pre-calculus 110 review - mrs. power...pre-calculus 110 review trigonometry (reference chapter 2,...
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Pre-Calculus 110 Review
Trigonometry (Reference Chapter 2, Sections 2.1 -2.2, pages 74-99)
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
1. For each angle listed below
a. Draw the angle in standard position and indicate in which quadrant the terminal arm is located.
b. Indicate the reference angle on your diagram.
c. State two other angles (other than the one in quadrant I) that have the same reference angle.
i. 245 ii. 305 iii. 100
2. Which angle in standard position has a different reference angle than all the others?
105 165 255 285
3. Determine the measure of the three angles in standard position, (90 360 ) that have a reference angles of
a. 65 b. 10 c. 85
4. Each point listed below is situated on the terminal arm of an angle in standard position. Determine the exact
trigonometric values for sin , cos , and tan .
a. P (3,-4) b. Q (-7, -24) c. A (-5, 12)
5. Find , 0 360 .
a. sin233 sin b. cos317 cos c. tan136 tan
d. sin98 sin e. cos261 cos f. tan76 tan
6. Determine the exact value of
a. sin120 b. cos270 c. tan300 d. sin330 e. cos315 f. tan210
7. is an angle in standard position. Find the two other primary trigonometric values if the terminal arm of is in
quadrant IV and 6
cos7
.
8. Consider in standard position such that 9
tan5
.
a. Determine the exact values for the other two primary trigonometric ratios.
b. Determine the possible value(s) of , if 0 360
9. Solve for where 0 360
a. 3
sin2
b. 1
cos2
c. tan 1 d. sin 1 e. 1
cos2
f. 1
tan3
10. Solve each equation, where 0 360 . Round to the nearest degree.
a. sin 0.45 b. cos 0.36 c. tan 2.41
Quadratic Functions
(Reference Chapter 3, Sections 3.1 – 3.3, pages 142-203)
Outcomes:
• Analyze quadratic functions of the form 2( )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 2y ax bx c to identify characteristics of the corresponding graph,
including vertex, domain and range, direction of opening, axis of symmetry, x- and y-intercepts, and to solve
problems.
1. Indicate the vertex, the equation of the axis of symmetry, direction of opening, the maximum or
minimum value, the domain and range for each of the following quadratic functions.
a. 22
( ) 2 93
f x x
b. 2( ) 2 8 5g x x x c. 24y x d. x-intercepts at -3 & 7, max value is 1
2. Determine the equation, both in vertex and standard form, for each of the following quadratic
functions.
a. Maximum point is at (-3,-5) and graph passes through
the point (-1,-21)
b. x-intercepts are at -2 and 4, with a minimum of -18
c. Graph of function is shown to the right
3. By inspection, determine the number of x-intercepts for each
quadratic function.
a. 2( ) 2( 5) 7f x x b. 21
( ) 4 13
g x x
4.
213 8
2y x
a. Sketch the graph of the quadratic function.
b. Identify the vertex, domain, range, axis of symmetry, x- and
y-intercepts, max/min value.
5. 2( ) 2 12 10f x x x
a. Sketch the graph of the function.
b. Identify the vertex, domain, range, axis of symmetry, x- and y- intercepts.
6. Determine the vertex form of the following quadratic functions by “completing the square”.
a. 2 5 29y x x b. 25 20 6y x x
7. A lifeguard at a public beach has 400 m of rope available to lay out a
restricted swimming area using the straight shoreline as one side of the
rectangle. What dimensions will maximize the swimming area?
8. Calculators are sold to students for $20 each. Three hundred students are
willing to buy them at that price. For every $5 increase in price, there are
30 fewer students willing to buy the calculator. What selling price will
produce the maximum revenue and what will the maximum revenue be?
9. A ligting fixture manufacturer has a daily costs of 2( ) 800 10 0.25C x x x , where C is the total daily cost in
dollars and x is the number of light fixtures produced. What is the minimum cost for daily production for this
company? How many fixtures are produced to yield a minimum cost?
Quadratic Equations (Reference Chapter 4, Sections 4.1-4.4, pages 206-262)
Outcomes:
• Factor polynomial expressions of the form: 2 , 0ax bx x a 2
2 3x x 2 2 2 2 , 0, 0a x b y a b 2 2
25 16x y 2( ( )) ( ( )) , 0a f x b f x c a 2
2( 5) 3( 5) 9x x 2 2 2 2( ( )) ( ( )) , 0, 0a f x b g y a b 2 2
9(2 1) 4( 2)x y
• Solve problems that involve quadratic equations.
1. Factor.
a. 25 20x x b. 2 12 32x x c. 23 14 5x x
2. Factor each of the following. Use substitution as a strategy to aid in the factoring.
a. 2
2 5 3 5 9x x b. 2
2 22 11 2 24x x x x c. 2 216
25 3 749
x y
3. Solve each quadratic equation by factoring.
a. 24 24p p c. 20.2 1.4 2 0x x e. 210 12 26q q
b. 2 5 36 0x x d. 22 14 16t t f.
220 6
7 7x x
4. A park has an area of 22484 m . The width is 8 m shorter than the length. Determine the dimensions of the park.
5. A mural is to be painted on a wall that is 15 m long and 12 m high. A border of uniform width is to surround the mural. If the mural is to cover 75% of the area of the wall, how wide must the border be?
6. Determine the number and nature of the roots.
a. 26 5 6 0x x b.
