investigation – key features of rational...

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.)


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


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


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


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.)


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


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


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


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:


a) 2 - x

3x-y

i) H. A. : V. A. :



negative intervals:



a) 1 x

2-y

i) H. A. : ii) V. A. :



negative intervals:




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 -


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 ____________


-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.


-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 ____________


4. Sketch 4)6)(x1)(x(x

1y

.

x

y


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


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


b) 65xx

15-2x-xf(x)

2

2

Aside: State the domain and range.

D = _____________________________

R = _____________________________


c) 5xx

xf(x)

2

2


Linear Oblique Asymptotes

Take a look at the graph of 1

12

xx

y .

Where are the asymptotes?


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.


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

.


2. Sketch 1x

1xy

2

.


3. Sketch 2x

125x2x-y

2



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



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?


b) 1x

1-xy

2

2

4. Determine where the following function crosses its linear oblique asymptote.

2xx

13x3xxy

2

23


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


2. Recall: Solving polynomial inequalities from Unit 3.

Solve x3 + 2x2 – 11x -12 < 0.


3. Solve for x.

a) 04-x

2x


b) 62x

4-2x


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


2. Recall: Solving polynomial inequalities from Unit 3.

Solve x3 + 2x2 – 11x -12 < 0.


3. Solve for x.

a) 04-x

2x


b) 62x

4-2x


Solving More Rational Inequalities

Solve the inequality. State the solution using interval notation.

5x

1

3-x

x

investigation – key features of rational...

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