros to help)
4. DRAW Asymptotes
5. GRAPH based on known values
or positive/negative sections
Math 3 Hon: Unit 4

4. Special Behavior in Rational Equations
#1: Vertical Asymptotes
x = a is a vertical asymptote if f(a) is undefined and a is a zero value of the denominator of f(x) only.
As x approaches a from the left or right side, f(x) approaches either ±∞ “Boundary you follow along”
Vertical Asymptotes: Zeros of Denominator that do not cancel

5. Special Behavior in Rational Equations
#2: Points of Discontinuity (Holes in Graph)
x = a is a point of discontinuity if f(a) is undefined
a is a zero value of the numerator and denominator of f(x).
Factor (x – a) can be reduced completely (cancel) from f(x)
HOLES: Zeros of Denominator that cancel

6. Special Behavior in Rational Equations
#3 Horizontal Asymptotes:
y = b is a horizontal asymptote if the end behavior of f(x) as x approaches positive or negative infinity is b.
Note: f(x) = b on a specific domain, but is predicted not approach farther left and farther right
Case 1: Degree of denominator is LARGER than degree of numerator
Horizontal Asymptote: y = 0 (x – axis)
Case 2: Degree of denominator is SAME AS degree of numerator
Horizontal Asymptote: y = fraction of LEADING coefficients
Case 3: Degree of denominator is SMALLER than degree of numerator
No Horizontal Asymptote: f(x) → ± ∞

7. Example 1: Sketch two possible graphs based on each description
Vertical: x = 2
Horizontal: y = - 3
Discontinuity: x = - 4
[2]
Vertical: x = -4, x = 0
Horizontal: y = 0
Discontinuity: x = 2

8. Example 2: Determine the asymptotes and discontinuity values for the given rational equation and plot them on the given axes
FACTOR NUMERATOR AND DENOMINATOR!!!! a)

9. Example 2 Continued b)

10. Example 2: Continued c)

11. Example 3: Determine the asymptotes and discontinuity values for the given rational equationb) a)

12. c) d)