Pre-AP Pre-Calculus
Summer Assignment
2015-2016
You must be able to complete this packet without the use of a calculator.
~~~ THERE WILL BE A TEST FRIDAY AUGUST 28th. ~~~
**You must show all work to receive any credit!**
(Answers, examples & review of concepts are located at the end of this packet.)
PART 1 Algebra Review
I. Equations of Lines
Write the equation of each line in slope-intercept form satisfying the given
conditions.
1. Slope is  2 and y-intercept is -1.
3
2. A horizontal line through (5, -1)
3. Through (5, -3) and (-7, -1).
4. Perpendicular to the line 3x – 4y = 7
through the point (6, -1).
5. Undefined slope with the same x-intercept as the line 3x + 2y = -6.
II. Solving Systems of Linear Equations
Solve using the indicated method.
6. 2x + 5y = 8 (Elimination)
6x + y =10
8. 3x + 4y = 2 (Elimination)
2x + 5y = -1
7. 3x + y = 6
(Substitution)
6x – 5y = 12
9.
3x + 4y = 2 (Substitution)
2x + 5y = -1
III. Factoring Review
10. (x + 5)(2x – 3)
11. (2x – 5)2
12. (x2 – 2x + 1)(x + 3)
Factor completely. (Remember to factor out Greatest Common Factors first!)
13. x2 – 6x + 8
14. 27x2 – 48
15. -15x2 – 5x
16. 4x2 + 12x + 9
17. 2x2 – 3x – 9
18. x3 – 4x2 + 2x – 8
19. 8x3 – 10x2 – 3x
20. x3 – 8
21. x4 - 81
IV. Solving Equations
Solve for x. Use factoring to solve all quadratic equations
22. 3 x  1 x  5   10
23. 5x2 + 3x – 2 = 0
24. 3x2 – 48 = 0
25. 6x2 – 12 = 0
26. 2x3 – x2 – 18x + 9 = 0
27. x4 = 2x2 – 1
2
4
V. RATIONAL EXPRESSIONS
Simplify. Leave answers in factored form.
28.
x  3  x3  8
x 2  4 x 2  x  12
29.
x 2  4  2x 2  3 x  2
2x 2  5 x  2
4x 2  1
30.
32.
34.
36.
31. x  3  x
x  4 x 2
3  7
2xy 6x 2
4x 
x2  4
x3  1
x 2  5x  6
3
2
2
33.
x 2  3x  2
3x 4  10x 2  8 3x  6

35.
x2
15x 2  10
x 2  5x  6
x x2


x 2  4x  4
5
2
x x6
37.
x2  9
x 3 3 x
3
8

2x  10 3x  15
Simplify. Remember to multiply top & bottom by the Common Denominator first & cancel any
factors that can cancel.
1
38.
y2
3
2

39. x  1 x  1
2
2

x 1 x
2
x
y
1
x
VI. RATIONAL EQUATIONS
Solve each rational equation.
40. 6  8  3
x x 5
41. x  2  30  1
x  3 x2  9 x  3
42. x  3  2x  1
x2  1
x 1
PART 2 Characteristics of Functions
(These are examples of the types of problems you might expect on the test, not every case for every type of problem.
You can also look online for additional review.)
Use the graph of the given function to answer the questions.
y
1.
4
2
x
-10
-8
-6
-4
-2
2
4
6
8
10
-2
-4
-6
a) Domain: ________________
b) Range: _________________
h) Where do the minimum value(s) occur,
if any? _______________
c) What is f(-3)? ____________
i) What are the minimum value(s)? ______
d) Where is f(x) = -3? ____________
Give intervals where f(x) is:
e) What is f(0)? ____________
j) Inc: ____________________
f) Where do the maximum values occur, if
any? _____________
k) Dec: ___________________
g) Where is f(x) = 0? _______
m) Where is f(x) ≥0? _______________
l) Constant: _______________
2. Be able to justify your answer to any of these questions. For example, justify your answer to d, and m
3.
y
a) Domain: ________________
4
f(x)
b) Range: _________________
3
c) What is f(2)? ____________
d) Where is f(x) = -1? ____________
2
e) What is f(-1)? ____________
1
x
-2
-1
1
-1
-2
2
3
4
5
f) Where is f(x) increasing? __________
g) Where is f(x) > 0? _______
h) Is f(x) one to one? ___________
i) What are the roots of f(x)? __________
Find the domain and intercepts for the following functions.
