Precalc Fall 2016
Sections P.3, 1.2, 1.3, P.4, 1.4, P.2 (radicals/rational exponents), 1.5, 1.6, 1.7, 1.8, 1.1, 2.1, 2.2
P.3 Polynomials and Special products
I Polynomial definition (p. 28)
an x n + an −1 x n −1 + ... + a1 x1 + a0 x 0
an x n + an −1 x n −1 + ... + a1 x + a0
Exponents must be counting numbers (1, 2, …) and the coefficients can be any real number
Standard form = highest degree first followed by descending powers
4 x 2 − 5 x 7 − 2 + 3x = −5 x 7 + 4 x 2 + 3x − 2
Vocab: Standard form, lead coefficient, terms, degree
II Grouping
When adding and subtracting, group like terms
(
) (
Ex 2a (show all steps!) 5 x3 − 7 x − 3 + x 3 + 2 x 2 − x + 8
(
) (
Ex 2b 7 x 4 − x 2 − 4 x + 2 − 3 x 4 − 4 x 2 + 3 x
III Multiplying Polynomials
A) Distributive property 5(x+7)
)
x(x-7y)
)
-2(x-2)
B) FOIL ( 2 x − 4 )( x + 5 ) step up from distributive property
C) Multiplying polynomials bigger than 2x2
(x
2
− 2 x + 2 )( x 2 + 2 x + 2 )
D) Special Products (p. 30)
+
− =
−
±
=
±2 +
±
=
±3
+3
±
1.2 Linear Eqns in One Variable
I Equations and solutions of equations
Identity Eqn:
−9=
+3
−3
How many solutions?
Conditional Eqn:
− 9 = 0 is true for x=3 and x=-3 only
Contradiction Eqn:
+7 =
+5
How many solutions?
II Linear Eqns in one variable
p. 87 Standard form:
+
=0
Linear equations have exactly ____ solution(s)
Equivalent Equations: have same solution(s)
2 −
= 4 is equivalent to
2 = 6 is equivalent to x=3
=4
III Equations that lead to linear equations
EX 3 Fraction equations
Ex 4
Extraneous solutions
+
=2
=
+3
−4 =
−2
+4
−1
−
IV Intercepts algebraically: to find the -intercept, you plug in ___ for ____ and to find the -intercpt you plug
in ___ for __.
Ex: Find the intercepts for 3 + 2
+ 5 = 15
1.3 Modeling Linear Equations
I Intro to problem solving – page _______
Key words for addition:
Key words for subtraction:
Key words for multiplication:
Key words for division:
Draw the picture, list what you have, what you are looking for, choose the formula, double check your
answers, state answer clearly.
II Using math modeling
Money and Percent #49
Dimensions #57
Distance formula d=rt #63
Similar triangles (height problems) #69
III Mixture Problems
Inventory #73
IV Common formulas
Box on page 101, has formulas for what?
Ex 9 Cat food tin
Which formula do we need?
Why?
P.4 Factoring Polynomials
I Common Factors – read p. 37
A) Definitions
a. Completely factored:
b. Prime polynomial:
B) Common factors – distributive property in reserve direction (Find GCF)
a. Ex 1a 6 x3 − 4 x
b. Ex 1c ( x + 2 )( 2 x ) + ( x + 2 ) 3
C) Special Factoring techniques
a. P. 38 for table
b. New special factoring forms
i.
Ex 3 ( x + 2 ) − y 2
2
u 3 + v3 = ( u + v ) ( u 2 − uv + v 2 )
u 3 − v3 = ( u − v ) ( u 2 + uv + v 2 )
ex 4
16 x 2 + 24 x + 9
x 2 − 10 x + 25
sign note:
#50 y 3 +
8
125
II No special pattern, trinomials with binomial factors
Ex 7 x 2 − 7 x + 12
2 x 2 + x − 15
III Factoring by grouping – read p. 41 paragraph
Ex #72 x3 + 5 x 2 − 5 x − 25
#80
2x2 + 9x + 9
IV Application: Geometric modeling
p.44 #131
1.4 Quadratic Equations and Applications
I Factoring
Standard form for a Quadratic equation is _______________________
Highest exponent is ____, so the degree of this polynomial is ____________
Quadratic equations are _________________________________________________.
Solving a quadratic equation by factoring only works if ___________________________
Ex 1 2
+9 +7=3
II Extracting Square Roots
If the quadratic equation is
= , then its equivalent equation is
This is a special product:
−
=
So answers are
+3
= 25
4 +7
= 44
5
−2
+ 100 = 2
II Completing the Square
+
is completed by adding to both sides of the equation
+
So
+
!
