MHF 4U1 - FINAL EXAM REVIEW Complete the following in your notebook. Polynomials Functions 1. For each of the following graphs, decide if: (a) the function is even or odd degree, (b) if the leading coefficient is positive or negative, (c) the function is even or odd.

-3 -2 -1 1 2 3

-4

-2

2

4

6

8

10

12

-4 -3 -2 -1 1 2 3

-12

-10

-8

-6

-4

-2

2

4

6

-5 -4 -3 -2 -1 1 2 3 4 5

-2

2

4

6

8

10

12

2. Sketch each of the functions.

(a)

€

f (x) = (x −1)(x +2)2 (b)

€

g(x) = −x 2(x −1)(x + 3) (c)

€

h(x) = (x + 4)(x −1)3

(d)

€

f (x) = −(x)(x −1)(x − 4) (e)

€

g(x) = x 2(x −2)3 (f)

€

h(x) = x(x − 3)(x +2)2

3. (a) Determine an equation to represent the graph of the polynomial function. (b) Find the cubic function that has x-intercept 2, –1, and 1; and passes through the point (4, – 6) 4. (a) Determine the remainder by using (i) long division, and (ii) the remainder theorem. f(x) = 2x4 + x3 – 3x2 + x – 4 is divided by (x – 2) (b) Explain how you know when a binomial is a factor of a polynomial. 5. When

€

f (x) = 2x 3 −5x 2 + px − 3 is divided by x + 2, the remainder is 3. Determine the value of p. 6. Factor each polynomial (do these totally by hand, no TI-84+): (a)

€

f (x) = x 3 − 3x +2 (b)

€

g(x) = x 4 −18x 2 +81 7. Solve each of the following equations (do these totally by hand, no TI-84+):: (a)

€

x 3 − 4x 2 + x +6 = 0 (b)

€

x 4 + x 3 −15x 2 +23x −10 = 0 8. Solve each of the following inequalities (it is not necessary to expand these!): (a)

€

(x −1)(x + 3)(x −5) ≤ 0 (b)

€

x(x +2)2(x − 3) > 0

-4 -3 -2 -1 1 2 3 4 5 6

-8

-6

-4

-2

2

4

6

8

10

12

-4 -3 -2 -1 1 2 3 4 5 6

-8

-6

-4

-2

2

4

6

8

Rates of Change 1. For each of the following, use the graph to determine: (a) the average rate of change between x = 0 and x = 3 (b) the instantaneous rate of change at x = 1

-2 -1 1 2 3 4 5 6

-5-4-3-2-1

123456789

-2 -1 1 2 3 4 5 6

-4

-3

-2

-1

1

2

3

4

2. For each of the following functions, determine: (a) the average rate of change between x = 3 and x = 7 (b) the instantaneous rate of change at x = 5.

(i)

€

f (x) = x 3 − 4x 2 +5x −1 (ii)

€

f (x) =2x −13x +5

(iii)

€

f (x) =log xx 2

3. For the function

€

y = 2sin(1.5θ) , where θ is measured in degrees, determine: (a) the average rate of change between θ =15o and θ =30 o. (b) the instantaneous rate of change at θ =10o.

4. Given the functions:

€

f (x) = x 2 − 3x − 4,

€

g(x) = x +1,

€

h(x) =2xx −5

:

(a) graph f(x), g(x) and (f + g)(x) (b) determine (f o g)(x), (h o g)(x) Rational Functions 1. For each of the following functions, (a) determine the x– and y- intercepts (b) determine any point discontinuities (c) determine the equations of any vertical and/or horizontal asymptotes (d) sketch the function.

(i)

€

y =1x −6

(ii)

€

y =x 2 −16x − 4

(iii)

€

y =x +2x 2 − 4

(iv)

€

y =2x + 4x −5

(v)

€

y =3

x 2 + 4

2. Solve the following rational equations:

(a)

€

5x +1

+43

=x +1x −1

(b)

€

3x +22x +1

=3x +1x −1

−13

(c)

€

8(x −1)x 2 − 4

=4

x −2

3. Solve the following rational inequalities:

(a)

€

1x − 4

<−2x +2

(b)

€

4 − xx

≥ x +2 (c)

€

x − 3x +2

≥x − 4x +1

Exponential & Logarithmic Functions 1. Write in exponential form:

(a) log41024 = 5 (b) log648 =

€

12⎛

⎝ ⎜

⎞

⎠ ⎟ (c) log359049 = 10

2. Write in logarithmic form: (a) 63 = 216 (b)

€

2435 = 3 3. Evaluate: (a) log3 531441 (b) log4 ⎟

⎠

⎞⎜⎝

⎛40961

(c) log246

(d) log2

€

12⎛

⎝ ⎜

⎞

⎠ ⎟ 7

4. Solve each equation to 4 decimal places. (a)

€

4 x = 20 (b)

