Math 11 Advanced Review Questions
1. Write each expression in simplest radical form.
a) 2 600
c)
2 3
8
b) 5 2 + 8 8
d) (2 + 2 3 )(5 − 3 )
2 3
e)
2 − 3
1
3
5
2
f)
2
6
2 −
1
8
3
7
2
2. Fully factor the following.
a) 12 x 2 − 5 x − 2
b) 100a 2 − 64b 2
c) 8 x 2 + 23 x + 15
d) 2 x 3 y + 9 x 2 y − 5 xy
e) x 2w 2 − y 2 z 2
f) 9 x 2 − 17 x + 8
g) x 2 − 12 x + 20
h) ( x − 4)2 − 9
i ) 16 x 2 − 25w 2
j) 2 x 2 − 9 x − 68
3. Rewrite each equation in transformational form AND state the transformations
present.
2 3

b) y = −6 + − sin x + 80° 
a) y = 3 cos( 4 x − 120°) − 12
3 5

4. Compute the following. Use fractions and show all of your steps.
3   3 1
1
− − 2 − −  − 4 ÷ − 
2   5 3
3
5. Graph the following function using a mapping rule. Show all of your work and
draw at least two cycles.
1
2
− ( y − 2) = cos ( x + 10°)
3
3
6. For each of the functions below write a transformed sine function and a
transformed cosine function.
3
y
sine:
a)
2
cosine:
1
x
− 7 2 0− 6 3 0− 5 4 0− 4 5 0− 3 6 0− 2 7 0− 1 8 0−
0− 9 0
−1
90 180 270 360 450 540
1
y
sine:
x
b)
−90
−60
−30
30
60
90
120
150
180
−1
cosine:
−2
−3
−4
−5
4
c)
y
sine:
3
2
1
−π
x
π
−1
2π
cosine:
−2
−3
−4
−5
−6
d)
4
y
sine:
2
cosine:
x
2π
7. The average depth of water at the end of a dock is 6 feet. This varies 2 feet in
both directions with the tide. Suppose there is a high tide at 4 AM. If the tide goes
from low to high every 6 hours, with t = 4 corresponding to 4 AM.
a) Sketch a well labelled graph to represent this situation. Draw at least two
cycles.
b) Write a function describing the depth of the water as a function of time.
c) What will be the height of the tide at 11:00 am? Use your function and show all
of your work and mathematical reasoning.
d) What will be the first two times that the height of the tide is 7 feet? Use your
function and show all of your work and mathematical reasoning.
8. When a spaceship is fired into orbit from a site such as Cape Canaveral in
Florida, which is not on the equator, it goes into an orbit that takes it alternately
north and south of the equator. Its distance from the equator is approximately a
sinusoidal function of time.
Suppose that a spaceship is fired into orbit from Cape Canaveral. Ten minutes
after it leaves the Cape, it reaches its farthest distance north of the equator, 4000
kilometres. Half a cycle later it reaches its farthest distance south of the equator,
also 4000 kilometres. The spaceship completes an orbit once every 90 minutes.
Let y be the number of kilometres the spaceship is north of the equator (you must
consider distances south of the equator to be negative). Let x be the number of
minutes that have elapsed since liftoff.
a) Sketch a well labelled graph of this situation.
b) Write a function to represent of kilometres the spaceship is north of the
equator versus number of minutes that have elapsed since liftoff.
c) Use your equation to predict the distance of the spaceship from the
equator when the time is 25 minutes, 41 minutes, and 163 minutes.