Precalculus
Math 170
Fall 2016
Read this worksheet and be sure you can do ALL the problems, and get help if needed!
Number systems – Counting (or Natural), Whole, Integers, Rational, Irrational, Real, Complex
Notation
1) Roster notation: a list of elements
EX) {0, 1, 2, 3, 4, . . . }
2) Description: describes the set
EX) Whole Numbers
3) Set Builder: a math sentence describing the elements
EX) {x| x < 4}
[use when can’t list all the elements]
reads: all x such that x is less than 4
4) Interval: use ( ) when you do not include the endpoint
Use [ ] when you do include the endpoint For ex) (a, b) = {x| a < x < b}
[a, b) = {x| a ≤ x < b}
(-∞, b] = {x| x ≤ b}
Ex 1) Graph [1, 2)
Ex 2) Graph (-∞, -5]
Sets – Unions and Intersections
Union (U) – A U B is the set of all elements in A together with all the elements in the set B.
Ex 3) A = {x ≤ 5} & B = {x<3} What is A U B?
Intersection (∩) - A ∩ B is the set of all elements in common to both sets A and B.
Ex 4) A = {x ≤ 5} & B = {x<3} What is A ∩ B?
Exponent Rules
1) aman = am + n
2) am/an = am – n
3) (am)n = am n
4) (ab)n = anbn
5) (a/b)n = an/bn
6) a-n = 1/an
7) a0 = 1 (a≠0)
Ex 5) x4 x7
when multiplying with the same base, add the exponents
when dividing with the same base, subtract the exponents
when raising a power to an exponent, multiply the exponents
when multiplying inside parenthesis, distribute the exponent
when dividing inside parenthesis, distribute the exponent
negative exponent means reciprocal
anything to zero power (other than zero) equals one
Ex 6) (g4)5
Rational Exponents - am/n =
n
am
Factoring
S2 – T2 = (S + T)(S – T)
Difference of Squares
Ex 7) 4-2
Ex 8) (y-2 x z4)-3
Ex 9) 41/2 =
Ex 10) 82/3 =
S3 – T3 = (S – T)(S2 + ST + T2)
Difference of Cubes
Fractions
A . C = AC
A ÷ C = AD
Always reduce!
B D
BD
B D
BC
Ex 11) ¾ ÷ ¼
Ex 12) x2 + 2x - 3 . 3x + 12_
x–1
x2 + 8x + 16
Ex 13) 3
+ x
x–1
x+2
Ex 14)
1
x2 – 1
_
2
(x + 1)2
S3 + T3 = (S + T)(S2 – ST + T2)
Sum of Cubes
To Add or Subtract Must Find LCD!!
Compound Fractions – multiply every term by the LCD to clear fractions
x
1
y
Ex 1)
y
1
x
Rationalizing Denominators – multiply the numerator & denominator by the conjugate
Ex 2) Rationalize
3
1 + √2
Solving Quadratic Equations
Methods : 1) Factoring (Remember to set equal to zero)
Ex 3) x2 + 5x + 6 = 0
2) x2 = c, then square root both sides (Rem to add ±)
Ex 4) x2 = 7
3) Complete the square
Ex 5) x2 – 8x + 13 = 0
4) Quadratic Formula
Ex 6) 3x2 – 5x – 1 = 0
Equations Involving Radicals – Isolate the radical, then square both sides. Remember to check solutions!!!