24 9 12x x c. 23 2 10x x
7. Solve.
a. 27 6 5x x b.
23 4 11 0a a
8. A field is in the shape of a right triangle. The fence around the perimeter of the field measures 40 m. If the
length of the hypotenuse is 17m, find the lengths of the other two sides.
9. A diver jumps upward from a platform. The height, h(t), in metres, of the diver above the water is modeled by 2( ) 5 10 40h t t t , where t is the time in seconds after she leaves the board.
a. Find the diver’s maximum height above the water. How long does it take the diver to reach this height?
b. How long is it before the diver enters the water?
c. At what time(s) is the diver at a height of 42m?
d. How high is the platform above the water?
Systems of Equations
(Reference Chapter 8, Sections 8.1-8.2, pages 424 -460)
Outcomes:
• Solve, algebraically and graphically, problems that involve systems of linear-quadratic and quadratic-quadratic
equations in two variables.
1. Solve each of the following systems of equations using the corresponding graphs.
a. 2
3 4
1
y x
y x
b.
2
2
2 2
4 2
y x x
y x x
2. Determine the values of m and n if (3, 14) is a solution to the following system of equations: 2
2
32
5
mx y
mx y n
3. Solve each of the following linear-quadratic systems of equations.
a. 2
9 4
3 7
x y
y x x
b.
2
2 3
3 5
y x
y x x
c.
2
2 4 3
4 2 7
x x y
x y
4. The sum of two whole numbers is 17. Thirteen less than three times the square of the smaller yields the larger
number. Determine the two numbers.
5. Solve each system of equations algebraically.
a.
2
2
7 2
4 3
y x x
x x y
b.
25 3 3
22 4 2
x y x
x x y
6. Jack is skeet shooting. The height of the skeet is modelled by the equation 25 32 2h t t , where h represents the height in metres t seconds after the skeet is released. The path of Jack’s bullet is modelled by the equation
31.5 1h t with the same units. How long will it take for the bullet to hit the skeet? How high off the ground will the skeet be when it is hit?
Linear and Quadratic Inequalities (Reference Chapter 9, Sections 9.1-9.3, pages 464-505)
Outcomes:
• Solve problems that involve linear and quadratic inequalities in two variables.
• Solve problems that involve quadratic inequalities in one variable.
1. Determine the inequality that corresponds to the graph illustrated to the
right.
2. Sketch the graph of each of the following linear inequalities.
a. 2 3y x b. 6 3y x c. 4 3 12x y
3. Solve.
a. 3 2 7 0x x b. 23 9 6x x c.
211 10 6x x d. 22 2 7x x
4. A firework is shot straight up into the air with an initial velocity of 500 feet per second from 5 feet off the
ground. The equation 2( ) 16 500 5h t t t models the path of the firework, where t is the time in seconds and
h(t) is the height of the firework in feet. Determine when the firework will be higher than 2000 ft. Extension: When will the firework be below 2000 ft?
5. State the inequality represented by the graph to the right. Express your final answer in standard form.
6. Sketch the graphs of the following inequalities.
a. 2
3 4y x b. 2 4 5y x x c.
24 4 3y x x
7. A garden is to be in the shape of a rectangle. The length of the garden must be 4 m longer than the width. The
gardener wants the area of the garden to be no greater than 480 square meters. Find all the possible dimensions.
Radical Expressions and Equations
(Reference Chapter 5, Sections 5.1-5.3, pages 272 -307)
Outcomes:
• Solve problems that involve operations on radicals and radical expressions with numerical and variable
radicands.
• Solve problems that involve radical equations (limited to square roots).
1. Write each radical as an entire radical.
a. 5 3 b. 3b b c.
33 5k k d. 3 42 3r r
2. Write each radical in simplest form.
a. 300 b. 450 c. 3 160 d. 4 324
3. Simplify.
a. 6 80 2 20 b. 5 12 2 27 c. 5 175
2526 8
d. 3 32 375 3 648
4. State the side length of a square with an area of 21573cm in simplified radical form.
5. Write 9 54 32x y in simplest form. Identify the values of the variable for which the radical represents a real
number.
6. Simplify.
a. 3 5 6 4 24 b. 2 7 3 5 4 5 7 c. 3 3 3 32 4 5 2 3 2 4
7. Simplify.
a. 24 14
8 2 b.
9 5
4 3 c.
7 5
4 2 3 d.
5 3 2
2 3 2
8. State the restrictions on x in each of the following:
a. 2 1x b. 3
9 x
9. Square the expression 3 2 5 x .
10. Solve each of the following. Check for extraneous roots.
a. 4 2 3 12x b. 2 1 9x x
c. 1 6 1p p d. 1 3 8 4 7z z
Rational Expressions and Equations
(Reference Chapter 6, Sections 6.1- 6.4, pages 310 -355)
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 mononials,
binomials or trinomials) • Solve problems that involve rational equations (limited to numerators and denominators that are monomials,
binomials or trinomials)
1. State the non-permissible value(s) in each rational expression.
a. 2
5
3
w
gj b.
3 2
4 5
x
x x
c.
2
4 3
3 10
p
p p
2. Simplify each rational expression. State the non-permissible values.
a. 3 12
32 8
x
x
b.
2
2
2 3
2 3
r r
r r
c.
2 2
2 2
4
2 5 3
x y
x xy y
3. Simplify. State the non-permissible values.
a. 2 2
2
2 1
1 6
x x x
x x x
b.