€

2(3.5)x = 32 (c)

€

log x 3 = 3log12−2log8

(d)

€

log2(x − 3) + log2(x + 3) = 4 (e)

€

2log3 x − log3(x −2) = 2 (f)

€

53−x =1125

(g) 44 32 =−x (h) ( )( )xxx 2131 3392 −+ =

5. Re-write y = 12t with a base of 3. 6. The value, A, of an investment after t years is given by A = 1280(1.085)t. (a) What is the initial value of the investment? (b) Determine the value of the investment after 10 years. (c) Determine how long it will take for the investment to double in value. 7. The level of a certain toxin in a lake is increasing by 20% per year. The current concentration is 50 parts per million (p.p.m.). (a) Write an equation relating the concentration in p.p.m, C, as a function of time in years, t. (b) Determine the concentration of the toxin in 5 years from now. (c) In how many years from now will the concentration reach 600 p.p.m. ? 8. The power source used by satellites is called a radioisotope. The power output of the radioisotope is given by the equation P = 50(0.996)t, where P is the power, in watts, and t is the time, in years. If the equipment in the satellite needs at least 15W of power to function, for how long can the satellite operate before needing recharging? Trigonometry (An Introduction to Radiians) 1. Convert into radians: (a) 45o (b) – 120 o (c) 360 o (d) 30 o

2. Convert the following into degrees: (a) 2π (b)

€

2π5

(c)

€

−π12

3. Determine the exact value of: (a) sin210o (b) cos

€

5π4

(c) tan

€

7π6

(d) csc

€

−3π2

(e)

€

cos π3

⎛

⎝ ⎜

⎞

⎠ ⎟ × sin

π6⎛

⎝ ⎜

⎞

⎠ ⎟ −cos2

π4⎛

⎝ ⎜

⎞

⎠ ⎟

4. Determine all values for θ,

€

0 ≤ θ≤ 2π. (a) sin θ = 0.9205 (b) cos θ = -0.3420 (c) sec θ = 1.1547 (d) cot θ = - 3.7321 5. Write in the following in terms of its co-related angle:

(a) sin 150 o (b) cos 100 o (c) sin

€

3π8

⎛

⎝ ⎜

⎞

⎠ ⎟ (d)

€

cos 7π8

⎛

⎝ ⎜

⎞

⎠ ⎟

6. If sin θ = 0.9659, what is the value of : (a)

€

cos π2

+θ⎛

⎝ ⎜

⎞

⎠ ⎟ (b)

€

cos π2−θ

⎛

⎝ ⎜

⎞

⎠ ⎟

Trigonometric Functions 1. For each of the following: (a) Complete a table of values for the “key” points. (b) Sketch the starting function. (c) Write a mapping formula. (d) Determine the translated “key” points. (e) Sketch the new graph. Be sure to fill the grid with as many cycles as will fit. Complete parts (a), (c) and (d) in your notebook. Parts (b) and (d) can be done on the grids below.

(i)

€

y = 3sin x +π6

⎛

⎝ ⎜

⎞

⎠ ⎟ −5 x scale:

€

π6

(ii)

€

y = −4cos 2x − 3π2

⎛

⎝ ⎜

⎞

⎠ ⎟ x scale:

€

π4

2. Determine the equation of a trig function that models each graph.

-2

-1

1

2

3

4

x axis scale: 1 square =

€

π4

-4

-3

-2

-1

1

2

3

4

x axis scale: 1 square = 2π

3. Solve for x,

€

0 ≤ x ≤ 2π. (a) 2sin x = 3 (b) -3cos x + 1 = 0 (c) sin2x – 1 = 0 (d)

€

sin2 x + 3sin x +2 = 0 (e) cos2x – cosx = 0 (f) 6sin2x + sinx – 1 = 0 4. Express as a single trigonometric function: (a)

€

cos5a cos2a − sin5a sin2a (b)

€

cos 3x cos x + sin3x sin x (c)

€

sin6m cos2m −cos6m sin2m (d)

€

sinm cos 3m +cosm sin3m (e)

€

2sin(3x) cos(3x) (f)

€

cos2(6θ) − sin2(6θ) (g)

€

1−2sin2(3x)

5. If x is in the interval

€

π2,π

⎛

⎝ ⎜

⎞

⎠ ⎟ and y is in the interval

€

π, 3π2

⎛

⎝ ⎜

⎞

⎠ ⎟ and

54cos −

=x and 125tan =y , evaluate: (a)

€

sin(x + y) (b)

€

cos(x − y) (c)

€

sin(2x) (d)

€

cos(2y) (e) tan (2x) 6. Prove the following identities: (a)

€

sin x tan x = sec x −cos x (b)

€

sin x(csc x − sin x) = cos2 x

(c)

€

(1+ sin x)(1− sin x) =1

sec2 x (d)

€

csc2 x + sec2 x = csc2 x sec2 x