Rational Functions

MHF4U – Final Exam Review

Rational Functions 1. 2.

3.

Find the vertical and horizontal asymptotes for f ( x ) =

x2 . Find the domain, intercepts and vertical and Given: f ( x ) = 3 x − 2 x 2 − 5x + 6 horizontal asymptotes. Sketch the graph. Given: g ( x ) = a. b. c. d.

4.

x−2 x + 5x + 6 2

Determine x and y intercepts. State the domain. State any asymptotes. Are there any holes? x−3 x2 − 9 Find the x and y intercepts. Find the domain. Find any asymptotes. Are there any holes?

Given: a. b. c. d.

5.

x+2 3x − 2

Create a function that has a graph with the given features. a. Vertical Asymptote at x = −2 and horizontal asymptote at y = 1 b. Two vertical asymptotes at x = 3 and x = −4 , with a hole at x = 1 c. Vertical Asymptote at x = −3 , horizontal asymptote at y = 0 , no x-intercept and a y intercept at –2 d. No vertical asymptote and x-intercepts at 2 and –1 x 2 − 16 . Find the domain, the intercepts, the location of any x asymptotes and describe the function’s behaviour near these asymptotes.

6.

Graph f ( x ) =

7.

Simplify and state restrictions. a.

8.

x +1 x − x x +1

b.

Solve and state any restrictions.

x −1 x + 3 = x −3 x + 4 5 d. 0 = −2 x2 + 4

a.

b.

x − x3 x2 − x ÷ x 2 − 2x − 3 2x − 6 x 1 2 + = 2 x −1 x +1 x −1

e. 0 = 2 x( x 2 + 1)

−

1 2

−1

c. 0 = 8πr − f. 0 = 2 −

2000 r2

10 x2

1

MHF4U – Final Exam Review 9. The estimate revenue and cost functions for the manufacture of a new product are R(x)=-2x2+15x and C(x)=5x+8. a) Express the average profit function AP( x ) =

P( x ) , in two different forms. x

b) Explain what can be determined from each form. c) What is the domain of the function in this context? d) What are the break-even quantities?

10. The value of a car, t years after it is bought, is modeled by V (t ) =

2100 + 8t + 150 . 1 + 0.5t

a) State the domain of V(t), determine any intercepts as asymptotes, and sketch the function. b) What will the car be worth in the long run? c) After how many years will the car be worth $1500? d) Find the average rate of change in the value of the car between 2 and 5 years and the instantaneous rate of change at 2 years. 11. Sketch f ( x ) = − x 2 + 9 and its reciprocal on the same grid. Answers: 1. 2.

3.

2 1 HA: y = 3 3 Domain: D= {x| x ∈ R, x ≠ −2, x ≠ 1, x ≠ 3} , x-intercept is 0 and y-intercept is 0, VA: x = 1, x = 3, x = −2 HA: y = 0 VA: x =

a) x-intercept is 2 and y-intercept is c) VA: x = −3, x = −2

4.

d) There are no holes.

a) There is no x-intercept and y-intercept is c) VA: x = −3

5.

HA: y = 0

−1 b) Domain D= {x| x ∈ R, x ≠ −3, x ≠ −2} 3

a) y =

x x+2

HA: y = 0 b) y =

1 3

b) Domain D = {x| x ∈ R, x ≠ ±3}

d) There is a hole at x = 3 .

x −1 ( x − 1)( x − 3)( x + 4 )

c) y =

−6 ( x + 3)

d) y = ( x − 2 )( x + 1)

2

MHF4U – Final Exam Review 6.

Domain: D = {x| x ∈ R, x ≠ 0} , x-intercepts are ± 4 , there is no y-intercept, VA: x = 0 , HA: none, Oblique Asymptote: y = x Behaviour of function near asymptotes:

x → +∞, y → x As

x → −∞, y → x x → 0 + , y → −∞ x → 0 − , y → +∞ 2x + 1 , x ≠ 0,−1 x( x + 1)

7.

a)

8.

a) x =

−5 , x ≠ 3,−4 3

d) x = ±

9.

b) − 2 , x ≠ 0,±1,3

b) x = −3 or x = 1 , x ≠ ±1

3 , no restrictions 2

a) AP( x ) =

− 2( x − 4)( x − 1) x

e) x = ±

3 , no restrictions 3

or AP( x ) = −2 x + 10 −

c) r = 4.3 , r ≠ 0

f) x = ± 5 , x ≠ 0

8 x

1 you can find h( x ) d) x = 4, x = 1

b) From the factored form, you can find zeros and from the form g ( x ) + oblique asymptotes

c) The Domain is: D = {x| x > 0, x ∈ R}

10. a) D = {t| t ≠ −2, t ∈ R} , t-intercept is –27.11 years and V-intercept is $2250, VA: t = −2 ,

HA: V ( t ) = 166 b) Worth $166 in long run c) 1.12 years after purchase d) Average Rate of Change between t=2 and t=5 is: -$147.86. Instantaneous Rate of Change at t=2 years is -$260.50

11.

3