MHF 4U1- ASSIGNMENT CHAPTER 3 B - Advanced Functions 12



MHF 4U1- ASSIGNMENT CHAPTER 3 B NAME:________________________

True/False

Indicate whether the statement is true or false.

____ 1. The solution to [pic] is [pic].

____ 2. The solution to [pic] is [pic].

Multiple Choice

Identify the choice that best completes the statement or answers the question.

____ 3. Solve the equation [pic].

|a. |x = –1 |c. |x = –5 |

|b. |x = 5 |d. |no solution |

____ 4. Solve the equation [pic].

|a. |x = 2 |c. |x = –2 |

|b. |x = [pic] |d. |no solution |

____ 5. What are the x-intercepts of the graph of [pic]?

|a. |–4, 5 |c. |4, –5 |

|b. |–7, 3 |d. |7, –3 |

____ 6. For what values of k does the graph of [pic] have no x-intercepts?

|a. |[pic] |c. |[pic] |

|b. |[pic] |d. |[pic] |

____ 7. Which of the following rational equations requires the use of technology to solve?

|a. |[pic] |c. |[pic] |

|b. |[pic] |d. |[pic] |

____ 8. Which of the following rational inequalities requires the use of technology to solve?

|a. |[pic] |c. |[pic] |

|b. |[pic] |d. |[pic] |

____ 9. Use the graph of [pic] to solve the equation [pic].

[pic]

|a. |[pic] |c. |[pic] |

|b. |[pic] |d. |no solution |

____ 10. Use the graph of [pic]to solve the inequality [pic].

[pic]

|a. |[pic] |c. |[pic] |

|b. |[pic] |d. |no solution |

Short Answer

11. Solve.

a) [pic]

b) [pic]

c) [pic]

d) [pic]

12. Solve each equation by graphing, using technology. Express your answers to two decimal places.

a) [pic]

b) [pic]

c) [pic]

d) [pic]

13. Solve each inequality by graphing, using technology. Express your answers to one decimal place.

a) [pic]

b) [pic]

c) [pic]

d) [pic]

Problem

14. A photographer uses a light meter to measure the intensity of light from a flash bulb. The intensity for the flash bulb, I, in lux, is a function of the distance from the light, d, in metres, and can be represented by [pic].

a) Determine the following, to two decimal places.

i) the intensity of light 3 m from the flash bulb

ii) the average rate of change in the intensity of light for the interval [pic]

iii) the approximate instantaneous rate of change in the intensity of light at exactly 3 m from the flash bulb

b) What does the sign of your answer to part a), subpart iii), indicate about the light intensity?

15. Ingrid is docking a motorboat. She turns off the power and lets the boat coast toward the dock. The distance, d, in metres, between the boat and the dock as a function of time, t, in seconds, is given by [pic].

a) What is the average velocity of the boat during the time interval from when Ingrid turns the boat off to when it meets the dock?

b) Determine the velocity of the boat when Ingrid turns off the power, to one decimal place.

c) Determine the velocity of the boat when it meets the dock, to one decimal place.

d) Sketch a graph of the function. On the same set of axes, sketch a graph of the speed of the boat in relation to the time.

16. Write an equation for the graph of the rational function shown. Explain your reasoning.

[pic]

17. Determine an equation in the form [pic] for a function that has asymptotes with equations [pic] and [pic] and a y-intercept of 2. Sketch a graph of your function.

18. Explain the difference between the solution to the equation [pic] and the solution to [pic].

19. Write an equation for a rational function whose graph has all of the indicated features.

• vertical asymptote with equation x = 3

• horizontal asymptote with equation y = 2

• hole at x = 1

• no x-intercepts

20. a) Use the asymptotes and intercepts to make a quick sketch of the function [pic] and its reciprocal, [pic], on the same set of axes.

b) Describe the symmetry in the graphs in part a).

c) Determine the equation of the mirror line in your graph from part a).

d) Determine intervals of increase and decrease for both f and fR. How do the sets of intervals compare?

e) Would the pattern from part d) occur for all pairs of functions [pic] and [pic]? Explain why or why not.