3
4. f ( x ) 
x-intercept(s):
2x 2  7x  15
Domain:
y-intercept:
5. g(x) = 3x  5
Domain:
y-intercept:
x-intercept(s):
Answer the questions about the given function.
6. f(x) = x  4
a) Domain: ___________
b) What is f(x + 5)? _________
c) Where does f(x) = 5? _________
d) What are the intercepts of f? _________
Determine algebraically whether each of the following functions is even, odd, or neither.
3
7. a) f(x) =
b) f(x) = 5  x
2
x x
Determine from the graph whether each of the following functions is even, odd, or neither.
8. a)
b)
10
y
10
y
8
8
6
6
4
4
2
2
x
x
-10
-8
-6
-4
-2
2
4
6
8
-10
-8
-6
-4
-2
2
4
6
8
10
10
-2
-2
-4
-4
-6
-6
-8
-8
-10
-10
Perform the following operations for the given functions.
x
x
9. f(x) =
g(x) =
x3
x 1
a) (f – g)(x)
b) (g ◦ f)(x)
Analyze the graph at the right.
y
10. a) (f – g)(-1) = _________
f(x)
4
b) (f + g)(0) = _________
c) (f ◦ g)(3) = ________
2
g(x)
-4
-2
2
4
-2
d) (g ◦ f)(7) = ________
e) (f ◦ g)(-3) = ________
-4
-6
g) Where is (f – g)(x) = 0?
f) (g ◦ f)(2) = ________
h) Where x is (f – g)(x) > 0?
i) Where is (f – g)(x) ≤ 0?
6
8
10
x
Graph the reflections of f(x).
11.
y
x
8
a)
y
y
8
6
6
4
4
2
-6
-4
y
x
y
2
x
-8
x
-2
2
4
6
8
x
10
-8
-6
-4
-2
2
-2
-2
-4
-4
-6
-6
-8
-8
-10
-10
f(x)
4
6
8
10
X-Axis y = ______
b)
c)
y
x
8
y
y
8
6
6
4
4
2
2
x
x
-8
-6
-4
-2
2
4
6
8
-8
10
2
-4
-4
-6
-6
-8
-8
-10
Y=X
d)
y
x
8
6
4
2
x
-4
-2
-2
-10
-6
-4
-2
Y-Axis y = _______
-8
-6
-2
2
-2
-4
-6
-8
-10
Origin
4
6
8
10
y
4
6
8
10
Graph the transformations of g(x).
12.
y
a)
10
x
8
10
y
4
g(x)
2
-6
-4
y
6
4
-8
x
8
6
-10
y
-2
2
4
6
8
2
x
x
10
-10
-8
-6
-4
-2
-2
2
4
6
8
10
-2
-4
-4
-6
-6
-8
-8
-10
-10
y = g(x−2) − 5
b)
y
10
x
y
c)
10
8
8
6
6
4
4
y
x
2
2
x
x
-10
-8
-6
-4
y
-2
2
4
6
8
-10
10
-8
-6
-4
-2
2
-2
-2
-4
-4
-6
-6
-8
-8
-10
-10
y = 2g(2x)
4
6
8
10
y = −g(x+1) + 2
Describe each transformation and write an equation for the new graph in terms of f(x).
13.
a)
b)
y
y
y
4
4
4
2
2
2
f(x)
-2
2
4
6
x
x
x
-4
-4
-2
2
4
6
-4
-2
2
-2
-2
-2
-4
-4
-4
-6
-6
-6
4
6
PART 3 Quadratic Functions
GRAPHING QUADRATIC FUNCTIONS
Describe and graph each transformation of f(x) = x2.