2 2
3 2 3 2
2 15 9 20
4 2 7 4
x x x x
x x x x x
c.
2 3
2 2 3 2
4 2
2 3 3
x x x
x x x x x x
4. Add or subtract. Give answers in simplest form. Identify all non-permissible values.
a. 2 2
3 1 2 2
2 2 2 8 6
x x
x x x
b.
2 2
2 2 1
4 12 4
x x
x x x
c. 2 2 2
5 2 3
1 4 3 2 3x x x x x
5. Simplify each expression. Identify all non-permissible values.
a.
4 4
31
12
rr
r r
b.
2
2 4
7 72
49 7
x xx
x x
6. Simplify each of the following expressions. Identify the non-permissible values.
a. 2
2 2
3 8 4 4 21
2 2 35 2 5 3
x x x x
x x x x x
b.
3 3 2
2 2
1 4 3 5 2
5 6 7 3 2 1
x x x x x x
x x x x x
7. When it was raining, Elliott drove for 120 miles. When the rain stopped, he drove 20 mph faster than he did
while it was raining. He drove for 300 miles after the rain stopped. If Elliott drove for a total of 10 hours, how
fast did he drive while it was raining?
8. Solve.
a. 2
2
2 5 17
1 1 1
x x
x x x
b.
5 26
1 1m m
c.
2
24 5 2 3
8 8
b b b
b b b b
9. A ski resort can manufacture enough machine-made snow in 12 hours to open its steepest run, whereas it would
take naturally falling snow 36 hours 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?
10. A positive integer is 4 less than another. The sum of the reciprocals of the two integers is 10
21. Find the integers.
11. Mark is kayaking in a river which flows downstream at a rate of 1mph. He paddles 5 miles downstream and then
turns around and paddles 6 miles upstream. The trip takes 3 hours. How fast can Mark paddle in still water?
Absolute Value Functions
(Reference Chapter 7, Sections 7.1- 7.3, pages 358 -391)
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:
1. Evaluate
a. 25 6 8 9 2 5 4
b. 37 0.4 5 2
c. 27 7 6
2
2. Write each of the following as a piecewise function.
a. 3 5y x b. 4 7y x c. 2 3 10y x x d.
26y x
3. Sketch the graph of each of the following absolute value functions. State the domain and range of each.
a. 3 4y x b. 2 3y x c. 2 4 12y x x d. 211
4y x
4. Solve.
a. 3 8 9 0x b. 4 8 8 3x x
c. 23 7 3 3x x x d. 21
2 2 22
x x x
SOLUTIONS :
Chapter 2 – Trigonometry
1.
a. 180 245 180 65R
2 other angles are :
1 180 180 65 115R
2 360 360 65 295R
b. 360 360 305 55R
2 other angles are :
1 180 180 55 125R
2 180 180 55 235R
c. 180 180 100 80R
2 other angles are :
1 180 180 80 260R
2 360 360 80 280R
2. If 105 ; 180 180 105 75R
If 165 ; 180 180 165 15R
If 255 ; 180 255 180 75R
If 285 ; 360 360 285 75R
Conclusion : 165 has a reference angle that is
different from the others.
3.
a. If 65R
1 2 3180 180 65 115 ; 180 180 65 245 ; 360 360 65 295R R R
b. If 10R
1 2 3180 180 10 170 ; 180 180 10 190 ; 360 360 10 350R R R
c. If 85R
1 2 3180 180 85 95 ; 180 180 85 265 ; 360 360 85 275R R R
4.
a. 223, 4 3; 4; 3 4 25 5P x y r .
Then4 3 4
sin ;cos ; tan5 5 3
b. 2 2
7, 24 7; 24; 7 24 625 25P x y r .
Then24 7 24
sin ;cos ; tan25 25 7
c. 2 25,12 5; 12; 5 12 169 13P x y r .
Then12 5 12
sin ;cos ; tan13 13 5
5.
a. 233 233 180 53 and 0 in quadrants III & IV R
360 360 53 307 sosin 233 sin 307 R
b. 317 360 317 43 and cos 0 in quadrants I & IV R
43 so cos317 cos 43 R
c. 136 180 136 44 and tan 0 in quadrants II & IV R
360 360 44 316 so tan136 tan 316 R
d. 98 180 98 82 and sin 0 in quadrants I & II R
82 so sin 98 sin82 R
e. 261 261 180 81 and cos 0 in quadrants II & III R
180 180 81 99 so cos261 cos99 R
f. 76 76 and tan 0 in quadrants I & III R
180 180 76 256 so tan 76 tan 256 R
6.
a. 120 is in quadrant II where sin 0 and 180 120 60 R .
3 3Since sin 60 , then sin120
2 2 .
b. 0
cos 270 0x
r r .
c. 300 is in quadrant IV where tan 0 and 360 300 60 R .
Since tan60 3, then tan300 3 .
d. 330 is in quadrant IV where sin 0 and 360 330 30 R .
1 1Since sin 30 , then sin 330
2 2 .
e. 315 is in quadrant IV so cos 0 and 360 315 45 R .
1 1Since cos 45 , then cos315
2 2 .
f. 210 is in quadrant III where cos 0 et 210 180 30 R .
1 1Since tan 30 , then tan 210
3 3 .
7.
a. 2 26Since cos and is in quadrant IV 6; 7; 7 6 13
7
xx r y
r
13 13Thus sin and tan
7 6
y y
r x.
