Download Pre-Calculus 110 – Exam Review Trigonometry (Reference

Pre-Calculus 110 – Exam Review
Trigonometry (Reference: Chapter 2, Sections 2.1 & 2.2)
Outcomes:

Demonstrate an understanding of angles in standard position, 0    360

Solve problems, using the three primary trigonometric ratios, for angles from 0 to 360 in
standard position.
Short Response:
1. Sketch an angle of 310 in standard position.
2. In which quadrant does the terminal arm of a 140o angle in standard position lie?
3. Which angle in standard position has a different reference angle than all the others?
a. 1250
b. 1550
c. 2050
d. 3350
4. What is the exact value of
a.
cos 150o?
b. sin 225
c. tan 270
5. Illustrate /explain how you would determine (without the use of a calculator) that cos180  1 .
Extended Response:
The point (-12, -15) is on the terminal arm of angle,  , in standard position.
1.
a. Determine the exact values of the sine, cosine and tangent of  .
b. Determine the value of  .
12
Consider A such that cos A 
2.
13
a. In which possible quadrant(s) is this angle?
b. If the sine of A is negative, in which quadrant is the angle?
c. Write exact values for the other two primary trigonometric ratios for
Solve for all values of
3.
sin  
 3
, for 0    360
2
cos  
1
2
, for 0    360
tan   0.4663
when
A in this quadrant.
Quadratics (Reference: Chapters 3 & 4)
Outcomes:

Factor polynomials such as
2x 2  x  3
25x 2  16y 2
2(x  5)2  3(x  5)  9
9(2x  1)2  4(y  2)2

Analyze quadratic functions of the form y  a(x  p )  q and determine the vertex, domain and
range, direction of opening, axis of symmetry, x-and y-intercepts.

Analyze quadratic functions of the form y  ax  bx  c to identify characteristics of the
corresponding graph, including the vertex, domain and range, direction of opening, axis of symmetry,
x-and y-intercepts, and to solve problems.

2
2
Solve problems that involve quadratic equations
Short Response:
1.
Write the equation of the axis of symmetry for f(x) = –6 (x – 3)2 – 7?
2.
What are the domain and range of y = 7 (x – 6)2 + 9?
3.
State the vertex and direction of opening of the quadratic function f(x) = –3(x + 2)2 – 9.
4.
What is the standard form of the function y = 2(x – 4)2 – 2?
5.
State whether y = 4x2 – 36x – 43 has a maximum or minimum value and identify the vertex
coordinates.
6.
Solve (x+4)(2x-9) = 0
7.
Determine the number and nature of the roots of x 2  3x  3  0
Extended Response:
1.
Determine the vertex form of the quadratic function a. y = x2 + 24x + 29?
2.
Determine the roots of the quadratic equation 0 = 0.2x2 – 1.4x + 2?
3.
Determine the exact, simplified, solutions to 2x 2  4x  3  0 .
4.
A rectangle with an area of 2504 cm 2 is x cm wide and (x+8) cm long. Find, to the nearest tenth of a
centimeter, the width and length.
5.
Sketch the graph of the quadratic function f (x )  2x 2  12x  10 . Identify the vertex, domain,
range, axis of symmetry, x-and y-intercepts.
6.
Factor 2(x  5)2  3(x  5)  9
7.
Factor 9(2x  1)2  4(y  2)2
8.
Use completing the square to convert y  5x 2  20x  6 to vertex form.
9. On a forward somersault dive, Joanelle’s height, h, in metres, above the water is approximately
modeled by the function
where t is the time in seconds after she leaves the board.
a.
b.
c.
d.
Find Joanelle’s maximum height above the water.
How long does it take her to reach the maximum height.
How long is it before she enters the water?
How high is the board above the water?
10. Two whole numbers differ by 3. The sum of their squares is 89. What are the numbers?
11. A garden measures 12 m by 8 m. A walkway of uniform width is to surround the garden. How wide
should the walkway be so that the walkway has the same area as the garden?
12. Determine the quadratic function in vertex form that has a maximum value located at (–3, –5) and
passes through (–1, –21).
Systems of Equations and Inequalities (Reference: Chapters 8 & 9)
Outcomes:



Solve, algebraically and graphically, problems that involve systems of linear-quadratic and quadraticquadratic equations in two variables.
Solve problems that involve linear and quadratic inequalities in two variables.
Solve problems that involve quadratic inequalities in one variable
Short Response:
1. Write the inequality that
represents the graph shown.
2.
The line y = 9x – 4 intersects the quadratic function y = x2 + 7x – 3 at one point. What are the
coordinates of this point of intersection?
3.
Which graph represents the system of equations:
y = –1.5x2 – 2x + 5
y = 2.3x + 0.2
Extended Response :
1. Solve algebraically each system of equations.
a.
b.
5x 2  3y  3  x
2x 2  x  4  2y
2x 2  4x  y  3
4x  2y  7
2. Solve the inequality 3x 2  9 x  6 .
3.
Write the inequality represented by
the graph to the right. Express your final
answer in standard form.
4. Graph the inequality y  5( x  3)2  4 .
Radical Expressions and Equations (Reference: Chapter 5)
Outcomes:
 Solve problems that involve operations on radicals and radical expressions with numerical and
variable radicands.
 Solve problems that involve radical equations.
Short Response:
1.
2.
State the side length of a square with an area of 1573cm 2 in simplified radical form.
Write 4 32x 9y 5 in simplest form. Identify the values of the variable for which the radical represents a
real number.
3.
Simplify
12 22
.
4 11
4.
Simplify
2 34
3
5.
State the restrictions on x in
2x  1 if the radical is a real number.
Extended Response:
1.
Simplify each of the following. Rationalize any denominators if necessary.
a. 2 50  3 18  4 32
b.
5
6
2.

1080 

3

7
2 9a
3
135
8
f.
7 11
9  33

g.
3 7 2 2
2 7 2
3 2 64 5
c.
d.

3
e.

7 4 3 2 7 5 3

Solve each radical equation. Remember to check for extraneous roots.
a. x  x  2  4
b.
2x  5  3x  2  1
Rational Expressions and Equations (Reference: Chapter 6)
Outcomes:
 Determine equivalent forms of rational expressions (limited to numerators and denominators that are
monomials, binomials or trinomials).
 Perform operations on rational expressions (limited to numerators and denominators that are
monomials, binomials or trinomials).
 Solve problems that involve rational equations (limited to numerators and denominators that are
monomials, binomials or trinomials).
Short Response:
1.
State the non-permissible value(s) for the rational expression
2.
What is the simplified version of the rational expression
3.
a b 
Simplify

3ab  3ab 
6a 4b 7
2
4
12
.
x 4
2
3 x  12
?
32  8 x
7 2
4 3
Extended Response:
6  x  2x 2
1. Simplify in lowest terms and identify the non-permissible values.
6x 2  x  15
2. Simplify in lowest terms and identify the non-permissible values for each of the following.
a.
x 2  9x
2x 2  x  15 2x 2  x  10


x 2  x  6 x 2  7x  10
x 2  25
b.
x 1
x 3
 2
x  x  12 x  7x  12
c.
x 3
8x  4
x 2  4x  21
 2
 2
x  2 x  2x  35 2x  5x  3
2
2
4

d. x  7 x  7
x
2

2
x  49 x  7
4.
Solve
y 5
5
7y  4

 2
y  4 y  2 y  6y  8
5.
A ski resort can manufacture enough machine-made snow in 12 h to open its steepest run, whereas it
would take naturally falling snow 36 h to provide enough snow. If the resort makes snow at the same time
that it is snowing naturally, how long will it take until the run can open?
6.
A marketing executive travelled 810 miles on a corporate jet in the same amount of time it took to
travel an additional 162 miles by helicopter. The rate of the jet was 360 mph greater than the rate of the
helicopter. Find the speed of the jet.
Absolute Value Functions (Reference: Chapter 7, Sections 7.1 - 7.3)
Outcomes:
 Demonstrate an understanding of the absolute value of real numbers.
 Graph and analyze absolute value functions (limited to linear and quadratic functions) to solve
problems.
Short Response:
7  73  ( 6)
1.
Evaluate
2.
Determine the solution 2x  8  6  3 .
3.
Which graph represents the solution to
x  7  6?
2
.
Extended Response:
1. Given the function
f (x )  2x  3
a. Sketch the graph of f(x).
b. State the domain and range of the function.
c. Express f(x) as a piecewise function.
2. Given the function f(x) = |x2 + 4x + 2|
a. Sketch the graph of the
function.
b. State the domain and range of
the function.
c. Express f(x) as a piecewise
function.
3. Solve the absolute value equations. Remember to check for extraneous solutions.
a. |4x + 8| = –8x + 3
b.