MHF 4U1- ASSIGNMENT CHAPTER 3 B

Answer Section

TRUE/FALSE

1. ANS: F

The solution is x < –5 or x > 2.

PTS: 1 DIF: 3 REF: Knowledge and Understanding

OBJ: Section 3.4 LOC: C4.1, C4.2 TOP: Polynomial and Rational Functions

KEY: rational inequality

2. ANS: F PTS: 1 DIF: 2 REF: Knowledge and Understanding

OBJ: Section 3.4 LOC: C4.1, C4.2 TOP: Polynomial and Rational Functions

KEY: rational inequality

MULTIPLE CHOICE

3. ANS: B PTS: 1 DIF: 1 REF: Knowledge and Understanding

OBJ: Section 3.4 LOC: C3.6 TOP: Polynomial and Rational Functions

KEY: rational equation

4. ANS: D PTS: 1 DIF: 2 REF: Knowledge and Understanding

OBJ: Section 3.4 LOC: C3.6 TOP: Polynomial and Rational Functions

KEY: rational equation

5. ANS: B PTS: 1 DIF: 2 REF: Knowledge and Understanding

OBJ: Section 3.4 LOC: C3.5 TOP: Polynomial and Rational Functions

KEY: rational function, x-intercept

6. ANS: A PTS: 1 DIF: 3

REF: Knowledge and Understanding; Thinking; Application OBJ: Section 3.4

LOC: C3.5 TOP: Polynomial and Rational Functions KEY: rational function, x-intercept

7. ANS: C PTS: 1 DIF: 2

REF: Knowledge and Understanding; Thinking OBJ: Section 3.4

LOC: C3.5, C3.6, D3.2

TOP: Polynomial and Rational Functions, Characteristics of Functions

KEY: rational equation, technology

8. ANS: D PTS: 1 DIF: 3

REF: Knowledge and Understanding; Thinking OBJ: Section 3.4

LOC: C3.5, C3.6, C4.1, C4.2, D3.2

TOP: Polynomial and Rational Functions, Characteristics of Functions

KEY: rational equation, rational inequality, technology

9. ANS: B PTS: 1 DIF: 1

REF: Knowledge and Understanding; Application OBJ: Section 3.4

LOC: C3.5, D3.2 TOP: Polynomial and Rational Functions, Characteristics of Functions

KEY: rational equation, graph

10. ANS: B PTS: 1 DIF: 1

REF: Knowledge and Understanding; Application OBJ: Section 3.4

LOC: C4.1, C4.2 TOP: Polynomial and Rational Functions KEY: rational inequality, graph

SHORT ANSWER

11. ANS:

a) [pic]

b) [pic] or x = –3

c) [pic]

d) No solution.

PTS: 1 DIF: 2 REF: Knowledge and Understanding

OBJ: Section 3.4 LOC: C3.6 TOP: Polynomial and Rational Functions

KEY: rational equation

12. ANS:

a) [pic] or [pic]

b) [pic] or [pic]

c) No solution.

d) [pic] or [pic]

PTS: 1 DIF: 2 REF: Knowledge and Understanding

OBJ: Section 3.4 LOC: C3.6, D3.2

TOP: Polynomial and Rational Functions, Characteristics of Functions

KEY: rational equation, technology

13. ANS:

a) [pic]

b) [pic] or [pic]

c) [pic] or [pic]

d) [pic] or [pic]

PTS: 1 DIF: 2 REF: Knowledge and Understanding

OBJ: Section 3.4 LOC: C3.6, D3.2

TOP: Polynomial and Rational Functions, Characteristics of Functions

KEY: rational inequality, technology

PROBLEM

14. ANS:

a) i) 1.11 lux

ii) –4.44 lux/m

iii) –0.74 lux/m

b) As the distance from the light increases, the intensity drops.