8. a. 2 29
Since tan and could be in quadrant II 5; 9; 5 9 1065
y
x y rx
9 5Thus sin and cos
106 106
y x
r r.
could also be in quadrant IV, thus 9 5
sin and cos106 106
y x
r r
b. 61 . Therefore =180 -61 =119 or =360 -61 =299R
9.
a. 1 3sin 60 and since sin 0, is in quadrants III or IV
2
R .
1 2Thus 180 180 60 240 and 360 360 60 300 R R .
b. 1 1cos 45 and since cos 0, is in quadrants I or IV
2
R.
1 2Thus 45 and 360 360 45 315 R R .
c. 1tan 1 45 and since tan 0, is in quadrants II or IV R .
1 2Thus 180 180 45 135 and 360 360 45 315 R R .
d. Since sin 90 1, 90 is the only solution since 90° is a quadrantal angle .
e. 1 1cos 60 and sincecos 0, is in quadrants II or III
2
R.
1 2Thus 180 180 60 120 and 180 180 60 240 R R .
f. 1 1tan 30 and since tan 0, is in quadrants I or III
3
R.
1 2Thus 30 and 180 180 30 210 R R .
10.
a. 1sin 0,45 27 and since sin 0, is in quadrants I or II R.
1 2thus 27 and 180 180 27 153 R R .
b. 1cos 0,36 69 and since cos 0, is in quadrants II or III R.
1 2Thus 180 180 69 111 and 180 180 69 249 R R .
c. 1tan 2,41 67 and since tan 0, is in quadrants II or IV R.
1 2Thus 180 180 67 113 and 360 360 67 293 R R .
Chapter 3 – Quadratic Functions
1.
a. 2
; 2; 9 so the vertex is at 2, 9 ;A.S.: 2; opens down3
a p q x
maximum @ 2 9; D : ; R : | 9f x R f x R f x
b.
288
2; 2; 5 3 so the vertex is at 2, 3 ;A.S.: 2;opens up2 2 4 2
a p q x
minimum @ 2 3; D : ; R : | 3g x R g x R g x
c. 1; 0; 4 so the vertex is at 0,4 ;A.S.: 0; opens down a p q x
maximum @ 0 4; D : ; R : | 4y x R y R y
d. 3 7
0; 2; 1; so the vertex is at 2,1 ;A.S.: 2; opens down2
a p q x
maximum @ 1 when 2; D : ; R : | 1y x x R y R y
2.
a. , 1, 21 ; , 3, 5 x y p q
2
2
2 2 2 2
21 1 3 5 16 4 so 4
4 3 5 (vertex form)
standard form: 4 3 5 4 6 9 5 4 24 36 5 4 24 41
a a a
y x
y x x x x x x x
b. 2 4
, 2,0 ; 1; 182
x y p q
2
2
2 2 2 2
0 2 1 18 18 9 so 2
2 1 18 (vertex form)
standard form: 2 1 18 2 2 1 18 2 4 2 18 2 4 16
a a a
y x
y x x x x x x x
c. , 0,2 ; , 3,8 x y p q
2
2
2 2 2 2
22 0 3 8 6 9 so
3
23 8 (vertex form)
3
2 2 2 2standard form: 3 8 6 9 8 4 6 8 4 2
3 3 3 3
a a a
y x
y x x x x x x x
3. a. Since a > 0 & q < 0, f(x) has 2 x-intercepts b. Since a < 0 & q < 0, g(x) has 0 x-intercepts
4. Vertex (3,8), domain { | }x x R , range { | 8, }y y y R , AS: x=3,
x-intercepts -1 and 7, y- intercept 7
2, max value y = 8
5. Vertex (3,-8), domain { | }x x R , range { | 8, }y y y R , AS: x=3,
x-intercepts 1 and 5, y- intercept 10
6. a.
2
2
2
2
( ) 5 29
( ) 5 29
25 25( ) 5 29
4 4
5 91 -5 91( ) ; Vertex ,
2 4 2 4
f x x x
f x x x
f x x x
f x x
b.
2
2
2
2
2
2
( ) 5 20 6
( ) 5 4 6
( ) 5 4 4 4 6
( ) 5 4 4 4 -6
( ) -5 -2 20 6
( ) 5( 2) 14
(2,14)
f x x x
f x x x
f x x x
f x x x
f x x
f x x
Vertex
7.
2
2
2
2
2
width
400 2 length
A = x(400-2x)
400 2
2 400
2( 200 )
2( 200 10000 10000)
2( 100) 20000
x
y x
A x x
A x x
A x x
A x x
A x
Dimensions are 100 m by 200 m. Maximum enclosed area is 220000m .
8.
2150 900 6000R x x
Max. R is @ 900
32 300
bx
a
; R = 7350
Maximum revenue is $7350 and it occurs when x = 3 which would bring the price of the calculators up to $35.
There would be 3 price increases of $5. They would have 210 sales instead of 300.
9.
2( ) 800 10 0.25
10 cost @ 20
2 0.5(20) $700
C x x x
bMinimum x
aC
20 light fixtures can be produced for a minimum cost of $700.
