Embed Size (px)
Citation preview
1A Rational Functions.notebook
1
November 24, 2014
Nov 1311:19 AM
Reciprocal Functions
Apr 108:35 AM
The Hyperbola
• as x approaches 0, the graph approaches the y-axis (x=0),a vertical asymptote, related to the roots of the denominator
Domain: x ∈ (∞, 0) ∪ (0, ∞)
• as x approaches ±∞, the graph approaches the x-axis (y=0), a horizontal asymptote, related to limits
Range: y ∈ (∞, 0) ∪ (0, ∞)
• inversely proportional relation• as x increases, y decreases
Reciprocal of Linear Functions
Chapter
9Characteristics y = x
Domain
Range
End behaviour
End behaviour
Behaviour at x = 0
Invariant points
Horizontal asymptote
Vertical asymptote
10 8 6 4 2 0 2 4 6 8 10
1098765432
12345678910
x
y
Reciprocal of Linear Functions
Chapter
9Characteristics y = x+2
Domain
Range
End behaviour
End behaviour
Behaviour at x = 0
Invariant points
Horizontal asymptote
Vertical asymptote
10 8 6 4 2 0 2 4 6 8 10
1098765432
12345678910
x
y
Reciprocal of Quadratic Functions
Chapter
9Characteristics
Domain
Range
End behaviour
End behaviour
Behaviour at x = 0
Invariant points
Horizontal asymptote
Vertical asymptote
10 8 6 4 2 0 2 4 6 8 10
1098765432
12345678910
x
y
Reciprocal of Quadratic Functions
Chapter
9Characteristics
Domain
Range
End behaviour
End behaviour
Behaviour at x = 0
Invariant points
Horizontal asymptote
Vertical asymptote
10 8 6 4 2 0 2 4 6 8 10
1098765432
12345678910
x
y
1A Rational Functions.notebook
2
November 24, 2014
Reciprocal of Quadratic Functions
Chapter
9Characteristics
Domain
Range
End behaviour
End behaviour
Behaviour at x = 0
Invariant points
Horizontal asymptote
Vertical asymptote
10 8 6 4 2 0 2 4 6 8 10
1098765432
12345678910
x
y
Reciprocal of Quadratic Functions
Chapter
9Characteristics
Domain
Range
End behaviour
End behaviour
Behaviour at x = 0
Invariant points
Horizontal asymptote
Vertical asymptote
10 8 6 4 2 0 2 4 6 8 10
1098765432
12345678910
x
y
Apr 108:35 AM
Are you seeing a pattern?
Nov 249:05 AM
Nov 249:06 AM Nov 249:06 AM
1A Rational Functions.notebook
3
November 24, 2014
Nov 249:07 AM Nov 249:07 AM
Nov 249:07 AM Nov 249:07 AM
Nov 249:07 AM Nov 249:08 AM
1A Rational Functions.notebook
4
November 24, 2014
Nov 1311:19 AM
Rational Functions
Apr 108:35 AM
Rational Functions
A rational function is a fraction made of polynomials.
, where q(x)≠0
• any roots that occur in the numerator and the denominator cause points of discontinuity (PoD)
• roots of the numerator are xintercepts• roots of the denominator are vertical asymptotes• substitute x=0 to find the yintercept• a horizontal / oblique asymptote is determined by
Apr 108:35 AM
Horizontal / Oblique Asymptote
There are three situations for determining the value of the horizontal / oblique asymptote:
1. If the highest exponent is in the denominator, then
HA: y = 0
2. If the highest exponent is in the numerator and the denominator, then
HA: y = ratio of the coefficients
3. If the highest exponent is in the numerator, then
OA: y = g(x)
Apr 108:35 AM
End Behaviour
A function may cross over a horizontal / oblique asymptote before .
To determine if a horizontal / oblique asymptote is crossed: • Substitute a large value for x and calculate the value of the
function.
• Compare the value of the function to the last detail of our graph to determine if the horizontal / oblique asymptote has been crossed.
