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Graphing Rational Equations By Hand

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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

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”

#1: x = a

#2: x = a

#3: x = a

#4: x = a

Examples: Zeros of Denominator that do not cancel

Vertical Asymptotes at x = 1 and x = -6

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 from f(x)

#4: x = a

#1: x = a

#2: x = a

#3: x = a

Example:Zeros of Denominator that cancel

Point of Discontinuity at x = -3

Vertical Asymptote at x = 5

Special Behavior in Rational Equations

#3 Horizontal Asymptotes:

- y = b is a horizontal 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 thandegree 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 thandegree of numerator

No Horizontal Asymptote: f(x) → ± ∞

Example 1:Identify the horizontal and vertical asymptotes, and any points of discontinuity.

0

0

[#1] Vertical:

Horizontal:

Discontinuity:

[#2] Vertical:

Horizontal:

Discontinuity:

0

Example 1:Continued

0

[#3] Vertical:

Horizontal:

Discontinuity:

[#4] Vertical:

Horizontal:

Discontinuity:

Example 1:Continued

0

[#5] Vertical: Horizontal:Discontinuity:

Example 1:Continued

0

[#6] Vertical: Horizontal:Discontinuity:

Example 2:Sketch two possible graphs based on each description

[1] Vertical: x = 2

Horizontal: y = - 3

Discontinuity: x = - 2

[2] Vertical: x = -2, x = 1

Horizontal: y = 0

Discontinuity: x = 2

Example 2:continued

[3] Vertical: x = -2

Horizontal: y = 3

Discontinuity: None

[4] Vertical: x = 2, x = -1

Horizontal: y = - 2

Discontinuity: x = 0