Download Things you should know for Pre-Calculus

THINGS YOU SHOULD KNOW
FOR
PRE-CALCULUS
Pre-AP Pre-Calculus
Mrs. Richter
2013 – 2014
Kinds of numbers
• Natural (counting) numbers: {1, 2, 3, 4, ….}
• Whole numbers: {0, 1, 2, 3, 4, ….}
• Integers: {…., -3, -2, -2, 0, 1, 2, 3, …}
• Rational number: can be written as a fraction
• Terminating number: 7/4 = 1.75
• Infinitely repeating: 4/11 = 0.363636…..
• Irrational: infinitely nonrepeating
• Example:
3=1.7320508….
π=3.14159265….
Bounded Intervals
Interval
Notation
Interval Type
Inequality
Notation
[a, b]
Closed
a≤x≤b
(a, b)
Open
a<x<b
[a, b)
Half-open
a≤x<b
(a, b]
Half-open
a<x≤b
Graph
Properties of Exponents
• 𝑢 𝑚 𝑢 𝑛 = 𝑢 𝑚+𝑛
• 53 ∙ 54 = 53+4 = 57
𝑢𝑚
• 𝑛
𝑢
0
𝑥9
• 4 = 𝑥 9−4
𝑥
• 80 = 1
=
𝑢𝑚−𝑛
•𝑢 =1
• 𝑢 −𝑛
=
1
𝑢𝑛
• 𝑦 −3
=
= 𝑥5
1
𝑦3
• (𝑢𝑣)𝑚 = 𝑢 𝑚 𝑣 𝑚
• (2𝑧)5 = 25 𝑧 5 = 32𝑧 5
• 𝑢 𝑚 𝑛 = 𝑢 𝑚𝑛
• 𝑥 2 3 = 𝑥 2∙3 = 𝑥 6
𝑢 𝑚
•
𝑣
𝑎 7
•
𝑏
=
𝑢𝑚
𝑣𝑚
=
𝑎7
𝑏7
Scientific Notation
Scientific
Notation
9.3 × 107 mi
5.3 × 10−5 g
Math work
Expanded form
93,000,000 mi
.000053 g
8,600 ft
3.49 × 10−8 in
Absolute Value
• The absolute value of a real number a is
𝑎, 𝑖𝑓 𝑎 > 0
𝑎 = −𝑎, 𝑖𝑓 𝑎 < 0
0, 𝑖𝑓 𝑎 = 0
Examples:
−4 , since -4<0, −4 = − −4 = 4
𝜋 − 6 , since 𝜋 − 6 is negative,
𝜋 − 6 = − 𝜋 − 6 = 6 − 𝜋 ≈ 2.858
Important formulas
𝑥1 − 𝑥2
2
• Distance Formula
•
• Midpoint formula
𝑥1 +𝑥2 𝑦1 +𝑦2
•
,
2
2
• Slope
•𝑚 =
Δ𝑦
Δ𝑥
=
+ 𝑦1 − 𝑦2
𝑦2 −𝑦1
𝑥2 −𝑥1
• Slope-intercept form • 𝑦 = 𝑚𝑥 + 𝑏
2
Solving Equations
• Graphically: use graphing feature on calculator
• Type equation into y=, make sure there is a 0 in the other y=
• Algebraically: without the use of a calculator
• Factoring
• Completing the square
• Quadratic Formula
Factoring
2𝑥 2 − 3𝑥 − 2 = 0
2𝑥 + 1 𝑥 − 2 = 0
2𝑥 + 1 = 0
𝑥−2=0
Completing the Square
• To solve 𝑥 2 + 𝑏𝑥 = 𝑐
𝑥2
𝑏
+ 𝑏𝑥 +
2
𝑏
𝑥+
2
2
2
𝑏
=𝑐+
2
𝑏2
=𝑐+
4
2
Completing the Square Example
• Solve 4𝑥 2 − 20𝑥 + 17 = 0
4𝑥 2 − 20𝑥 + 17 = 0 (divide by 4)
𝑥2
𝑥 2 − 5𝑥
17
− 5𝑥 + = 0 (subtract 17
)
4
4
17
= − (now complete the square)
4
Quadratic Formula
• From the form 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0, use the following formula
to find the values of x.
−𝑏 ± 𝑏 2 − 4𝑎𝑐
𝑥=
2𝑎
Quadratic Formula example
• Solve the equation 3𝑥 2 − 6𝑥 = 5
• First you must subtract 5 from both sides to get it into the correct
format
• 3𝑥 2 − 6𝑥 − 5 = 0 (now substitute into the QF)
Practice Problems
• Pg. 12-13: 47, 52, 57, 60, 69
• Pg. 20-23: 13, 21, 54, 63
• Pg. 29-30: 15, 19, 65
• Pg. 40: 21, 23, 25
• Pg. 50: (follow directions for each section) 3, 13, 15, 19,
23, 60
• Pg. 64: (solve inequalities by any method) 1, 11, 17, 23