" =
When the leading coefficient is one
Ex 3
When the leading coefficient is not 1: Ex 5 3
+2 −6 =0
−4 −5= 0
Inside an algebraic expression: Ex
IV Quadratic Formula: Given the standard form quadratic equation
and derive the quadratic equation:
=
!±√!
%
%&
+
Proof on page 111
Solutions to the quadratic formula are based on the discriminant,
1) It is positive:
2) It is zero:
3) It is negative:
________________
_______________
______________
Using the quadratic formula: Ex 6
V Applications Dimensions #114
+ # = 0, we can complete the square
−4 #
______________
_____________
______________
+3 −9= 0
falling time and position #119
P.2: Radicals
36
n
a m = ( n a )m
− 36
n
a * n b = n ab
125
64
n
a na
=
b
b
3
5
−32
4
−81
n
n m
a = nm a
( a)
n
n
=a
page 20, generalizations about nth root of numbers
if ' is even,
If ' is odd,
n
n
an = a
an = a
IV Simplifying Radicals
75x 3
3
−40x 6
3
16 x − 3 54 x 4
V Rationalizing Denominators and Numerators
5
(
√
√
24 3
2
3+ 7
*Always try to keep radicals out of the denominator!
The same technique works for numerators!
V Rational Exponents
1
a n = n a 1/n is the rational exponent
m
an =
( a)
n
m
m/n is the rational exponent
Ex 16c (radicals to exponents)
2x 4 x3
Ex17a (exponents to radicals)
(x
2
+ y2 )
3/ 2
1.5 Imaginary Numbers
I What are they?
Imaginary unit is ___________ = ________
Complex numbers: add real numbers to imaginary units ___ + ____
**Formal definition p. 122
Equality in Imaginary numbers
+ ) = # + ) means ________________________
II Operations with complex numbers
A) Addition/subtraction
Add or subtract the ______________, then add or subtract the _________________
Ex 1 a) 4 + 7) + 1 − 6)
c)3) − −2 + 3) − 2 + 5)
B) Multiplying
Ex 2b) 2 − ) 4 + 3)
d) 3 + 2)
III Complex Conjugates
Complex conjugates _____________ and __________ ______
when multiplied together produce _________, and eliminate ________________
*
Writing a quotient in standard form: Ex 4
*
IV Complex Equations of Quadratic Equations
**Always pull out ) first!
Simplifying Complex Radicals: Ex 5b √−48 − √−27
Complex numbers and quadratic equations Ex 6b 3
−2 +5=0
1.6 Other types of Equations
I Polynomial equations of degree 3 or higher
Solve by factoring (need =0) Ex 1 3 = 48
Like Ex 2, #14
+2
+3 +6=0
Some polynomials are Quadratic in nature (or type) i.e. __________________________
An example similar to Example 3, #20: 36, + 29, − 7 = 0
*Note: plug the original __________ solving for the original variable
II Equations involving radicals
Ex 4 √2 + 7 − = 2
√4 − 3 = √2 − 5 + 2
#52
+3
/
=8
III Equations with fractions or absolute value: key here is to multiply by the LCD of all fractions
0
+ =
#66
−
=1
/
In absolute value problems, we have to account for both the positive and negative outcomes. Beware of
extraneous answers
| + 1| = 5 + 10
#74 | + 6 | = 3 + 18
IV Applications – you are responsible for reading the examples on your own. There are many different
examples with all the types of equations we have discussed today.
1.7 Linear Inequalities in one variable
Read pages 140 and 141 on your own, will not cover in class, but use them a lot
I Venn diagram
Bounded
Unbounded
II Solving Linear equalities
When solving inequalities, you need to solve it as a _________________________________.
But, it takes __________________________ to satisfy it…make sure you sketch all parts!
Just like solving linear equations…Similar to Ex 2: #40 4
+1 <2 +3
Sign flips? Similar to Ex 3: #37 2 − 1 ≥ 1 − 5
Double inequalities! Whatever you do, do it to ALL sides!
Similar to Ex 4: #48 8 ≤ − 3 + 5 < −13
III Inequalities involving absolute value - Blue box page 144, it’s important.
Like when we solved absolute value eqns in 1.6, we need to solve them considering 2
outcomes…______________________and ______________________.