Chapter 4 – Quadratic Equations
1.
a. 25 20 5 4x x x x b. 2 12 32 4 8x x x x
c. 23 15 3 1 3
3 14 53
x xx x
5 3 1
3
x x 5 3 1x x
2.
a. Let 5 a x
2 22 6 2 3 2
2 5 3 5 9 2 3 92
a ax x a a
3 2 3
2
a a
3 2 3
3 2 3 5 3 2 5 3 5 3 2 10 3 8 2 7
a a
a a x x x x x x
b. 2Let 2 b x x
22 2 2
2 2 2 2
2 11 2 24 11 24 3 8
3 8 2 3 2 8 2 3 2 8 3 1 4 2
x x x x b b b b
b b x x x x x x x x x x x x
c. Let c 3 and 7x d y
2 2 2 216 16 4 4 4 4
25 3 7 25 5 5 5 3 7 5 3 749 49 7 7 7 7
4 4 4 45 15 4 5 15 4 5 19 5 11
7 7 7 7
x y c d c d c d x y x y
x y x y x y x y
3.
a. 24 24p p
24 24 0
4 6 0
0 or 6
p p
p p
p p
b. 2 5 36 0x x
9 4 0
9 or 4
x x
x x
c. 20.2 1.4 2 0x x
20.2 7 10 0
0.2 5 2 0
5 or 2
x x
x x
x x
d. 22 14 16t t
2
2
2 14 16 0
2 7 8 0
2 8 1 0
8 or 1
t t
t t
t t
t t
e. 210 12 26q q
2
2
10 26 12 0
2 5 13 6 0
5 15 5 22 0
5
52
q q
q q
q q
3 5 2
5
q q
0
3 5 2 0
23 or
5
q q
q q
f. 220 6
7 7x x
2
2
6 200
7 7
16 7 20 0
7
6 15 6 810
7 6
31
7
x x
x x
x x
2 5 2x 3 4
6
x
0
12 5 3 4 0
7
5 4 or
2 3
x x
x x
4. Let x be the length and x 8 be the width of the park. The area of the park is given by
28 8 2484A x x x x
2 8 2484 0
54 46 0
54 or 46
x x
x x
x x
The length of the park is 54 m and the width is 54 – 8 = 46 m
5.
The width of the mural is approximately 0.89 m.
6.
a. 2 26; 5; 6; 4 5 4 6 6 25 144 169 0a b c D b ac
2 real and distinct roots
b. 224; 12; 9; 4 12 4 4 9 144 144 0a b c D b ac
2 real and equal roots
c. 223; 2; 10; 4 2 4 3 10 4 120 116 0a b c D b ac
2 non-real roots
7.
a. 7; 6; 5a b c
22 6 6 4 7 54 6 176 6 4 11 3 2 11
2 2 7 14 14 7
b b acx
a
b. 3; 4; 11a b c
22 4 4 4 3 114 4 116
2 2 3 6
b b acx
a
(No real solutions)
8.
‘
9.
a. 10
12 10
bx
a
. The diver reaches the maximum height after 1 second. The maximum height is 45 m.
b.
2
2
5 10 40 0
5( 2 8) 0
5( 4)( 2) 0
4 and t = -2
t t
t t
t t
t
It takes the diver 4 seconds to reach the water.
c. 5; 7; 4a b c
2
2
5 10 40 42
5 10 2 0
10 100 400.225 and 1.77
10
The diver reaches a height of 42m at 0.225 s & again at 1.77 s.
t t
t t
t
d. 0 40 mh
Chapter 8 – Systems of Equations
1.
a. 1 1 2 2, 0, 4 and , 3,5x y x y b. 1 1 2 2, 2,6 and , 1 ,3x y x y
2.
2 32
9 14 32
2
mx y
m
m
2 6
2(9) 5(14)
52
mx y n
n
n
3.
a. 2 2
9 4 9 4
3 7 7 3
x y y x
y x x y x x
2
2
2
9 4 7 3
2 1 0
1 0
1
x x x
x x
x
x
If 1
9 1 4 5
Solution: 1,5
x
y
b. 2 2
2 3 3 2
3 5 3 5
y x y x
y x x y x x
2
2
3 2 3 5
6 7 0
7 1 0
7 or 1
x x x
x x
x x
x x
If 7
3 7 2 23
If 1
3 1 2 1
Solutions: 7, 23 et 1,1
x
y
x
y
c. 22 4 3
4 2 7
x x y
x y
2
2
2(2 4 3)
4 2 7
4 8 2 6
4 2 7
x x y
x y
x x y
x y
__________________
2
2
4 4 1
4 4 1 0
(2 1)(2 1) 0
1
2
x x
x x
x x
x
4. let x = smaller number and let y= larger number
2
2
2
17
3 13
3 13 17
3 30 0
3 10 3 0
10
3
x y
x y
x x
x x
x x
x or 3x
2
If 3
3 3 13 14
The two numbers are 3 and 14.
x
y
5.
a.
2 2
2 2
7 2 2 7
4 3 4 3
y x x y x x
x x y y x x
2 2
2
2 7 4 3
2 6 4 0
2 1 2 0
1 or 2
x x x x
x x
x x
x x
2
2
If 1
1 2 1 7 6
If 2
2 2 2 7 7
Solutions: 1,6 and 2,7
x
y
x
y
4 2 7
14 2 7
2
9
2
1 9Solution: ,
2 2
x y
y
y
b.
25 3 3
22 4 2
x y x
x x y
2
2
2
2
2(5 3 3 0)
3(2 2 4 0)
10 6 2 6 0
6 6 3 12 0
x y x
x y x
x y x
x y x
___________________
24 5 6 0
( 2)(4 3) 0
32
4
x x
x x
x x
Solutions to the system: 3 35
( 2, 7) and ,4 16
6.
