Rational Functions- MCR3U Test1

MCR3U - Rational Functions & Expressions Test

MCR3U – Rational Functions & Expressions

Unit Test
Time: 90–100 minutes
Total Marks: 100
Name: ____________________________

Instructions

  • Show all your work to receive full marks.
  • Simplify all answers where possible.
  • Use proper notation for rational expressions and exponents.
  • Calculators may be used unless otherwise noted.

Section A: Rational Function Transformations

[10 Marks]

Describe the vertical/horizontal shifts, reflections, stretches or compressions relative to \( f(x) = \frac{1}{x} \).

1. \( g(x) = \frac{-2}{3(x-4)} + 4 \)
2. \( h(x) = \frac{3}{\frac{1}{2}(x-4)} + 2 \)
3. \( k(x) = \frac{-1}{x+5} - 3 \)
4. \( m(x) = \frac{4}{-2(x-1)} + 1 \)
5. \( p(x) = \frac{-3}{\frac{x}{2} + 1} + 5 \)

Section B: Rational Exponents

[15 Marks]

Product Rule

1. \( x^{3/2} \cdot x^{5/4} \)
2. \( 8^{2/3} \cdot 8^{1/3} \)
3. \( y^{1/2} \cdot y^{3/2} \cdot y^1 \)

Quotient Rule

4. \( \frac{x^{7/3}}{x^{2/3}} \)
5. \( \frac{16^{5/4}}{16^{1/2}} \)
6. \( \frac{y^{9/2}}{y^3} \)

Power Rule

7. \( (x^{2/3})^{3/2} \)
8. \( (8^{1/3})^2 \)
9. \( (y^{5/4})^4 \)

Section C: Simplifying

1. \( \frac{x^2 - 9}{x^2 - 6x + 9} \)
2. \( \frac{x^2 - x - 6}{x^2 + x - 6} \)
3. \( \frac{4x^2 - 16}{2x^2 - 8} \)

Section D: Mult & Div

1. \( \frac{x^2 - 4}{x^2 + x} \cdot \frac{x+2}{x} \)
2. \( \frac{x^2 - 9}{x^2 + 5x + 6} \div \frac{x-3}{x+2} \)
3. \( \frac{2x^2}{x^2 - 1} \cdot \frac{x+1}{x} \)

Section E: Add & Sub

1. \( \frac{1}{x} + \frac{1}{x+2} \)
2. \( \frac{2}{x-1} - \frac{3}{x+1} \)
3. \( \frac{x}{x^2-4} + \frac{2}{x+2} \)

Section F: Applications

[12 Marks]

1. Profit Model: \( P(x) = \frac{100x}{x+5} - 20 \)

a) Find \( P(10) \).

b) Determine the horizontal asymptote and explain its meaning in context.

2. Speed Model: \( S(t) = \frac{180}{t+2} \)

a) Find the speed after 3 hours.

b) Determine the limiting speed as \( t \to \infty \) and interpret.

3. Geometry: Length \( L(x) = \frac{24}{x} \)

a) Find \( L(4) \).

b) Determine \( L(x) \) as \( x \to 0 \) and explain what happens.

Section G: Optional Challenge [+2 Marks]

Given \( f(x) = \frac{x^2 - 4}{x+2} \), perform long division or simplify to write it in the form: \( f(x) = \text{quotient} + \frac{\text{remainder}}{x+2} \)

© 2024 Math Department | Grade 11 University Functions

Answer Key

Section A

  1. Reflect x-axis, V. Comp (2/3), Right 4, Up 4
  2. V. Stretch (6), Right 4, Up 2
  3. Reflect x-axis, Left 5, Down 3
  4. Reflect x and y axes, V. Comp (2), Right 1, Up 1
  5. Reflect x-axis, V. Stretch (3), H. Stretch (2), Left 2, Up 5

Section B

  1. \( x^{11/4} \)
  2. \( 8^1 = 8 \)
  3. \( y^3 \)
  4. \( x^{5/3} \)
  5. \( 16^{3/4} = 8 \)
  6. \( y^{3/2} \)
  7. \( x \)
  8. \( 4 \)
  9. \( y^5 \)

Section C

  1. \( \frac{x+3}{x-3}, x \ne 3 \)
  2. \( \frac{(x-3)(x+2)}{(x+3)(x-2)}, x \ne -3, 2 \)
  3. \( 2, x \ne \pm 2 \)

Section D

  1. \( \frac{(x-2)(x+2)^2}{x^2(x+1)} \)
  2. \( 1, x \ne -2, -3, 3 \)
  3. \( \frac{2x}{x-1}, x \ne 0, \pm 1 \)

Section F (Applications)

1a) \( P(10) = 100(10)/15 - 20 \approx 46.67 \). 1b) \( y=80 \); long-term maximum profit.

2a) \( S(3) = 180/5 = 36 \). 2b) \( 0 \); speed stops as time goes to infinity.

3a) \( L(4) = 6 \). 3b) \( \infty \); as width approaches 0, length must be infinitely long.