FINAL EXAM REVIEW

MCT 4C

COLLEGE TECHNOLOGY

 


Polynomial Functions and Equations

        Factoring types

common, difference of squares, sum & difference of cubes, Trinomial (Type I and II), grouping, and factor theorem.

        Solving equations

quadratic equations: with and without the linear term, using factoring where possible and the quadratic formula where factoring is not possible, with real and complex solutions.

non-quadratic equations (i.e. linear, cubic, and quartic)

        Graphing polynomials with degrees 2, 3, and 4

        Solving non-linear inequalities

 

1.      How do you know if you can use trinomial factoring on an expression?

2.      How can you tell if a quadratic expression is factorable without actually factoring?

3.      How can you identify a difference of squares pattern?

4.      Explain how to identify a sum of cubes and provide an explanation as to how to factor it.

5.      Identify what type of factoring you would use on each of the following expressions and explain why you would use that method:

a) b) c)

 

6.      Factor Fully.

 

a) b) c)

 

d) e) f)

 

g) h) I)

 

7.      Factor each of the following fully. (Do NOT expand before factoring!)

a) b)

 

8.      What are the things you think is important to remember when solving a polynomial equation? If you were asked what type of factoring you would use first, what order would you put the different factoring skills in?

9.      Solve each of the following using the most efficient method. (No decimals)

a) b) c) d)

e) f) g) h)

i) j) k) l)

 

m) n) o)

 

10. Explain the steps required to sketch a polynomial function using its roots.

 

11. Sketch each of the following functions using the x-intercepts, and basic shape of the curve.

a) b) c)

 

12.

a) Find the family of quartic functions whose x-intercepts are 2, 1, 1 and 3.

 

b) Find the specific quartic function that has the intercept in a) and passes through the point (2, 6).

13.

a) Find the family of cubic functions whose roots are 2, 1 and 0.

 

b) Find the specific function that has the roots in part a) and passes through the point (2, 6).

 

14. Find the specific function that passes through the points (3, 0), (2, 0), (4,0) and (0, 4)

 

Exponents and Logarithms

 

 

        Simplifying using exponent rules

        Evaluating (EXACT VALUES) of exponential expressions

        Solving exponential equation Using same base, factoring, and logs.

        Evaluating (EXACT VALUES) of logarithmic expressions

        Convert from log to exponential form and from exponential to log form

        Write as a single logarithm/ expanding logs

        Solve Logarithmic equations Using same base, and exponential form

        Exponential Growth and Decay problems using logarithms

 

 

 


1.      Evaluate without a calculator:

 

a) b) c) d)

 

2.      Simplify each of the following.

a) b) c)

 

3.      Express as a single power with base 5:

 

4.      Write each logarithm in exponential form.

a) b) c)

 

5.      Write each exponential in logarithmic form.

a) b) c)

 

6.      Convert the following functions into its inverse function

 

a) b) c)

 

7.      Evaluate each logarithm without a calculator.

a) b) c) d)

 

e) f) g)

 

h) i)

 

 

8.      Expand each of the logarithms.

a) b)

 

 

9.      Write Each as a Single Logarithm:

a)     b)

 

10. Solve for the unknown. (Give to one decimal place if needed.)

 

a) b) c) d)

e) f) g)

h) i)

 

j) k)

 

 

 

 

 

11. If x equals , y equals , and z equals simplify the following into terms of m and n.

 

12. Show that

 

13. What are the restrictions on the following logarithms, and explain how you would convert them to exponential form.

 

a) b) c)

 

 

14. The population of a bacteria grows at 17% per day.

a)     How many days will it take for the bacteria to double in size?

b)     After 30 days, the bacteria have grown to a population to a size of 1.45 x 104. Approximately how many bacteria was there initially?

 

15. In 1992, the population of the Vancouver area was 1.69 million and was increasing at a rate of 1.7% per year.

 

a)     Write an equation to represent the population of the Vancouver area, P, as a function of the number of years, y, since 1992.

 

b)     Calculate the approximate year that the population will be double that of 1992.

 

16. In 1976, a research hospital bought half of a gram of radium for cancer research. Assuming

the hospital still exists, how much of this radium will the hospital have in the year 6836,

if the half life of the radium is 1620 years?

 

17. There are 50 bacteria present initially in a culture. In 3min., the count is 204800.

What is the doubling period?

18. In a recent dig, a human skeleton was unearthed. It was later found that the amount of 14C

in it had decayed to of its original amount. If 14C has a half life of 5760 years,

how old was the skeleton?

