MHF 4UI Unit 5 Day 1
Key Features of Rational Functions
Investigation – Key Features of Rational functions of the form bax
ny
and
baxmx
y
A rational function has the form _______________________________________________
The domain of a rational function is _____________________________________________
________________________________________________________________________
1. Sketch the graphs below. Label the asymptotes and intercepts, if any.
1
f xx
1
1
xxf
1
1
xxf
1
2
xxf
-5 -4 -3 -2 -1 1 2 3 4 5
-4
-2
2
4
x
y
-5 -4 -3 -2 -1 1 2 3 4 5
-4
-2
2
4
x
y
-5 -4 -3 -2 -1 1 2 3 4 5
-4
-2
2
4
x
y
-5 -4 -3 -2 -1 1 2 3 4 5
-4
-2
2
4
x
y
2. Complete the table below. Use interval notation for the intervals.
x
xf1
1
1
xxf
1
1
xxf
1
2
xxf
1
2
xxf
Domain
Range
Eqn of
Vertical
Asymptote
Eqn of
Horizontal
Asymptote
Positive
Intervals
Negative
Intervals
Increasing
Intervals
Decreasing
Intervals
x-intercept
y-intercept
PREDICT
3. For the function bx
nxf
for b,n R , n 0 , predict the following properties:
Domain: _____________________________
Range: ______________________________
Eqn of Vertical Asymptote: ____________
Eqn of Horizontal Asymptote: __________
4. Given the completed tables of values, sketch each function. Label the asymptotes.
a) 5
f x2x 4
x y
-1000 -0.002
-10 -0.2
-2 -0.6
0 -1.3
1.7 -8.3
2.3 8.3
4 1.3
6 0.6
1000 0.003
b) 4
f x2x 6
x y
-1000 0.002
-10 0.2
0 0.7
2 2
2.8 10
3.2 -10
5 -1
9 -0.3
1000 -0.002
a)
-10 -5 5 10
-10
-5
5
10
x
y
b)
-10 -5 5 10
-10
-5
5
10
x
y
5. For the function n
f xax b
for a,b,n R , a,n 0 , predict the following properties:
Domain: _____________________________
Range: ______________________________
Eqn of Vertical Asymptote: ____________
Eqn of Horizontal Asymptote: __________
General shape of graphs for bx
nxf
General shape of graphs for n
f xax b
6. Complete the table of values, where necessary, and sketch. Label the asymptotes.
a) 2x
f xx 3
x y
-1000
-10
-2
0
2
2.5
3.8
4
6
1000
b) 4x
f xx 3
x y
-1000 -3.988
-10 -3.1
-2 -1.6
0 0
2 8
2.2 11
4 -16
5 -10
9 -6
1000 -4.012
c) 3x
f xx 1
x y
-1000 3.003
-10 3.3
-6 3.6
-1.5 9
-1.2 18
-0.7 -7
0 0
3 2.25
9 2.7
1000 2.997
d) 5x
f xx 1
x y
-1000 -5.005
-6 -6
-3 -7.5
-2 -10
-0.6 7.5
0 0
1 -2.5
3 -3.75
9 -4.5
1000 -4.995
a)
-10 -5 5 10
-10
-5
5
10
x
y
b)
-10 -5 5 10
-10
-5
5
10
x
y
c)
-10 -5 5 10
-10
-5
5
10
x
y
d)
-10 -5 5 10
-10
-5
5
10
x
y
(3 dec. pl.)
(3 dec. pl.)