2
2
1. f(x) = 2x − 3
2. f(x) = (x – 2) − 7
Transformations:
10
1 
3. f(x) =  x   2
2 
Transformations:
2
Transformations:
y
10
y
10
8
8
8
6
6
6
4
4
4
2
2
2
x
-10
-8
-6
-4
-2
2
4
6
8
x
10
-10
-8
-6
-4
-2
2
4
6
8
-10
-4
-4
-2
2
-2
-4
-4
-6
-6
-6
-8
-8
-8
-10
-10
-10
10
y
10
8
8
8
6
6
6
4
4
4
2
2
-2
2
4
6
8
10
6
8
10
6. f(x) =
Transformations:
y
4
1 2
x
2
Transformations:
5. f(x) = -3(x + 1)2 + 2
y
2
x
-6
-6
-2
Transformations:
-8
-8
-4
4. f(x) = -2(x − 1)2 +7
-10
x
10
-2
10
y
x
-10
-8
-6
-4
-2
2
4
6
8
10
x
-10
-8
-6
-4
-2
2
-2
-2
-2
-4
-4
-4
-6
-6
-6
-8
-8
-8
-10
-10
-10
4
6
8
10
1
(x + 2)2 − 6
2
Transformations:
7. f(x) = (x + 3)2 + 6
Transformations:
10
9. f(x) = −x2 − 5
8. f(x) =
y
10
Transformations:
y
10
8
8
8
6
6
6
4
4
4
2
2
2
x
-10
-8
-6
-4
-2
2
4
6
8
x
10
-10
-8
-6
-4
-2
2
4
6
8
x
10
-10
-8
-6
-4
-2
-2
-2
-4
-4
-4
-6
-6
-6
-8
-8
-8
-10
-10
-10
Write the equation for each transformation of f(x) = x2.
10. ____________________
11. ____________________
y
10
8
8
6
6
4
4
2
y
2
x
-10
-8
-6
-4
-2
2
4
6
8
10
x
-10
-8
-6
-4
-2
2
-2
-2
-4
-4
-6
-6
-8
-8
-10
-10
12. ____________________
10
4
6
8
10
8
10
13. ____________________
y
10
8
8
6
6
4
4
2
y
2
x
-10
-8
2
-2
10
-6
-4
-2
2
4
6
y
8
10
x
-10
-8
-6
-4
-2
2
-2
-2
-4
-4
-6
-6
-8
-8
-10
-10
4
6
4
6
8
10
WRITING QUADRATIC FUNCTIONS
1. Analyze the function f ( x)  2x 2  16x  30 . Then graph.
10
A. Domain: __________
B. vertex: __________
C. x-intercepts: _______________
D. y-intercept: __________
E. Range: __________
F. Intervals where f is increasing: _________
G. Intervals where f is decreasing: __________
y
8
6
4
2
x
-10
-8
-6
-4
-2
2
4
6
8
10
-2
-4
-6
-8
Analyze the function algebraically.
-10
2. f(x) = -4x2 + 32x − 48
a) Vertex __________
b) X-Intercepts ______________
c) Y-Intercept _____________
d) Does f(x) have a maximum or minimum? ______
e) Where does the max or min occur? ___________
f) What is the max or min? ________
g) Domain _________
h) Range __________
i) Axis of Symmetry ____________
Use the given information to write the equation of each quadratic function.
3. Its graph is a parabola with x-intercepts (2, 0) and (–1, 0) and y-intercept (0, 6).
equation: _______________
4. The function has zeros (5, 0) and (1, 0) and f(0) = 1.
equation: ________________
5. Its graph is a parabola with vertex (4, 8) and passes through the origin.
equation: ________________
6. The maximum value of g is g(-1) = 6, and g(-3) = 4.
equation: _______________
7. The vertex is (4, -1) and contains the point (2, 3).
equation: _______________
8. The intercepts of the parabola are (-1, 0), (5, 0), and (0, 15).
equation: _______________
9. Write the equation for #6 in all three forms.
Standard _______________
Vertex _______________
Root _______________
ANSWERS - PAP PreCalculus Summer Assignment
Answers to Part #I
1.