 

 

 

 

 

 

 

Trigonometry in the Coordinate Plane

 

       

Exact Values

 
Evaluate Expressions

        Solving Trigonometric Equations Special Triangles and Unit Circle

Cast Rule

Factoring

Using Radians

        Prove Identities

        Graphing Trigonometric Functions ( or )

        Domain / Range

        Amplitude, Period, Phase Shift, Vertical Translation, Reflection (Vertical and Horizontal)

 

1. Evaluate the following: a) b) c)

2. If , make a diagram and find the other five ratios, if q is acute.

3. If and q is obtuse, find the exact values of the other five ratios

and the approximate value of q ,

 

4. Determine the exact value.

a) b) c) d)

e) f) 5sec30tan60

5. If , find all possible values of 0 q 360 .

 

6. Solve each of the following for . Where necessary, round to one decimal place.

a) b) c)

d) e) f)

g) h) i)

j) k)

 

  1. Prove the following identities.

a) b)

c) d)

e) f)

g) h)

 

  1. State the domain, range, period, amplitude, vertical translation and phase shift of each of the following:

a) b) c)

d) e) f)

 

  1. Sketch two cycles of each function in #2.

 

Applications of Trigonometric Ratios and Vectors

 

        Pythagorean Theorem

        Primary Trig ratios (sin x , cos x, and tan x) and Inverse Trig Ratios (cscx, secx, tanx)

Solving for the angle and solving for the side

        Sine Law Solving for the angle and solving for the side

        Cosine Law Solving for the angle and solving for the side

        Ambiguous Case of Sine Law

        Application questions -> Angle of elevation and depression.

        Multi-step questions (Indirect Measurement) you need to find more than one thing to get to the final answer

        Resultant and Equilibrant Vectors

        Resolving Vectors

        Applications of Vectors

 

 


1.      State the formula for each and describe when you should use them:

a)     a primary trig ratio

b)     the Pythagorean theorem

c)      the sine law

d)     the cosine law

 

2.      Solve for the unknown side. (Round to 2 decimals if necessary.)

 


a) b)

 

 

 

 

 

 

 

 

3.      Solve each of the triangles. (Round your angle to the nearest degree and the side length to 1 decimal place.) Give ALL possible solutions.

 

a)     Given DRST where ST = 9.0 m, S = 90o, and T = 46o.

b)     DJKL is given with JL = 5.0 cm, JK = 2.7 cm and K = 57o.

c)      Given triangle ABC where AB = 9.8 cm, A = 27o and C = 112o.

d)     DPQR has PQ = 7 cm, QR = 6 cm, and PR = 9 cm.

 

4.      Explain how you know when solving a triangle that you must check for the ambiguous case.

5.      A triangular park measures 250.0 m along one side. The other two sides form angles of 32o and 56o with the first side. Calculate the area of the park.

 

6.      Two boxes are 65 cm apart. The angle of depression from the top of taller box to the top of the shorter box is 22o. How high is the taller box given that the shorter box is 25 cm tall?

 

7.      Two office towers are 31.7 m apart. The angle of elevation from the top of the shorter tower to the top of the taller tower is 27o. The angle of depression from the top of the shorter tower to the base of the taller tower is 48.2o. Calculate the height of each tower.

 

8.      Explain the difference between vectors and scalars.

 

9.      How does a vector sum result in the zero vector? Explain using diagrams.

 

10. Label the following vector in two different ways:

 

11. Find the resultant vector of the forces 50 N at [E35N] and 65N at [W40N] graphically.

 

12. Given the following diagram, use components to determine the magnitude and direction of the resulting force:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

13. The pilot of an airplane that flies at 800 km/h wishes to travel to a city 800 km due east. There is a 80 km/h wind from the north east.

a)     What should the planes heading be?

b)     How long will the trip take?

 

Geometry and The Circle

 

        Calculating Perimeter, Area, Volume and Surface Area

        Arc Length and Chord of a Circle

        Determining the equation of a circle given radius and the coordinates of the centre

        Solving linear-quadratic systems by substitution to determine the point of intersection of a line and a circle

        Determining the equation of a tangent to a given circle

 

1.      Determine the equation of the circle with diameter of 8 units and centre at (-8,-1).

2.      Solve each system. No decimals allowed.

a) b) c)

 

3.      Determine the point(s) of intersection between the circle and the line

4.      What two numbers differ by 5 and have squares with a sum of 196?

5.      For a circle with radius 7 cm, determine the exact value of the arc length subtended by an angle of 315.

6.      The hypotheneuse of an isosceles right triangle is the chord of a circle. The equivalent sides in the triangle represent the circles radius, and the length of the chord is 12 cm. Determine the length of the arc subtended by the chord.

7.      Determine equations of the tangents with slope -1 to the circle x2+y2=6.