2
2
5 32 2 31.5 1
5 0.5 1 0
0.5 0.25 200.4 and 0.5
10
t t t
t t
t
The skeet was shot at 0.5 seconds after release at a height of 16.75m
Chapter 9 – Linear and Quadratic Inequalities
1. 2 points on the graph are (4,2) and (0,5)
5 2 3
0 4 4m
3
54
y x
2.
a. 2 3y x b. 6 3y x
c. 3 4 12x y
3.
a. 7
3 2 7 0 intercepts @ 3 and ; 3 2 7 opens up2
x x x x x y x x .
7
3 2 7 0 so 3 or 2
x x x x
b. 2 23 9 6 3 9 6 0 3 1 2 0 roots @ 1 and 2x x x x x x x x .
23 1 2 opens down, so 3 9 6 0 when 1 2y x x x x x .
c. 2 2 2 511 10 6 6 11 10 0 3 2 2 5 0 roots @ and
3 2x x x x x x x x .
2 2 5 26 11 10 opens up, so 6 11 10 0 when or
2 3y x x x x x x .
d.
22
2 27 7 4 2 24 7 33
2 2 7 2 7 2 02 2 2 4
b b acx x x x x
a
7 33 7 33roots @ 3.19 and 0.31
4 4x x
2 2 7 33 7 332 7 2 opens up, so 2 7 2 0 when
4 4y x x x x x
4.
2
222
16 500 5 2000
500 500 4 16 19954 500 12232016 500 1995 0
2 2 16 32
t t
b b act t t
a
4.70 and 26.6 2 216 500 1995 opens down, so 16 500 1995 0 when 4.70 26.6y x x t t t
The firework is above 2000 ft between 4.70 and 26.6 seconds.
Extension : The firework is below 2000 ft at times less than 4.7 seconds and between 26.6 seconds and 31.26 sec.
(Firework hits the ground at 31.26 seconds)
5.
6.
a. 2
3 4y x
1; 3; 4
0 5
a p q
y
2
2
2
2
0 3 4
0 6 9 4
0 6 9 4
0 6 5
0 5 1
5 or 1
x
x x
x x
x x
x x
x x
b. 2 4 5y x x
2
1
42
2 1
45 9
4 1
0 5
a
p
q
y
20 4 5
0 5 1
5 or 1
x x
x x
x x
c. 24 4 3y x x
2
4
4 1
2 4 2
43 4
4 4
0 3
a
p
q
y
20 4 4 3
0 2 3 2 1
3 1 or
2 2
x x
x x
x x
7.
Chapter 5 – Radical Expressions and Equations
1.
a. 25 3 5 3 75 b.
23 3 6 7b b b b b b b
c. 23 3 2 2 3 2 3 53 5 3 5 3 5 9 5 45k k k k k k k k k
d. 3 33 3 34 4 3 3 4 3 4 732 3 2 3 2 3 8 3 24r r r r r r r r r
2.
a. 2300 100 3 10 3 10 3 b.
2450 225 2 15 2 15 2
c. 3 33 3 3160 8 20 2 20 2 20 d.
44 44 4324 81 4 3 4 3 4
3.
a. 6 80 2 20 6 4 5 2 2 5 24 5 4 5 20 5
b. 5 12 2 27 5 2 3 2 3 3 10 3 6 3 4 3
c. 5 175 5 1 5 40 5 45
252 6 7 5 7 5 7 7 7 7 76 8 6 8 8 8 8 8
d. 3 3 3 3 3 3 32 375 3 648 2 5 3 3 6 3 10 3 18 3 8 3
4. 1573 121 13 11 13 cm (side length of the square)
5. 9 5 8 4 24 4 432 16 2 2 2 , 0, 0x y x x y y x y xy x y • • • • •
6.
a. 3 5 6 4 24 5 18 4 72 5 3 2 4 6 2 15 2 24 2 9 2
b. 2 7 3 5 4 5 7 8 35 12 5 2 7 3 35 5 35 60 14 5 35 44
c. 3 3 3 3 3 3 33 3 32 4 5 2 3 2 4 6 8 15 4 2 16 5 8 6 2 15 4 2 2 2 5 2
3 3 3 312 15 4 4 2 10 22 15 4 4 2
7.
a. 24 14 24 14
3 78 28 2
b.
9 5 3 9 15 9 15 3 15
4 3 12 44 3 3
c.
7 5 4 2 3 28 5 14 15 28 5 14 15 28 5 14 15 14 5 7 15
16 4 3 16 12 4 24 2 3 4 2 3
d. 5 3 2 2 3 2 10 3 5 6 2 6 2 30 2 7 6 32 7 6
4 3 2 12 2 102 3 2 2 3 2
8. a.
2 1 0
1
2
x
x
b. 3
9
9 0
9
9
x
x
x
x
9.
