Download PreCalc Academic - Garnet Valley School District

Pre-Calculus Academic
Summer Work and List of Topical Understandings
Students should practice each of the following skills to prepare for your Pre-Calculus class. Be
prepared to turn in your math packet on the first day of class. A graded assessment about these
skills will be given at the beginning of the semester. Complete as much of this packet on your
own as you can, then get together with a friend, e-mail your teacher, or “google” the topic.
SHOW YOUR BEST WORK.
Requirements
The following are guidelines for completing the summer work packet…
ü You must show all of your work. Use separate paper if necessary.
ü Be sure all problems are neatly organized and all writing is legible.
ü We expect you to come in with certain understandings that are prerequisite to Pre-Calculus.
A list of these topical understandings is below.
Topics within summer work…
v Exponent Rules
v Radical and Rationals
v Transformations of Functions
v Function Notation, Composite Functions and Inverse Functions
v All things Quadratic (solving, zeros, factoring, graphing, quadratic formula)
v Log Rules
v Triangle Trigonometry including right triangles, law of sines and law of cosines, evaluating
trig. functions and solving trig. equations
v Graphing Functions
v Interpreting and comprehending word problems
v Graphing, simplifying expressions, and solving equations of the following types:
trigonometric, rational, absolute value, logarithmic, exponential, polynomial/power, and
radical.
Finally, we suggest not waiting until the last two weeks of summer to begin on this packet. If
you spread it out, you will most likely retain the information much better. Once again this is
due, completed with quality, on the first day of class. It is intended to help you be successful in
the coming year.
GarnetValleyHigh School
Practice Set of Required Math Skills for
Pre-Calculus
Skill 1: Exponent Rules
All students should be able to complete operations involving fractions and recall exponent facts
quickly and accurately without the use of a calculator.
Evaluate if x = 8, y = -2 and
2
1.
5.
⎛ y ⎞
⎜ ⎟
⎝ 5 ⎠
-34
3
4
2.
16
6.
(-3)-4
3.
(x – y)-2
4.
(xyz)0
7.
x4 = 16
8.
x3 = -343
Skill 2: Exponent Rules Continued
All students should be able to apply the rules of exponents to simplify and expand expressions,
evaluate expressions and to solve equations. (Give answers with no negative exponents)
3
n −3
1.
x ⋅x
4.
y 12
y3
5.
7.
t −5 v −2
t 6 v −7
8.
2.
2
10.
(x + 4 )
13.
⎛ 6 ⎞
⎜ ⎟
⎝ 5 ⎠
⎛ 49 ⎞
⎜ ⎟
⎝ 4 ⎠
y 4+ 2 x
y2
−
1
2
4 3
(− 2x )
3.
x 5 y −3
w 0 z −1
t −1 − v
v
6.
x
x +y
9.
−2
−3 2
11.
(x − 5)
12.
(− 2 x )
14.
−
1
2 x −2
15.
(3x )
−2
−3
x9
−3 −2
Skill 3: Radicals and Rationals
All students should be able to simplify radicals and perform operations on radicals. (Give
answers with rationalized denominators)
Simplify, add or subtract.
1.
300
2.
192
2
3.
18
2
4.
3 1
÷
2 2
5.
x
2x
−
4+ y 4+ y
6.
2
x
+
x + 2x 4x + 8
7.
2 4
−
t v
8.
5
3
+
x −4 x−2
2
2
x
x+3
x+2
x−
9.
x 1
+
y x
x2 + y
xy
11.
10.
6
1
−
x + 2 x − 15 x − 3
1
−1
x+5
12.
3x 2 − 5 x − 2
( x2 − 4) (6 x + 2)
2
Skill 4:Transformations of Functions.
Students should be able to identify, graph and write equations of parent functions based on
vertical and horizontal shifts, reflections over the x and y axis, and vertical and horizontal
stretches/compressions.
List the transformations of the parent function.
1.
f(x) = 2|x + 3| - 4
2.
g(x) =
2x
3.
h(x) = (-x – 1)2
Use the function f(x) shown below to graph the transformations in problems #4 - 8
y
4
4.
g(x) = f(x – 1)
5.
g(x) = f(x) + 1
6.
g(x) = ½f(x)
7.
g(x) = f(1/2x)
8.
g(x) = -½f(x + 2)
3
2
1
x
−4
−3
−2
−1
1
−1
2
3
4
−2
−3
−4
9.
Write the function that results from shifting f(x) = x3 , 5 units to the right, 2 units down,
reflected over the x-axis, and a vertical stretch of 4.
Skill 5: Function Notation, Composite Functions and Inverse Functions.
Students should be able to comfortably evaluate functions, operations of functions, composition
of functions. Students should be able to find an inverse function and any necessary domain
restrictions so the inverse is itself a function.
Evaluate for the given functions.
1.
⎧ 2 x + 4 x < −2
f ( x) = ⎨
⎩−3x + 2 x ≥ −2
a. f(1)
Given f ( x) =
b. f(-3)
2.
c. f(-2)
⎧ 3
⎪ x − 1 if
g ( x) = ⎨ 2
⎪ x + 2 if
⎩
a. g(0)
x≤2
x>2
b. g(2)
c. g(6)
5
, g(x) = x – 6 and h(x) = x2 – 4x – 12, find each function or value.
x+3
3.
(f – g)(2)
4.