7. Complete the table below. Use interval notation for the intervals.
a) 2x
f xx 3
b) 4x
f xx 3
c)
3xf x
x 1
d)
5xf x
x 1
Domain
Range
Eqn of Vertical
Asymptote
Eqn of Horizontal
Asymptote
Positive
Intervals
Negative
Intervals
Increasing
Intervals
Decreasing
Intervals
x-intercept
y-intercept
8. Without sketching, complete the table below:
2
3
xx
xf 5
8
xx
xf
Domain
Range
Eqn of Vertical
Asymptote
Eqn of Horizontal
Asymptote
x-intercept
y-intercept
9. In general, for the function mx
f xx b
, b,m R , m 0 , predict the following:
Domain: ________________________
Range: _________________________
Eqn of Vertical Asymptote: __________
Eqn of Horizontal Asymptote: ________
General shape of graphs for mx
f xx b
10. Given the completed tables of values, sketch each function. Label the asymptotes.
a) 5x
f x2x 6
x y
-1000 2.492
-10 1.9
-2 1
0 0
2.4 -10
4 10
6 5
10 3.6
1000 2.508
b) 4x
f x3x 9
x y
-1000 -1.337
-10 -1.9
-3.5 -9.3
-2.6 8.7
-2 2.7
0 0
2 -0.5
9 -1
1000 -1.329
c) 3x
f x2x 1
x y
-1000 1.501
-5 1.7
-0.6 9
-0.4 -6
0 0
2 1.2
8 1.4
1000 1.499
d) 2x
f x5x 10
x y
-1000 -0.401
-6 -0.6
-2.1 -8.4
-1.9 7.6
0 0
3 -0.24
10 -0.3
1000 -0.399
a)
-10 -5 5 10
-10
-5
5
10
x
y
b)
-10 -5 5 10
-10
-5
5
10
x
y
c)
-10 -5 5 10
-10
-5
5
10
x
y
d)
-10 -5 5 10
-10
-5
5
10
x
y
11. Complete the table below. Use interval notation for the intervals.
a) 5x
f x2x 6
b) 4x
f x3x 9
c)
3xf x
2x 1
d)
2xf x
5x 10
Domain
Range
Eqn of Vertical
Asymptote
Eqn of Horizontal
Asymptote
Positive
Intervals
Negative
Intervals
Increasing
Intervals
Decreasing
Intervals
x-intercept
y-intercept
12. Without sketching, complete the table below:
3x
f x2x 4
2xf x
3x 9
4xf x
5x 20
Domain
Range
Eqn of Vertical
Asymptote
Eqn of Horizontal
Asymptote
x-intercept
y-intercept
13. In general, for the function mx
f xax b
, a,b,m R , a 0,m 0 , predict the following:
Domain: ________________________
Range: _________________________
Eqn of Vertical Asymptote: __________
Eqn of Horizontal Asymptote: ________
General shape of graphs for mx
f xax b
MHF 4UI Unit 5 Day 1
HW - Rational Functions of the form n
f xax b
and mx
f xax b
Complete the tables below:
4f x
x 6
4f x
3x 6
2xf x
x 6
2xf x
3x 6
Write in
Factored
Form
Domain
Range
Eqn of
Vertical
Asymptote
Eqn of
Horizontal
Asymptote
x-intercept
y-intercept
Sketch
Positive
Intervals
Negative
Intervals
Increasing
Intervals
Decreasing
Intervals
y
x
y
x
y
x
y
x
MHF 4UI Unit 5 Day 2
Key Features of Rational Functions Investigation – Key Features of Rational functions of the form mx n
yax b
1. Complete the table of values, where necessary, and sketch. Label the asymptotes.
a) x 1
f x2x 4
x y
-1000
-8
-3
-1
0
1
1.8
2.2
3
5
9
1000
b) x 1
f x2x 4
x y
-1000 0.501
-7 0.6
-3 1
-2.1 5.5
-1.9 -4.5
-1 0
0 0.25
3 0.4
8 0.45
1000 0.4995
c) 2x 4
f x0.5x 2
x y
-1000 -4.008
-10 -5.3
-6 -8
-5 -12
-4.5 -20
-3.4 9.3
-2 0
0 -2
2 -2.7
4 -3
1000 -3.992
d) 2x 2
f x0.5x 1
x y
-1000 -3.988
-10 -3
-1 0
0 2
1.1 9.3
4 -10
5 -8
10 -5.5
1000 -4.012
a) b)
-10 -5 5 10
-10
-5
5
10
x
y
-10 -5 5 10
-10
-5
5
10
x
y
c) d)
-10 -5 5 10
-10
-5
5
10
x
y
-10 -5 5 10
-10
-5
5
10
x
y
(3 dec. pl.)
