Final Exam Prep for Written Part

Exam "Prep" for Written Portion of Final Exam

A. Transformations:

1.

What equation results when the function

f

(x)

x x2 3

is translated 4 units to the left

and expanded vertically by a factor of 3?

2. Write the equation of the function that results when you start with y x2 , compress it

horizontally

by

a

factor

of

1 2

, shifts it 3 units to the right and 4 units downward and

then take its absolute value.

3.

For the function

f

(x)

x

x 3

, determine the equation that defines the inverse function

f 1 (x) .

4. Given (m, n) is on the graph of y f (x) , what is the image of this point when the

function is transformed to

y

f

[

1 2

(x

2)]

.

B. Trig Functions, Applications and Identities

5. A Ferris wheel has a diameter of 60 m and its centre is 32 m above the ground. It rotates once every 48 seconds. Lauren gets on the Ferris wheel at its lowest point and then the wheel starts to rotate. a) Determine the sinusoidal equations of least phase shift that gives Lauren's height, h , above the ground as a function of the elapsed time, t, where h is in meters and t is in seconds. b) Determine the first time when Lauren will be 17 m above the ground in the first rotation of the Ferris wheel.

6. At a seaport, the depth of the water, d , in meters at time, t hours, during a certain day

is given by:

d (t)

3.4 sin

2

(t

7.00) 12

2.8

.

a) Determine the first time low tide occurs.

b) Determine the depth of water at 6:30 pm.

7. A sine function has a maximum point at (4, 32) and the nearest minimum point to the right is (16, 18) . Determine sine equation of least phase shift for this function.

8. Solve the following equation algebraically and give exact values where possible. 2 sin x 3sin x cos x where 0 x 2

9. Solve 2 cos 2 x cos x 1 0 algebraically over the set of real numbers. Give solution using exact values.

10. Determine the general solution for b) 3sin(5x) 1 101. Solve: 2 cos2 3x 1 where 0 x 2

11. Solve cos 2 cos over the set of real numbers. Give exact value solutions.

12. Prove the following Identities and state any restrictions.

a)

sin 2x 1 cos 2x

cot

x

b)

sec

1

tan

1 sin cos

c)

sin 2 cos

cos 2 sin

csc

122.

Given

sec

5 4

and

tan

0,

find

the

exact

value

of

a)

sin

.

and

b)

sin 2

123. Find the exact value of

2 sin

7 6

cos2

5 4

tan

sec

4 3

csc

2

tan

3 4

C. Logs, Exponents and Applications 13. The population of Canada is 30 million people and is growing at an annual rate of 1.4%. The population of Germany is 80 million and is decreasing at an annual rate of 1.7%. In how many years will the population of Canada be equal to the population of Germany. Give

answer as an exact value and then round to two decimal places.

14. A population of frogs doubles every 20 weeks. If the population is currently 400 frogs, in how many weeks will the population reach 10 000? Give answer as exact value and then to the nearest week.

15. If 3 150 grams of a radioactive substance decay to 450 grams in 73 weeks, determine the half-life of the substance. Give answer as exact value and then to the nearest week.

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