(h – g)(x)
5.
⎛ g ⎞
⎜ ⎟ (8 )
⎝ h ⎠
6.
(gh)(5)
7.
( g o f ) (−2)
8.
h(g(x))
Find the inverse function for each f(x). List any domain restrictions to f(x) so the inverse is a
function.
2
9.
12.
f ( x) = x − 12
f ( x) = x 2 − 4
3
11.
f(x) = 3x
14.
f(x) = log2x
Skills 6: Quadratics.
Students should be able to factor quadratic and polynomial expressions, solve quadratic
equations, find the zeros and graph quadratics.
Factor:
1.
b 2 + 11b − 26
4.
5x2 − 7 x − 6
7.
9 + 8x – x2
2.
5.
8.
Solve:
10.
3x(x – 1) – x(x – 8) = 3
12.
4 x 2 = 20
x 4 − 81
100 x 2 − 75
x3 − 64
3.
6.
9.
11.
13.
36m 2 − 25n12
3x 2 − 5 x − 2
4 x2 + 8x + 4
2 x 2 − 7 x = 15
x2 + 2x + 3 = 0
14.
16.
x 2 − 144 = 0
x3 – 9x2 – x + 9 = 0
15.
17.
x3 − 5 x = 0
x 2 + 4 x = −12
Graph the following.Give the coordinates of the vertex and the zeros.
18.
f(x) = -2(x – 3)2 + 4
19.
g(x) = ½x2 – 4x + 3
Graph the following.
20.
y > x2 + 2x + 1
Skill 7: Log Rules and Equations
All students should be able to apply the rules of logarithms to simplify and expand expressions,
evaluate expressions and to solve equations. (Give answers with no negative exponents)
1
1.
2.
3.
log10
log 2 16 + log 2
lne 5
4
4.
log1 5. log m x + log m 5 = log m 10 6. log 6 x + log 6 (2 x + 1) = 2
Express as a single logarithm
7. log a 4 + log a n
8. log x − log 5
Express in terms of log m and log n.
11. log m n
12. log n(10 m )3
9. 5 log 2 w
10. ln v − 2(ln 4 + ln u )
Skill 8: Triangle Trigonometry
Students should be familiar with right triangle trigonometry, law of sines, law of cosines,
trigonometric functions, inverse trig. functions, special right triangles and the unit circle.
Find the exact value of the 6 trigonometric functions for each angle.
π
2
1.
2.
3.
θ = −45°
tan ϑ = −
2
5
90° < ϑ < 180°
4.
θ = 210°
7.
Find the exact vales of the 6 trig.
functions of θ below.
10
5.
θ=−
π
4
6. θ = − 420°
8. Find the exact values of the 6 trig.
of an angle whose Terminal side
passes through (3, −8) .
6
Solve the following equations. Give all answers in the interval ⎡⎣0o,360o
9.
sinx = ½
10.
tanx = 0
11.
2cos2x – 1 = 0
)
Evaluate. Give all answers in the interval [0, 2π )
⎛
3 ⎞
arcsin ⎜⎜ −
⎟⎟
⎝ 2 ⎠
Solve the triangle (find all missing angles or sides)
15.
InΔABC , b = 23.4, c = 14.7, ∠C = 37.2°
12.
arccot(-1)
13.
14.
16.
In ΔABC, a = 14 cm, b = 9 cm, c = 6 cm
17.
A lighthouse keeper observes that there is a 3° angle of depression to a sinking ship. If
the keeper is 19 m above the water, how far is the sinking ship from shore?
arccos(−1)
Find the missing sides of the 45-45-90 or 30-60-90 triangles.
45-45-90
1.
2.
3.
4.
5
16
6
30-60-90
5.
7.
6.
9.
12
60
10
6
8.
5
Skill 9: Graphing Functions
Students should be able to sketch graphs of non-linear functions, by using a table of values and
through transformations.
Graphing: Sketch a graph, showing the coordinates of two or more important points or
asymptotes.
1
1.
2.
3.
y= x
y=
y = 2x
x
x
⎛ 1 ⎞
x
4.
5.
6.
y = ⎜ ⎟
y=e
y = x2 + 4
⎝ 3 ⎠
2
7.
8.
9.
y = (x + 3)
y = 2( x − 4) 2 − 1
y = −2x 2
10.
13.
16.
y = x 2 − 4x + 3
y = sin x
π ⎞
⎛
y = 2 cos⎜ x − ⎟
4 ⎠
⎝
11.
y = 2− x+4
12.
y = log 2 x
14.
y = cos x
15.
y = tanx
17.
y = 3 + sin 2 x
Selected Answers:
Skill 1: #2) 8
#5) -81
Skill 2: #2) 2/7
#10) x2 + 8x + 16
x
x+3
Skill 4: #1) Left 3, down 4, vertical stretch of 2
Skill 3: #4)
3
#14) −
x2
2
#9)
#2)
horizontal compression of 2
Skill 5: #1c) 8
#11) f −1 ( x) = log 3 x
#7) -1
Skill 6: #8) (x – 4)(x2 + 4x + 16)
Skill 7: #2) 5
#7) log a 4n
Skill 8: #3)
sin
π
cos
π
tan
2
π
2
2
=1
=0
= und
π
csc
sec
2
π
2
cot
#15) x = 0, + 5
=1
= und
π
2
=0
#9) x = 30 o, 150 o