(3 dec. pl.)
2. Complete the table below. Use interval notation for the intervals.
x 1
f x2x 4
x 1f x
2x 4
2x 4f x
0.5x 2
2x 2f x
0.5x 1
Domain
Range
Eqn of Vertical
Asymptote
Eqn of Horizontal
Asymptote
Positive
Intervals
Negative
Intervals
Increasing
Intervals
Decreasing
Intervals
x-intercept
y-intercept
3. Without sketching, predict the entries for the table below:
x 3
f x4x 8
3x 6f x
0.5x 2
Domain
Range
Eqn of Vertical
Asymptote
Eqn of Horizontal
Asymptote
x-intercept
y-intercept
4. In general, for the function mx n
f xax b
, predict the following:
Domain: _______________________________
Range: ________________________________
Eqn of Vertical Asymptote: ______________
Eqn of Horizontal Asymptote: ____________
General shape of graphs for mx n
f xax b
MHF 4UI Unit 5 Day 2
HW - Rational Functions of the form mx n
f xax b
Complete the tables below:
2x 2
f x3x 6
2x 2f x
3x 6
2x 2f x
3x 6
2x 2f x
3x 6
Write in
Factored
Form
Domain
Range
Eqn of
Vertical
Asymptote
Eqn of
Horizontal
Asymptote
x-intercept
y-intercept
Sketch
Positive
Intervals
Negative
Intervals
Increasing
Intervals
Decreasing
Intervals
y y y y
x x x x
2x 2
f x3x 6
2x 2f x
3x 6
2x 2f x
3x 6
2x 2f x
3x 6
Write in
Factored
Form
Domain
Range
Eqn of
Vertical
Asymptote
Eqn of
Horizontal
Asymptote
x-intercept
y-intercept
Sketch
Positive
Intervals
Negative
Intervals
Increasing
Intervals
Decreasing
Intervals
y y y y
x x x x
MHF 4UI Unit 5 Day 3
Horizontal Asymptotes
For the rational function g(x)
f(x)y , the horizontal asymptote can be determined from
either:
if the degree of the numerator = degree of the denominator, the asymptote is
if the degree of the numerator < degree of the denominator, the asymptote is
Examples:
g(x) of tcoefficien leading
f(x) of tcoefficien leadingy
y = 0
MHF 4UI Unit 5 Day 3
Sketching Rational Functions
1. For each of the following functions,
i) determine the x- and y-intercepts
ii) sketch
iii) determine the positive/negative intervals
iv) determine the increasing/decreasing intervals
a) 6 - 3x
84xy
i) horizontal asymptote :
vertical asymptote :
ii) x-intercept y-intercept iii) sketch
iv) positive intervals:
negative intervals:
v) increasing intervals:
vi) decreasing intervals:
MHF 4UI Unit 5 Day 3
a) 2 - x
3x-y
i) H. A. : V. A. :
ii) x-intercept y-intercept iii) sketch
iv) positive intervals:
negative intervals:
v) increasing intervals:
vi) decreasing intervals:
a) 1 x
2-y
i) H. A. : ii) V. A. :
ii) x-intercept y-intercept iii) sketch
iv) positive intervals:
negative intervals:
v) increasing intervals:
vi) decreasing intervals:
MHF 4UI Unit 5 Day 4
Graphing Functions Using the Big/Little Concept
The concept: consider,
10, 100, 1 000, 10 000, …
numbers getting larger
and the reciprocals,
10
1,
100
1,
000 1
1,
000 10
1, …
numbers getting smaller
the result:
and also,
0.1, 0.01, 0.001, 0.0001, …
numbers getting smaller
and the reciprocals,
0.1
1,
0.01
1,
0.001
1,
0.0001
1, …
(10), (100), (1 000), (10 000) …
numbers getting bigger
the result:
Conclusion:
useful for graphing
reciprocal functions
# small# big
1
# big# small
1
as n +, n
1 0+ as n -,
n
1 0-
as n 0+, n
1 + as n 0-,
n
1 -
MHF 4UI Unit 5 Day 4
Graphing Reciprocal Functions 1. Given the graph y = x + 2, graph the reciprocal function
2
1
xy .