Nov 191:34 PM
Example:
PoD: x=
xint: x=
VA: x=
yint: y=
HA/OA: y=
Domain:
Range:
Nov 191:34 PM
Example:
PoD: x=
xint: x=
VA:x=
yint: y=
HA/OA: y=
Domain:
Range:
1A Rational Functions.notebook
5
November 24, 2014
Nov 191:34 PM
Example 2:
PoD: x=
xint: x=
VA: x=
yint: y=
HA/OA: y=
Domain:
Range:
Nov 191:34 PM
Example 2:
PoD: x=
xint: x=
VA:x=
yint: y=
HA/OA: y=
Domain:
Range:
Nov 191:34 PM
Example 3:
PoD: x=
xint: x=
VA: x=
yint: y=
HA/OA: y=
Domain:
Range:
Nov 191:34 PM
Example 3:
PoD: x=
xint: x=
VA:x=
yint: y=
HA/OA: y=
Domain:
Range:
Nov 172:40 PM
3 1 7 103 30
1 10 40
Nov 191:34 PM
Example 4:
PoD: x=
xint: x=
VA: x=
yint: y=
HA/OA: y=
Domain:
Range:
1A Rational Functions.notebook
6
November 24, 2014
Nov 191:34 PM
Example 4:
PoD: x=
xint: x=
VA:x=
yint: y=
HA/OA: y=
Domain:
Range:
Nov 191:34 PM
Example 5:
PoD: x=
xint: x=
VA: x=
yint: y=
HA/OA: y=
Domain:
Range:
Nov 191:34 PM
Example 5:
PoD: x=
xint: x=
VA:x=
yint: y=
HA/OA: y=
Domain:
Range:
Nov 191:34 PM
Example 6:
PoD: x=
xint: x=
VA: x=
yint: y=
HA/OA: y=
Domain:
Range:
Nov 191:34 PM
Example 6:
PoD: x=
xint: x=
VA:x=
yint: y=
HA/OA: y=
Domain:
Range:
Nov 191:34 PM
Worksheet
1A Rational Functions.notebook
7
November 24, 2014
Nov 191:34 PM
Q1:
PoD: x=
xint: x=
VA: x=
yint: y=
HA/OA: y=
Nov 191:34 PM
Q2:
PoD: x=
xint: x=
VA: x=
yint: y=
HA/OA: y=
Nov 192:41 PM Nov 191:34 PM
Q3:
PoD: x=
xint: x=
VA: x=
yint: y=
HA/OA: y=
Nov 191:34 PM
Q4:
PoD: x=
xint: x=
VA: x=
yint: y=
HA/OA: y=
Nov 191:34 PM
Q5:
PoD: x=
xint: x=
VA: x=
yint: y=
HA/OA: y=
1A Rational Functions.notebook
8
November 24, 2014
Nov 202:39 PM Nov 191:34 PM
Q6:
PoD: x=
xint: x=
VA: x=
yint: y=
HA/OA: y=
Nov 191:34 PM
Q7:
PoD: x=
xint: x=
VA: x=
yint: y=
HA/OA: y=
Nov 191:34 PM
Q8:
PoD: x=
xint: x=
VA: x=
yint: y=
HA/OA: y=
Nov 191:34 PM
Q9:
PoD: x=
xint: x=
VA: x=
yint: y=
HA/OA: y=
Nov 191:34 PM
Q10:
PoD: x=
xint: x=
VA: x=
yint: y=
HA/OA: y=
1A Rational Functions.notebook
9
November 24, 2014
Nov 191:34 PM
Extras from text
Nov 154:06 PM
Nov 154:07 PM Nov 154:07 PM
Nov 154:07 PM Nov 154:08 PM
1A Rational Functions.notebook
10
November 24, 2014
Nov 154:08 PM Nov 154:09 PM
Nov 154:09 PM Nov 154:10 PM
Nov 154:10 PM Nov 154:11 PM
1A Rational Functions.notebook
11
November 24, 2014
Nov 202:12 PM
p.142 Q25
PoD: x=xint: x=yint: y=VA: x=HA/OA: y=
Nov 211:28 PM
PoD: x=xint: x=yint: y=VA: x=HA/OA: y=
Nov 211:33 PM
PoD: x=xint: x=yint: y=VA: x=HA/OA: y=
Nov 211:37 PM
PoD: x=xint: x=yint: y=VA: x=HA/OA: y=