26. x = -3, ½, 3
2. y = -1
27. x = 1, -1
3.
28.
4.
29.
5. x = -2
6. (3/2, 1)
7. (2, 0)
30.
31.
8. (2, -1)
9. (2, -1)
32.
10. 2x2 + 7x – 15
33.
11. 4x2 – 20x +25
12. x3 + x2 – 5x + 3
13. (x – 2)(x – 4)
14. 3(3x + 4)(3x – 4)
15. -5x(3x + 1)
16. (2x + 3)2
17. (2x + 3)(x – 3)
18. (x – 4)(x2 + 2)
x+3
34.
35.
36.
37.
38.
19. x(2x – 3)(4x + 1)
39.
20. (x – 2)(x2 + 2x + 4)
40. x = 3, -10/3
21. (x2 + 9)(x + 3)(x – 3)
41. x = -7 (3 is extraneous)
22. x = 7
42. x = -4/3 (1 is extraneous)
23. x = -1, 2/5
24. x =
25. x =
Answers to Part #2
1. a) [-7, ), x ≠ 0
b) [-5, )
c) -2
d) 1, 5
e) Does Not Exist
f) None
g) -5, 6
h) x = 3
i) -5
j) (3, )
k) (-7, -4), (0, 3)
l) (-4, 0)
m) [-7, -5], [6, )
2. d) f(x) = -3 at x = 1 and 5 because
f(1) = -3 and f(5) = -3.
m) f(x) ≥ 0 when the graph is above
or touches the x-axis.
3. a) [-2, 4]
b) [-2, 4]
c) -2
d) .5, 3.5
e) 4
f) (-2, -1), (3, 4)
g) [-2, 0), 4
h) No
i) (0, 0)
10. a) -3
b) 1
c) -3
d) 2
e) -2
f) 0
g) -2 and 6
h) (-, -2), (6, )
i) [-2, 6]
11. a)
6.
7.
9.
6
4
4
2
2
x
-8
-6
-4
-2
2
4
6
8
x
10
-8
-6
-4
-2
-2
-2
-4
-4
-6
-6
-8
-8
-10
-10
y
c)
2
4
6
8
2
4
6
8
10
y
d)
8
8
6
6
4
4
2
2
x
-8
-6
-4
-2
12. a)
2
4
6
8
x
10
-8
-6
-4
-2
-2
-2
-4
-4
-6
-6
-8
-8
-10
-10
10
y
b)
8
10
y
6
4
4
2
2
x
-8
-6
-4
-2
c)
2
4
6
8
10
-2
2
-4
-6
-8
-8
-10
-10
y
2
x
-4
-4
-6
4
-6
-6
-4
6
-8
-8
-2
8
-10
x
-10
-2
10
-2
10
8
6
4. x ≠ 5, 
5.
y
8
6
-10
3
1

None  0, 
2
5

5 
All Reals  ,0  (0, 5)
3 
a) [-4, )
b) x  9
c) 21
d) (-4, 0) (0, 2)
a) Neither
8. a) Odd
b) Even
b) Even
 4x
a)
( x  3)( x  1)
x
b) 
3
b)
y
8
2
4
6
8
10
-2
-4
-6
-8
-10
13. a) x-axis reflection
shift down 2
y = −f(x) – 2
b) y-axis reflection
vertical stretch by 2
shift down 2
y = 2f(-x) – 2
4
6
8
10
Answers to Part #3 GRAPHING
10
1.
y
2.
8
y
10
3.