Similar to Ex 5a | − 5| < 2 can be read as __________________
#62 | − 7| < −5
#64 | − 8| ≥ 0
IV Applications #127 p. 149
1.8 Other types of inequalities
I Polynomial inequalities
Key numbers (p.150): ____________________________________ ________________
_________________ __________________________________________________________.
Test intervals (p.150): _____________________________________________________
_____________________________________________________________________________
Ex:
−2 −3 =
+1
− 3 , so its zeros are ____________and ____________
The zeros divide it into test intervals, which are _______________________________________
On a test interval, the function is either all positive or all negative.
Solving a polynomial inequality
Similar to Ex 1&3, #22
>2 +8
Use table to solve and look at all the pieces!
Similar to Ex2, #28 2
+ 13
− 8 − 46 ≥ 6
We can check out solutions algebraically and graphically:
Algebraically: plug in an x-value into the original inequality
Graphically: sketch the graph of the polynomial and see what happens with the graph and the x-axis.
Unusual solution sets Ex4
+2 +4>0
+2 +1≤0
Try #30
+3 +8>0
+3 +5<0
−4 +4>0
II Rational Inequalities
Key numbers in a rational expression are ______________________________________
______________________________________________________________________________
/
Ex 5
6
≤3
III Finding the domain of an expression
All we are doing is building on the stuff that we learned in P.5 when we covered rational expressions
Similar to Ex7: #64 7
0
Section 1.1 Graphs of Equations.
I Is it a solution?
Ex1 y=10x-7 points
(2,13)
(-1,-3)
II Key top graphing: plug in x-values and produce solutions
Ex 2 y=7-3x
x
Y=7-3x
(x,y)
-1
0
1
2
Ex 2
x
=
−2
Y=x^2-2
(x,y)
-1
0
1
2
III Finding Intercepts: always put answer in __________________
To find x intercept, set y=0
=
−8
4
^3 − 8 −
+ 32 = 64
4
^3 − 8 −
+ 32 = 64
To find y intercept, set x=0
=
IV Symmetry p. 80-81
−8
replace something, if get same as started with then it is symmetric
Respect to x-axis
plug in ______ for y, simplify
Respect to y-axis
plug in _____ for x, simplify
Respect to origin
plug in ____ for x, _____ for y, simplify
Ex. 5 Find symmetry algebraically
=2
Symmetric to x-axis?
Symmetric to y-axis?
Symmetric to origin?
Sketch the graph of the eqn:
= | − 1|
EX 7
Check for symmetry first and intercepts
x
-2
-1
0
1
2
y
V Circles
(x,y)
Standard form is….
−ℎ
+
−:
=;
with radius ; and its center at ℎ, :
If 3,4 is a point on a circle and its center is −1, 2 , what is the equation of the circle?
To find radius use distance formula
Given
+8 +
==
−
+
−
− 2 = 32, write it in standard form of a circle by completing the square twice
2.1 Linear equations in 2 variables
I Using Slope Slope-intercept form:
If we plug in x=0…
>
Graphing: slope is rise over run, so if you know m= ,
then you go to the right ___________ and up ______.
Horizontal lines ____________________
Graph y=-1
Vertical lines _______________________
Graph x=4
Graphing a linear equation: x+y=2
Context is important, do not confuse the
point x=0 with the line x=0 (or y=4 with the
point y=4)
3x+1=y
II Finding slope
p.172 Slope (for non-vertical lines) is ______ over ______, or m= __________
Order is important!
Like Ex 2: find slope between (4,5) and (-2,5)
Can zero be on the bottom? __________
if you get zero on the bottom, then the graph is a _________________.
Picture of positive, negative, and horizontal slopes…bottom page 172
III Writing Linear Equations in two variables
Point slope form is…___________________ this is a tool to help us get to ________________
#66: If line goes through the points (4,3) and (-4,-4), then what is its equation in slope intercept form?
IV Parallel and Perpendicular lines
Parallel Lines have the _______________________________________
Perpendicular lines have slopes as __________________________________
6
i.e. if slope was / then the perpendicular line would have a slope of ________
Note, the perpendicular line of a verical line ( = 5) would be a horizontal line (and vice versa)!
Like Ex 4: #88 Find parallel line and perpendicular line to
+
= 7, −3,2
V Applications
Slope can represent ________________________________________ in the real world
p. 182 #130: The University of Florida had enrollments of 46,107 students in 2000 and 51,413 students in
2008.
(a) What was the average annual change in enrollment from 2000 to 2008?