3 2 5 3 2 5
9 6 5 6 5 4( 5)
9 12 5 4 20
4 12 5 11
x x
x x x
x x
x x
10.
a. 4 2 3 12x
2
2
2 3 3
2 3 3
2 3 9
2 12
x
x
x
x
6
if 6
4 2 6 3 4 12 3 4 9 4 3 12
6 is a solution
x
x
x
b. 2 1 9x x
2 2
2
2
2
2 1 9
2 1 9
4 1 81 18
4 4 81 18
22 85 0
5 17 0
x x
x x
x x x
x x x
x x
x x
5 or 17
if 5
5 2 5 1 5 2 4 5 2 2 5 4 9
5 is a solution
if 17
17 2 17 1 17 2 16 17 2 4 17 8 25 9
17 is an extraneous root
x x
x
x
x
x
c. 1 6 1p p
2 2
22
1 6 1
1 6 2 6 1
2 6 6
6 3
6 3
p p
p p p
p
p
p
6 9
3
if 3
Left side 3 1 4 2
Right side 3 6 1 9 1 3 1 2
3 is a solution
p
p
p
p
d. 1 6 2 5 4z z
2 2
2 2
2
2
1 6 2 5 4
1 2 6 2 6 2 5 4
2 6 2 2 2
6 2 1
6 2 1
6 2 2 1
4 5 0
5 1 0
z z
z z z
z z
z z
z z
z z z
z z
z z
5 or 1
if 5
Left side 1 6 2 5 1 6 10 1 16 1 4 5
Right side 5 4 5 5 20 25 5
5 is a solution
if 1
Left side 1 6 2 1 1 6 2 1 4 1 2 3
Right side 5 4 1 5 4 1 1
1 is an extraneous root
z z
z
z
z
z
Chapter 6 – Rational Expressions and Equations
1.
a. 0; 0g j b. 0; 5x x
c. 2 3 10 5 2 5; 2p p p p p p
2.
a.
3 43 12 3; 4
32 8 8 4 8
xxx
x x
b. 2
2
3 12 3
2 3
r rr r
r r
2 3 1r r
3 3; 1;
2 3 2
rr r
r
c. 2 2
2 2
24
2 5 3
x yx y
x xy y
2
2
x y
x y
2; 2 ; 3
33
x yy x x y
x yx y
3.
a. 2 2
2
22 1
1 6
x xx x x
x x x 1x
1 1
x x
3 2 x x
21 or ; 3; 1; 2
3 3
x x x xx x x
x x
b. 2 2
3 2 3 2
52 15 9 20
4 2 7 4
xx x x x
x x x x x
2
3x
x
4
x
x
4x
2 1
5
x
x
4x
2
2
3 2 1 2 7 3 1 or ; 0; 4; 5;
4 4 2
x x x xx x x x
x x x x
c. 2 3
2 2 3 2
24 2
2 3 3
xx x x
x x x x x x
2x
x 2x
x
3
2
x
x
3x
2x 3x
23 3 or ; 0; 2; 3
3 3
x x x xx x x
x x
4.
a.
2 2
2 1 3 1 3 2 1 13 1 2 2 3 1
2 2 2 8 6 2 1 1 2 3 1 2 1 1 3 2 3 1 1
x x x x xx x x
x x x x x x x x x x x x x
2 2 2 5 1 13 8 3 2 4 2 5 4 1
2 1 1 3 2 1 1 3
x xx x x x x x
x x x x x x
2 1x
5 1; 3; 1
2 1 31 3
xx x
x xx x
b.
2 2
2 1 2 1 2 1 62 2 1 1
4 12 4 6 2 2 2 6 2 2 2 2 6
x x x x xx x x
x x x x x x x x x x x x x
2 2 2 22 6 4 7 6 2
6 2 2 6 2 2
xx x x x x x
x x x x x x
1
6 2
x
x x
1
; 6; 26 22
xx x
x xx
c. 2 2 2
5 2 3 5 2 3
1 4 3 2 3 1 1 3 1 3 1x x x x x x x x x x x
5 3 2 1 3 1 5 15 2 2 3 3
1 1 3 3 1 1 3 1 1 1 1 3
6 20; 3; 1
1 1 3
x x x x x x
x x x x x x x x x x x x
xx x
x x x
5.
a.
2 2
4 4
4 12 ( 12) 4( 3) 12 4( 3) ( 12) 4( 12)31 3 ( 12) 3 ( 12) 3 ( 4)( 3) 3( 4)
12
r r r r r r r r r rrr r r r r r r r r r r
r r
0, 12, 3, 4r
b.
2
2 42 7 4 7 2 72 4 27 7
2 7 7 7 7 7 7 7 7 7
49 7
7 72 14 4 28 2 14 2 42 2 42 14; 7,
7 7 7 7 7 7 3 14 3 14 3
x x x xxx xx x x x x x x x x x
x x
x xx x x x x xx
x x x x x x x x
6.
a. 2
2 2
4 2 13 8 4 4 21 3
2 2 35 2 5 3 2
xx x x x x
x x x x x x
7x
7
5
x
x
3x
2 1x 3x
3 4
2 5
x
x x
2 23 5 4 2 8 15 4 8 4 23 1; 2; 7; 5; 3;
2 5 5 2 2 5 2 5 2
x x x x x x x xx x x x x
x x x x x x x x
b.
3 3 2
2 2
1 4 3 5 2 1
5 6 7 3 2 1 5
xx x x x x x x
x x x x x x
2 2x x
7 1x x
3 1x
1x
x 3 1x 2x
1 2
5 7
x x
x x
2 21 7 2 5 8 7 3 10 17 11
5 7 7 5 5 7 5 7
15, 1,7,0, 2,
3
x x x x x x x x x
x x x x x x x x
x
7.
8.
a.