8
-8
-6
-4
6
6
4
4
4
2
2
-2
2
2
x
6
8
10
-10
-8
-6
-4
-2
2
4
6
8
10
-8
-4
-8
-6
-6
-6
-8
-8
-8
-10
-10
-10
y
10
5.
y
10
6.
8
6
6
4
4
4
2
2
2
4
6
8
10
-10
-8
-6
-4
-2
2
4
6
8
10
-8
-6
-4
-2
-4
-6
-6
-6
-8
-8
-8
-10
-10
-10
y
10
8.
y
10
9.
8
6
6
4
4
4
2
2
10
4
6
8
2
4
6
8
10
y
2
x
-10
-8
-6
-4
-2
2
4
6
8
10
x
-10
-8
-6
-4
-2
-2
-2
-2
-4
-4
-4
-6
-6
-6
-8
-8
-8
-10
-10
-10
10. f(x) = (x + 6)2 – 4
2
8
6
8
y
-2
-4
6
10
x
-10
-4
4
8
2
-2
2
6
x
-2
-2
4
8
6
8
-4
2
-4
10
-6
-2
-2
x
-10
-4
-4
-2
7.
-6
-2
8
-6
-8
-4
x
-10
x
-10
-2
10
4.
4
y
8
6
x
-10
10
11. f(x) = -3(x – 5)2 + 9 12. f(x) = -x2 + 4 13. f(x) =
1
(x – 3)2 - 4
2
Answers to Part #3 WRITING QUADRATIC FUNCTIONS:
1a) All Reals b) (-4, 2) c) (-3, 0) (-5, 0) d) (0, -30) e) (-, 2] f) (-, -4) g) (-4, )
2. a) (4, 16), b) (6, 0)(2, 0) c) (0, -48) d) Max e) max @ 4 f) max is 16 g) All Reals h) (-,
16] i) x = 4
3. f(x) = -3(x – 2)(x + 1) 4. f(x) = 1/5(X – 5)(X – 1) 5. f(x) = -½(x – 4)2 + 8
6. f(x) = -½(x + 1)2 + 6 7. f(x) = (x – 4)2 – 1 8. f(x) = -3(x + 1)(x – 5) 9. f(x) = x2 – 8x + 15
f(x) = (x-4)2 – 1 f(x) = (x – 3)(x – 5)
10
Algebra Review
I. Lines
Slope-Intercept Form
y = mx +b
Slope Formula
m=
Horizontal Lines
Vertical Lines
Parallel Lines
Perpendicular Lines
To find y-intercept…
To find x-intercept…
Slope = 0
(y = #)
Slope is Undefined. (x = #)
Slopes are the same.
Slopes are opposite reciprocals.
Set x = 0 & solve for y.
Set y = 0 & solve for x.
y 2  y1
x 2  x1
II. Solving Systems of Linear Equations
EXAMPLES:
Elimination
x+y=7
2x + y = 5
Substitution
x+y=7
2x + y = 5
Add the
equations:
-2(x + y = 7)
2x + y = 5


-2x – 2y = -14
2x + y = 5
-y = -9
y=9
Plug the value of x into one of the
equations:
x + (9) = 7
x = -2
Solution: (-2, 9)
Solve one equation for a single
variable:
x+y=7

y = -x
+7
Plug the expression for y into the
other equation and solve for x.
2x + (-x +7) = 5
2x – x + 7 = 5
x+7=5
x = -2
Solution: (-2, 9)
III. Factoring Review
Multiplying Polynomials
Perfect Square
(a+b)2 = a2 + 2ab + b2
Trinomials
(a-b)2 = a2 – 2ab + b2
(-2) + y = 7
y=9
FACTORING: Special Cases
Perfect Square
a2 – b2 = (a + b)(a – b)
Binomials
Sum of Two Cubes
a3 + b3 = (a + b)(a2 – ab + b2)
Difference of Cubes
a3 – b3 = (a – b)(a2 + ab + b2)
Factor by Grouping
Example:
X3 + 2x2 + 3x + 6
= x2(x + 2) + 3(x + 2)
= (x + 2)(x2 + 3)
IV. Solving Equations
Remember:
Multiplying everything by the common denominator & cancelling will get rid of any
pesky fractions!
