solving for discontinuities algebraically
DESCRIPTION
Solving for Discontinuities Algebraically. 16 – 17 November 2010. Always Factor!. The 1 st step → always factor the numerator and the denominator!!! Goal: Get matching factors in numerator and denominator. Vertical Asymptotes. Occur when the denominator equals zero . - PowerPoint PPT PresentationTRANSCRIPT
Solving for Discontinuities
Algebraically
16 – 17 November 2010
Always Factor!
The 1st step → always factor the numerator and the denominator!!!
Goal: Get matching factors in numerator and denominator
1
)1)(5(1
542
x
xxy
x
xxy
Vertical Asymptotes
Occur when the denominator equals zero. Step 1: Factor the numerator and the
denominator Step 2: Set the denominator equal to zero Step 3: Solve for x Step 4: Write your answers in the form x =
Example:
x
xx
xxy
x
xxy
1
011
)1)(2(1
22
Your Turn:
Complete problems 1 – 5 on the “Solving for the Discontinuities of Rational Equations” handout.
Removable Discontinuities
Occur when Shortcut!
Factors that occur in both the numerator and the denominator
0
0y
1
)1)(5(1
542
x
xxy
x
xxy
Removable Discontinuities, cont.
Step 1: Factor the numerator and the denominator
Step 2: Identify factors that occur in both the numerator and the denominator
Step 3: Set the common factors equal to zero
Step 4: Solve for x Step 5: Write your answers in the form x =
Example:
)2(
)2)(2(2
42
x
xxy
x
xy
2:
02
xHole
x
Your Turn:
Complete problems 6 – 10 on the “Solving for the Discontinuities of Rational Equations” handout.
Vertical Asymptote vs. Removable Discontinuity
Algebraically, they act similarly
Consider:
3
3
2
)2(
)2)(2(
)2(
44
x
xxy
x
xxy
Vertical Asymptote vs. Removable Discontinuity, cont.
3
3
2
)2(
)2)(2(
)2(
44
x
xxy
x
xxy
!!!0
00
484
)22(
4)2(4)2(
2
3
3
2
y
y
y
x
Vertical Asymptote vs. Removable Discontinuity, cont.
Think-Pair-Share
1. 30 sec – Individually think about why the equation has a vertical asymptote instead of a removable discontinuity.
2. 1 min – Talk about this with your partner.
3. Share your reasoning with the class.
Vertical Asymptote vs. Removable Discontinuity, cont.
)2)(2)(2(
)2)(2(
)2(
)2)(2(
)2(
44
3
3
2
xxx
xxy
x
xxy
x
xxy
2:
022
1
xVA
xx
y
Vertical Asymptote vs. Removable Discontinuity, cont.
Depends on: How many times a factor occurs Where the factor occurs
Removable Discontinuity → the multiplicity of the factor in the numerator ≥ the multiplicity of the factor in the denominator
Vertical Asymptote → the multiplicity of the factor in the numerator < the multiplicity of the factor in the denominator
Vertical Discontinuity vs. Removable Discontinuity, cont.
Common Factor: Common Factor:
Multiplicity Greater in Numerator or Denominator?
Multiplicity Greater in Numerator or Denominator?
Type of Discontinuity: Type of Discontinuity:
1
)1)(2(
x
xxy
)8)(8)(2(
8
xxx
xy
Your Turn:
Complete problems 11 – 15 on the “Solving for the Discontinuities of Rational Equations” handout.
Homework
In Precalculus textbook, pg. 290: 7 – 12 Hint! You will need to use the quadratic
formula for #8.
Horizontal Asymptotes
Occurs when the degree of the numerator ≤ the degree of the denominator
If n = m → HA:
If n < m → HA: y = 0
If n > m → HA doesn’t exist
0
0
...
...
bb
aay
m
n
b
ay
Example 1
If n = m → HA:
If n < m → HA: y = 0
If n > m → HA doesn’t
exist
b
ay
7
372
x
xy
0: yHA
Example 2
If n = m → HA:
If n < m → HA: y = 0
If n > m → HA doesn’t
exist
b
ay
72
1323
4
xx
xy
HA: none
Example 3
If n = m → HA:
If n < m → HA: y = 0
If n > m → HA doesn’t
exist
b
ay
712
32423
23
xx
xxxy
3
1
12
4: yHA
Your Turn:
Complete problems 11 – 15 on the “Solving for the Discontinuities of Rational Equations” handout.