2
2
2 5 17
1 1 1
x x
x x x
2
2
2 2
2
2 5 17
1 1 1 1
2 1 5 1 17
2 2 5 5 17
7 12 0
3 4 0
3 or 4
x x
x x x x
x x x x
x x x x
x x
x x
x x
1
So 3 is a solution
and 4 is a solution.
x
x
x
b.
5 26
1 1m m
2
2
5 1 2 1 6 1 1
1 1 1 1 1 1
5 1 2 1 6 1 1
5 5 2 2 6 6
6 7 3 0
2 3 3 1 0
3 1 or
2 3
m m m m
m m m m m m
m m m m
m m m
m m
m m
m m
1
3So is a solution
2
1and is a solution.
3
m
m
m
2
2
2
120 30010
20120( 20) 300 10 ( 20)
120 2400 300 10 200
0 10 220 2400
0 22 240
0 ( 30)( 8)
30 8
Elliott drove 30 mph while it was raining.
x xx x x x
x x x x
x x
x x
x x
x x
c. 2
24 5 2 3
8 8
b b b
b b b b
2 2
2 2
2
24 5 2 3
8 8
5 2 3 824
8 8 8
24 5 2 13 24
8 8 8
24 5 2 13 24
9 0
9 0
0 or 9
b b b
b b b b
b b b bb
b b b b b b
b b b b b
b b b b b b
b b b b b
b b
b b
b b
0; 8
so 0 is extraneous
and 9 is a solution.
b b
b
b
9.
112 36
x x
3 36
36 36 36
4 36
9 hours
x x
x
x
10.
The two positive numbers are 7 and 3.
Time to open hill Fraction completed in
1 hour
Fraction of work
completed in x hours
Machine-made snow 12 1
12
12
x
Natural snow 36 1
36
36
x
Together x 1
x 1
11.
Mark paddles at 4mph.
Distance (miles) Speed (mph) Time (h)
With the current 5 x+1 5
1x
Against the current 6 x–1 6
1x
Total 4
Chapter 7 – Absolute Value
1.
a. 25 6 8 9 2 5 4 31 17 3 4 31 17 3 4 21
b. 37 0.4 5 2 7 4.6 8 7 4.6 8 32.2 8 40.2
c.
27 7 ( 6) 7 49 6 7 43 7 43 5025
2 2 2 2 2
2.
a.
53 5 if
5 33 5 0 invariant point @ ;
533 5 if
3
x x
x x y
x x
b.
74 7 if
7 44 7 0 invariant point @ ;
744 7 if
4
x x
x x y
x x
c. 2 3 10 5 2 0 invariant points @ 5 and 2 x x x x x x
2
2
2
3 10 if 5 or 2 3 10 opens up,
3 10 if 2 5
x x x xy x x y
x x x
d. 26 6 6 0 invariant points @ 6 x x x x
2
2
2
6 if 6 6 6 opens down, y
6 if 6 or 6
x xy x
x x x
3.
a. Sketch the graph of 3 4y x and reflect the piece that is negative.
D : ; R : | 0x R y R y
b. Sketch the graph of 2 3y x and reflect the piece that is negative.
D : ; R : | 0x R y R y
c. Sketch the graph of 2 4 12 y x x and reflect the piece that is
negative.
D : ; R : | 0x R y R y
d. Sketch the graph of 211
4y x and reflect the parts that are
negative.
D : ; R : | 0x R y R y
4.
a. 3 8 9 0 3 8 9x x No solution
b. 4 8 8 3x x
4 8 if 2 (case 1)4 8 0 invariant point @ 2; 4 8
4 8 if 2 (case 2)
x xx x x
x x
Case 1 2 x
4 8 8 3
12 5
5
12
5 is a solution
12
x x
x
x
x
5
12x
Case 2 2 x
4 8 8 3
4 8 8 3
4 11
11
4
11 is an extraneous root
4
x x
x x
x
x
x
c. 23 7 3 3x x x
73 7 if (case 1)
7 33 7 0 invariant point @ ; 3 7
733 7 if (case 2)
3
x x
x x x
x x
Case 1 7
3 x
2
2
1
2
3 7 3 3
4 0
2 2 0
2
2 is a solution
2
2 is a solution
x x x
x
x x
x
x
x
x
2x
Case 2 7
3 x
2
2
2
2 2
3 7 3 3
3 7 3 3
6 10 0
D 4 6 4 1 10 4 0
since D 0, no real solutions for Case 2
x x x
x x x
x x
b ac
d. 212 2 2
2x x x
2
22
2
2
2
2
12 2 4 2
21 4 2 82 2 0 2 2 2
12 2 12
2
1 invariant points @ 2 2 2 0.83 and 2 2 2 4.83; 2 2 opens up
2
12 2 if (2 2 2) or (2 2 2) (case 1)
1 22 2
122 2 if
2
b b acx x x
a
x x y x x
x x x x
x x
x x
(2 2 2) (2 2 2) (case 2)x
Case 1 0.83 or x 4.83x
2
2
2
1
1
12 2 2
2
4 4 4 2
2 8 0
4 2 0
4
4 is an extraneous root
2
2 is a solution
x x x
x x x
x x
x x
x
x
x
x
Case 2 0.83 x 4.83
2
2
2
1
1
12 2 2
2
4 4 2 4
6 0
6 0
0
0 is a solution
6
6 is an extraneous root
x x x
x x x
x x
x x
x
x
x
x
2; 0x x