V. RATIONAL EXPRESSIONS
RATIONAL EXPRESSIONS





Factor any thing that can factor.
Simplify as much as possible.
Each successive step must be equal to the step before.
Reduce if possible.
Leave your answer in factored form.
VI. RATIONAL EQUATIONS
RATIONAL EQUATIONS


Note domain restrictions on x.
Multiply both sides of the equation by the Least Common Denominator (LCD) to cancel
all denominators.
 Solve.
 Each successive step is an equivalent equation.
Check for extraneous solutions.
#1
Functions
Function: A correspondence which Domain: Set of all possible
assigns each x-value to exactly
x-values for a given function.
one y-value.
Possible restrictions to the domain:
a
1) f(x)  , b  0
One-to-One Function: A function
b
which assigns each y-value to
exactly one x-value.
2) f(x)  a, a  0
Even Function: A function that has
3) f(x) = logb a, a > 0
y-axis symmetry. f(x) = f(-x)
Odd Function: A function that has
origin symmetry. f(-x) = −f(x)
4) real-world situations
(e.g., distance cannot be
negative, number of
people must be whole
numbers, etc.)
Range: Set of all possible function Intercepts: Points where the graph
values (y-values).
crosses the axes.
Usually best to determine from a
graph of the function.
X-Intercepts: Point(s) where y = 0.
Other names: roots, zeros
Y-Intercept: Point where x = 0.
#2
Reflections
Y-Axis Reflection
X-Axis Reflection
y = f(-x)
x changes sign
y = –f(x)
y changes sign
Origin Reflection
Y = X Reflection
f(-x) = –f(x)
x and y change sign
x = f(y)
x and y are switched
Absolute Value Reflections
f(x) 
f(x)
if f(x) ≥ 0
–f(x) if f(x) < 0
 
f x 
f(x)
if x ≥ 0
f(-x) if x < 0
The part of the graph
below the x-axis is
reflected up.
The part of the graph left
of the y-axis is a reflection
of the right side.
#3
Transformations
y  a f  b(x  h)  k


Changes to y:
Changes to x:
a - Vertical Stretch or Shrink
multiply y-values by a
a < 0 x-axis reflection
a > 1 vertical stretch
0 < a < 1 vertical shrink
b - Horizontal Shrink or Stretch
k - Vertical Shift
add k to y-values
k > 0 shift up
k < 0 shift down
h - Horizontal Shift
(x – h) add h to x-values
divide x-values by b
b < 0 y-axis reflection
b
> 1 horizontal shrink
0<
b
(x +
< 1 horizontal stretch
shift right
h) subtract h from x-values
shift left
#4
Linear Function
f(x) = x
6
y
4
2
x
-6
-4
-2
2
-2
4
6
x
-4
-2
0
2
4
y
-4
-2
0
2
4
-4
-6
Domain: (-, )
Origin Symmetry
Range: (-, )
Odd Function
One-To-One
#5
Quadratic Function
f(x) = x2
6
y
x
-3
-2
-1
0
1
2
3
4
2
x
-6
-4
-2
2
4
6
-2
-4
-6
y
9
4
1
0
1
4
9
Domain: (-, )
Y-Axis Symmetry
Range: [0, )
Even Function
Three Forms of the Quadratic Function
Standard Form: f(x) = ax2 + bx + c
Vertex Form: f(x) = a(x − h)2 +k
Vertex:
 b  b 


 2a , f   2a  




Vertex: (h, k)
 r  r  r  r 
Root Form: f(x) = a(x − r1)(x − r2) Vertex:  1 2 , f  1 2  
 2 
 2
Height of a Projectile (feet)
h(t) = -16t2 + v0t + s0