(b) Use the average annual change in enrollment to estimate the enrollments in 2002 (cut back on how
much for sake of time)
(c) Write an equation of a line that represents the data in terms of year T what t=0 is 2000. What’s the
slope? Interpret the slope in context of the problem
2.2 Functions
I Intro
A function is (p. 185) ____________________________________________________________________
Characteristics of a function are…(blue box p.185)
p. 186 blue box – Four ways to represent a function
Determining a function visually __________________
Draw an example of a function
Draw an example that is not a function
Finding functions algebraically…rearrange to solve for Y
Ex 2a + = 1
Ex2b – +
II Function Notation
Input x
Output f(x) “f of x”
Evaluating a function
Ex 3 A
=− +4 +1
A) G(2)
Piecewise functions
Like ex 4: @
Equation @
=1
=1−
B) G(-3)
C ) G(x+2)
+ 1, ≤ 1
=B
C Evaluate at x=0, 1, 3
3 + 2, > 1
Finding values when @
Ex 5a −2 + 10
= 0 set eqn equal to zero, you are finding the X values
Ex 5b @
=
−5 +6
Finding values when @
=A
Set eqns equal to each other
Ex 6a @
=
+ 1 and A
=3 −
III Domains of functions
Have student read the paragraph p.190 starting at the title and ending at Ex 7
We are just building on what we already know
Ex 7: a) @: { −3,0 , −1,4 , 0,2 , 2,2 , 4, −1 }
b) A
= 6
c) Volume of a sphere:
= G;
d) ℎ
= √4 − 3
IV Applications
#93 Path of a Ball: The height y (in feet) of a baseball thrown by a child is
=
H
+ 3 + 6 where x is the
horizontal distance (in feet) from where the ball was thrown. Will the ball fly over the head of another child 30
feet away trying to catch the ball? (he holds a baseball glove at 5 feet tall)
V Difference Quotients
Ex11: For @
=
− 4 + 7, find
I
J
J
I
A summary of terms can be found on page 193, this is expected that you know and I will be using them
without going over them again
Review Outline Exam 1 (Sections 1.1-1.8, 2.1-2.4)
Section 1.1
Determine if a point is a solution to a graph.
Graph an equation by plotting points. (*Must
show your x/y chart!)
Find the x and y intercepts of a graph.
(*Intercepts must be written as coordinate
points!)
Determine both algebraically and graphically if a
graph is symmetrical about the x-axis, the yaxis, or the origin.
Section 1.2
Solve a linear equation
Solve an equation with a fractional expression.
Solve an equation with extraneous solutions.
Find the x and y intercepts of an equation.
Section 1.3
Write and solve a mathematical model.
Section 1.4
Solve a quadratic equation by factoring, using
square roots, completing the square, and using
the quadratic formula.
Determine the number of solutions to a
quadratic equation by using the discriminant.
Section 1.5
Add, subtract, and multiply complex numbers.
Use the complex conjugate to write a complex
quotient in standard form.
Solve a quadratic equation with complex
solutions.
Section 1.6
Solve a polynomial equation with a degree of
three or higher.
Solve an equation involving radicals. (*You
must check your solutions!)
Solve an equation involving fractions. (*You
must check your solutions!)
Solve an equation with absolute value. (*You
must check your solutions!)
Section 1.7
Represent an inequality on a number line.
Represent an inequality in interval notation.
Solve linear inequalities in one variable.
Solve double inequalities.
Solve linear inequalities involving absolute
value.
Section 1.8
Find key numbers.
Identify and test the test intervals for solutions
sets. (*You must show your work for your
tests!)
Solve polynomial inequalities and write your
answers in interval notation.
Solve rational inequalities and write your
answers in interval notation.
Find domain for a square root function. (*Write
an inequality to do this.)
Section 2.1
Graph a line in slope-intercept form (y = mx+b).
Find the slope of a line. (*Slope formula must
be memorized! m =
y2 − y1
)
x2 − x1
Write the equation of a line in point-slope form
( y − y1 = m( x − x1 ) ).
Write the equation of a line in slope-intercept
form.
Describe what it means for two lines to be
parallel.
Describe what it means for two lines to be
perpendicular.
Write the equation of a line parallel or
perpendicular to a given line.
Section 2.2
Determine if a relation is a function.
Use function notation and evaluate a function.
Evaluate a difference quotient.
Evaluate a piecewise function.
Find values for which f(x)=0.
Find values for which f(x)=g(x)
Find the domain of a function.