x 2 xy 2
1
xy
-10
-3
-2.1
-2
-1.9
-1
6
-10 -5 5
-5
5
x
y
For the reciprocal function 2
1
xy
x-intercept ___________
y-intercept ___________
vertical asymptote _____________
horizontal asymptote ____________
MHF 4UI Unit 5 Day 4
-5 -4 -3 -2 -1 1 2
-1
1
2
3
4
5
6
7
8
9
10
x
y
2. Given the graph for the quadratic function 2y x 4x 4 , graph the reciprocal function
21
yx 4x 4
on the same grid using the BIG/LITTLE concept.
Write in factored form:
x-intercept ___________ vertical asymptote _____________
y-intercept ___________ horizontal asymptote ____________
3. Given the graph for a quadratic function below, graph the reciprocal function on the same grid
using the BIG/LITTLE concept.
MHF 4UI Unit 5 Day 4
-5 -4 -3 -2 -1 1 2 3 4 5 6 7
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
10
x
y
x-intercept ___________ vertical asymptote ________________________
y-intercept ___________ horizontal asymptote ____________
MHF 4UI Unit 5 Day 4
4. Sketch 4)6)(x1)(x(x
1y
.
x
y
MHF 4UI Unit 5 Day 5
x
y
x
y
x
y
x
y
x
y
x
y
Holes
Investigation
1. Graph.
a) f x x 4 b) g x x 2 c) 1
h xx 1
2. Graph using the given table of values to plot the points.
a) 2x 3x 4
f xx 1
b)
2x 4f x
x 2
c)
2
3
x 1f x
x 1
x y
-6 -2
-4 0
-2 2
-1 3
0 4
0.9 4.9
0.99 4.99
1
1.01 5.01
1.1 5.1
2 6
x y
-6 -4
-4 -2
0 2
1 3
1.9 3.9
1.99 3.99
2
2.01 4.01
2.1 4.1
3 5
4 6
x y
-3 -0.5
-2 -1
-1.5 -2
-1.1 -10
-1.01 -100
-1
-0.99 100
-0.9 10
-0.5 2
0 1
1 0.5
MHF 4UI Unit 5 Day 5
Summary
Let (x + k) represent a factor in both the numerator and denominator of a rational
function.
kxg(x) k)(x
f(x)k)(xh(x)
d
c
;
If the factor completely cancels out of the denominator ( dc ), state the
restriction and note that there is a hole at x = -k. The y-coordinate can be
determined by substituting x = -k into the simplified h(x).
or
If the factor does not completely cancel (c < d), there is a vertical asymptote at x
= -k.
1. Determine the asymptotes and holes, if any, for each function, then sketch.
a) 96x-x
3-xf(x)
2
MHF 4UI Unit 5 Day 5
b) 65xx
15-2x-xf(x)
2
2
Aside: State the domain and range.
D = _____________________________
R = _____________________________
MHF 4UI Unit 5 Day 5
c) 5xx
xf(x)
2
2
MHF 4UI Unit 5 Day 6
Linear Oblique Asymptotes
Take a look at the graph of 1
12
xx
y .
Where are the asymptotes?
