MCR 3UProblems Involving Function Transformations
- Rivers located near an ocean experience a large wave called a tidal bore due to the tides. The speed, v, in kilometers per hour, of the tidal bore in a river is a function of the depth, d, in metres, of the river. The function is .
a)Determine the domain and the range of this function.
b)Make a table of values and graph the function.
- The value, V, in dollars, of an n-year old car is given by .
a)How much was this car worth when it was first purchased?
b)Determine the value of the car after i)10 yearsii)12 years
c)How long would it take the car to depreciate to a value of $2000?
d)Is a function? Justify your answer.
- The amount, A, in dollars, that needs to be invested at an interest rate I to have $100 after 1 year is given by the relation . Note that i must be expressed in decimal form.
a)Determine the domain and the range for this relation.
b)Graph the relation.
c)How much money needs to be invested at 5% to give $100 after 1 year?
d)What rate of interest is required if $90 is invested?
- On Earth, the time t in seconds taken for an object to fall from a height, h, in metres, to the ground is given by the formula . On the moon, the formula changes to .
a)Determine the domain and the range of each relation.
b)Graph both relations on the same set of axes. Compare the graphs and describe any similarities or differences.
c)Determine the difference between the time it takes for an object to fall from a height of 25 m on Earth and the time it takes on the moon. Justify your answer.
- A small skateboard company is trying to determine the best price for its boards. When the boards are priced at $80, 120 are sold in a month. After doing some research, the company finds that each increase of $5 will result in selling 15 fewer boards.
a)Write an equation to represent revenue, R, in dollars as a function of x, the number of $5 increases in price.
b)State the domain and the range of the revenue function.
c)Determine the inverse of the revenue function. What does this equation represent in the context of the question? State the domain and range of the inverse.
d)Determine the number of $5 increases for a revenue of $8100.
- The value, V, in thousands of dollars, of a certain car after t years can be modeled by the equation .
a)Sketch the graph of this relation.
b)What was the initial value of this car?
c)What is the projected value of this car afteri)1 year?ii)2 years?iii)10 years?
- Two skydivers jump out of a plane. The first skydiver’s motion can be modeled by the function . The second skydiver jumps out a few seconds later with a goal of catching up to the first skydiver. The motion of the second skydiver can be modeled by . For both functions, the distance above the ground is measured in metres and the time is the number of seconds after the second skydiver jumps.
a)Graph the functions on the same set of axes.
b)Will the 2ndskydiver catch up to the 1stbefore they have to open the parachutes at 800 m?
c)State the domain and range of these functions in this context.
- The equation can be used to convert between Celsius and Fahrenheit temperatures, where x is the temperature in ºCelsius, and y is the temperature in ºFahrenheit.
a)Find the inverse of the equation. What does it represent? What do the variables represent?
b)Graph the original and inverse equations on the same set of axes.
c)What temperature is the same in Celsius and Fahrenheit? Explain how you know.
- At the traffic safety bureau, Matthew determines that for a car traveling at approximately 100 km/h, the distance, in metres, it takes to stop once the brakes are applied is approximately given by , where t is the time in seconds.
a)Find the inverse of this function. What does this represent in the context of the question?
b)In the context of the question, what are the domain and range of the original function and its inverse?
c)Compare how long it takes to stop in the first 20 m of braking, the second 20 m of braking, and the third 20 m of braking.
- A rock is thrown from the top of 100-m cliff. Its height, h, in metres, after t seconds can be modeled approximately by the function .
a)Graph the function and states its domain and range.
b)Determine the inverse of and state its domain and range. Explain what this inverse represents in the context of the question.
c)A second rock is thrown upward off the cliff. Its height, h, in metres, after t seconds can be modeled approximately by the function . Solve for t by finding the inverse. Then, determine at what time the rock will hit the ground.
Answers
1. a) D:R: / 2. a) $24000 b) i) $3090.90 ii) $2769.12 c) 22 years
d) Sample answer: relation is a function because for each value in the domain there is exactly one value in the range / 3. a) D:
R:
c) $95.24d) 11.1%
4. a) D: R: same on E & m
b) on the Moon, the graph has been stretched vertically
c) E: M: ; on the moon, the gravity is weaker so will pull the object with less force / 5. a)
b) D: R:
c); the number of $5 increases as a function of the revenue, R in dollars d) 2
6. b) $11666.67c) i) $8750ii) $7000iii) $2692.31
7.b) No c) g D: R:
d) h D: R: / 8. a)
c)
9. a) ; time (s) req’d for a car @ 100km/h to stop once brakes are applied for certain distances (m)
b) D: R:
10. a) D: R: b) D: R:
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