Ex 7) Solve 2 x  1  2  x
Inequalities
Linear –
Ex 8) Solve and graph 3x < 9x + 4
Ex 9) Solve and graph 4 ≤ 3x – 2 < 13
Quadratic – Steps: 1) Get everything to one side and zero on the other
2) Determine when the factors will be zero
3) Determine signs and make a chart for the factors
EX) Solve x2 – 5x + 6 ≤ 0
(x – 2)(x – 3) ≤ 0
x - 3 = 0 when x = 3
x – 2 = 0 when x = 2
Ex 10) x >
2
x–2 –
x–3 –
(x – 2)(x – 3) +
3
+
–
–
+
+
+
Since we want the product
to be less than zero, we
want the interval btwn 2 & 3
[2, 3]
2_
x–1
Solving Absolute Values
1) Less than
|x–c|<b
2) Greater than | x – c | > b
3) Equal to
|x–c|=b
Ex 1) Solve |3x + 2| ≥ 4
=>
=>
=>
-b < x – c < b
x – c > b OR x – c < -b
x – c = b OR x – c = -b
Ex 2) Solve |5x – 2| < 7
Ex 3) Solve |2x + 3| = 4
Some Formulas to Remember
The distance between the points A = (x1, y1) and B = (x2, y2) is d(A, B) = √ (x2 – x1)2 + (y2 – y1)2
x & y-intercepts
x-intercept: where the graph crosses the x-axis
y-intercept: where the graph crosses the y-axis
To find - set y = 0 & solve for x.
To find - set x = 0 & solve for y.
Ex 4) Find the x & y-intercepts of y = x2 - 2
Circles – all points equidistant from a given point (called the center)
An equation of a circle with center (h, k) and radius, r, is
(x – h)2 + (y – k)2 = r2
Ex 5) Find the equation of a circle with radius of 3 and center (-2, 5).
Ex 7) Graph (x – 2)2 + (y + 1)2 = 16
Ex 6) Graph x2 + y2 = 25
Lines – graphical representations of solution sets to linear equations
Ex 8) Find the slope between points (-2,1) and (-5,3)
Two forms of the Equation of a line
Slope-Intercept Form
y = mx + b
Slope = ------------ = -------------------
Point-Slope Form
y – y1 = m(x – x1)
Ex 9) Find the slope intercept form of the line that passes through (1, -6) and has slope of -1.
Special Lines
Vertical Lines x = #
Horizontal
Ex 1) Graph y = -4
Ex 2) Graph x = 3
y=#
Parallel – two lines that never cross & have the same slope
Perpendicular – two lines that intersect to form right angles & have opposite, reciprocal slopes
Ex 3) Write an equation of a line through (2, 6) and parallel to the line
x + 2y = 5
Ex 4) Show that the points P (3, 3), Q (8, 17) and R (11, 5) are the vertices of a right triangle.
Applications
Use a method you like and stick with it.
Read & Understand
Draw Picture or Table or Graph
Define Variables
Find Equation
Solve
Answer
Check
Ex 5) A rectangular building is 3 feet longer than it is wide. If the area is 10 sq.ft., find the dimensions.
Ex 6) A manufacturer of soft drinks advertises their OJ as “naturally flavored” although it only contains 5%
OJ. A new regulation calls for 10% juice. How much pure OJ must be added to 900 gal. of OJ to meet the
regulation.
Ex 7) A dam is built on a river to create a reservoir. The water level, w, in the reservoir is given by the
equation: w = 4.5 t + 28 where t is the number of years since the dam was constructed and w is measured
in feet. What do the slope and the y-intercept of this graph represent?
Ex 8) An economist models the market for wheat by the following equations
Supply:
y = 8.33 p – 14.58
Demand:
y = -1.39 p + 23.35 where p is the price per bushel (in dollars) and
y is the number of bushels produced and sold (in millions).
a) At what point is the price so low that no wheat is produced?
b)
At what point is the price so high that no wheat is sold?
Ex 9) A carnival has two plans for tickets.
Plan A: $5 entrance fee and $0.25 per ride OR Plan B: $2 entrance fee and $0.50 per ride.
How many rides would you have to take for Plan A to be less expensive?
Trigonometry Review
Ex 1) Evaluate sin (300)
Ex 3) Evaluate tan (
π
2
)
Ex 5) sin2x + cos2x = ______
3π
Ex 2) Evaluate cos (
4
Ex 4) Evaluate sec (
)
2π
3
)