MHF 4UI Unit 5 Day 6
There is a vertical asymptote at 1x .
There is NO HORIZONTAL ASYMPTOTE
There is another asymptote along the line 1 xy . This
is called a LINEAR OBLIQUE ASYMPTOTE.
MHF 4UI Unit 5 Day 6
Linear Oblique Asymptotes Note: A horizontal asymptote shows the trend of the end behaviours – the value that y
approaches as x - and as x +
A curve may “follow a pattern” as x - and as x + and not necessarily a specific y-
value. The curve may be approaching a linear oblique asymptote.
For rational functions, a linear oblique asymptote occurs when the degree of the
numerator is exactly one more than the degree of the denominator.
Note: If there is a linear oblique asymptote, there isn’t a horizontal asymptote and vice-
versa. That is, you can have one or the other, or neither, but not both.
1. Determine the equation of the linear oblique asymptote of 1x
1xy
2
.
MHF 4UI Unit 5 Day 6
2. Sketch 1x
1xy
2
.
MHF 4UI Unit 5 Day 6
3. Sketch 2x
125x2x-y
2
MHF 4UI Unit 5 Day 7
Sketching Rational Functions
Warm up: Asymptotes
1. State a possible equation of the rational function with the following features:
a) vertical asymptote x 3 , horizontal asymptote y 2 , x-intercept -1
b) vertical asymptote x 5 , horizontal asymptote y 1 , x-intercepts -6, 8
c) vertical asymptotes x 9 and x 7 , no horizontal asymptote, x-intercepts 4, 8
d) vertical asymptote x 3 , horizontal asymptote y 0 , x-intercepts 1, -2,
hole at x 5
2. Find the equation of the linear oblique asymptote.
a) 22x 4
yx 1
b)
4
y 3x 92x 7
MHF 4UI Unit 5 Day 7
Sketching Rational Functions
3. Sketch the following functions. Label all key features (asymptotes, intercepts, holes).
a) 2
2
2)-(x
3x2x-y
Where does the function cross the
horizontal asymptote?
MHF 4UI Unit 5 Day 7
b) 1x
1-xy
2
2
4. Determine where the following function crosses its linear oblique asymptote.
2xx
13x3xxy
2
23
MHF 4UI Unit 5 Day 8
Solving Rational Inequalities
Warm Up
1.
i) State increasing and decreasing interval(s).
increasing: _________________________________________________
decreasing: _________________________________________________
ii) State positive and negative interval(s).
positive: _________________________________________________
negative: _________________________________________________
iii) State domain and range.
Domain: _________________________________________________
Range: _________________________________________________
y = - 4
x = 3
x
y
MHF 4UI Unit 5 Day 7
2. Recall: Solving polynomial inequalities from Unit 3.
Solve x3 + 2x2 – 11x -12 < 0.
MHF 4UI Unit 5 Day 7
3. Solve for x.
a) 04-x
2x
MHF 4UI Unit 5 Day 7
b) 62x
4-2x
MHF 4UI Unit 5 Day 8
Solving Rational Inequalities
Warm Up
1.
i) State increasing and decreasing interval(s).
increasing: _________________________________________________
decreasing: _________________________________________________
ii) State positive and negative interval(s).
positive: _________________________________________________
negative: _________________________________________________
iii) State domain and range.
Domain: _________________________________________________
Range: _________________________________________________
y = - 4
x = 3
x
y
MHF 4UI Unit 5 Day 8
2. Recall: Solving polynomial inequalities from Unit 3.
Solve x3 + 2x2 – 11x -12 < 0.
MHF 4UI Unit 5 Day 8
3. Solve for x.
a) 04-x
2x
MHF 4UI Unit 5 Day 8
b) 62x
4-2x
MHF 4UI Unit 5 Day 9
Solving More Rational Inequalities
Solve the inequality. State the solution using interval notation.
5x
1
3-x
x