PTS: 1 DIF: 2

REF: Knowledge and Understanding; Application; Communication

OBJ: Sections 3.2, 3.5 LOC: C2.1, C3.7, D1.4, D1.5, D1.6, D1.7, D1.8, D1.9, D2.2

TOP: Polynomial and Rational Functions, Characteristics of Functions

KEY: word problem, reciprocal of quadratic function, rate of change

15. ANS:

a) 1.75 m/s toward the dock

(Ingrid turns off the boat at t = 0 and the boat hits the dock when d = 0. Setting d = 0 gives that t = 10 when the boat hits the dock. To find the average velocity, determine [pic] for [pic].)

b) 6.1m/s toward the dock

(This is the instantaneous velocity at t = 0.)

c) 0.5 m/s toward the dock

(This is the instantaneous velocity at t = 10.)

d)

[pic]

The thicker line is the graph of the speed of the boat with respect to its starting position. It is decreasing because the speed of the boat decreases once power has been cut. Students should not be expected to find this graph precisely but should intuit its basic shape.

Note: Although [pic] is negative, this derivative does not represent the speed of the boat, only the rate of change in the distance between the boat and the dock, which is shrinking. In fact, the equation for the speed of the boat with respect to its starting position is the exact opposite of this: [pic], which is the equation of the thicker line graphed above. Again, students are not expected to be this precise.

PTS: 1 DIF: 3 REF: Knowledge and Understanding; Application

OBJ: Sections 3.3, 3.5

LOC: C2.2, C2.3, D1.3, D1.4, D1.5, D1.6, D1.7, D1.8, D1.9, D2.2

TOP: Polynomial and Rational Functions, Characteristics of Functions

KEY: linear expressions in numerator and denominator, rate of change, graph

16. ANS:

[pic] is one possible equation. Any function of the form [pic] is a reasonable candidate since it is difficult to tell from the graph how stretched the function is.

PTS: 1 DIF: 3 REF: Knowledge and Understanding; Thinking; Application

OBJ: Section 3.2 LOC: C2.1, C2.3, D3.1

TOP: Polynomial and Rational Functions, Characteristics of Functions

KEY: reciprocal of quadratic function

17. ANS:

[pic]

[pic]

(Since there is a vertical asymptote at x = –1, the denominator must be a multiple of x + 1. Since there is a horizontal asymptote at [pic], the ratio [pic] = [pic]. If you pick c = 4, then d = 4 and a = 3. To ensure a y-intercept of 2, d = 4 and b = 8.)

PTS: 1 DIF: 2 REF: Knowledge and Understanding; Thinking; Application

OBJ: Section 3.3 LOC: C2.2, C2.3 TOP: Polynomial and Rational Functions

KEY: linear expressions in numerator and denominator, graph

18. ANS:

The equation [pic] has one solution, x = –6. However, there are infinitely many solutions to [pic], with x = –6 being one of them, acting as a boundary on the interval of solutions for x.

PTS: 1 DIF: 2 REF: Knowledge and Understanding; Communication

OBJ: Section 3.4 LOC: C3.5, C4.1 TOP: Polynomial and Rational Functions

KEY: rational inequality, rational equation

19. ANS:

There are many possible answers. One is [pic].

PTS: 1 DIF: 4 REF: Knowledge and Understanding; Thinking; Application

OBJ: Section 3.5 LOC: C3.5, C3.7 TOP: Polynomial and Rational Functions

KEY: rational function, hole

20. ANS:

a) For [pic]: asymptotes [pic]; intercepts [pic]

For [pic]: asymptotes [pic]; intercepts [pic]

[pic]

b) There is reflective symmetry in this graph, about a vertical mirror line that runs exactly halfway between the two vertical asymptotes.

c) x = –1

d) For f: increasing for [pic], decreasing for no values of x

For fR: increasing for no values of x, decreasing for [pic]

Where f increases, fR decreases, except at [pic] and [pic], the locations of the vertical asymptotes.

e) Yes, this pattern would occur for all such pairs of functions. Values that are growing larger grow smaller when reciprocated, and vice versa.

PTS: 1 DIF: 4

REF: Knowledge and Understanding; Application; Thinking; Communication

OBJ: Section 3.3 LOC: C2.2, C2.3, D3.1

TOP: Polynomial and Rational Functions, Characteristics of Functions

KEY: linear expressions in numerator and denominator, graph, increasing, decreasing

NOT: Students can use a graphing calculator to confirm their graphs.

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