Download 2 - MissLottMathClass

Do Now!!
In your composition notebook THOROUGHLY
explain how you would solve the equation
below:
-3(x – 3) ≥ 5 – 4x
EOC Review
Objective
SWBAT make connections with content from
Chapters 1 – 4.
1-1 Variables and Expressions
A variable is a symbol, usually a letter, that
represents values of a variable quantity. For
example, d often represents distance. An
algebraic expression is a mathematical phrase
that includes one or more variables. A
numerical expression is a mathematical
phrase involving numbers and operations
symbols, but no variables.
1-1 Variables and Expressions
What is an algebraic expression for the word
phrase 3 less than half a number x?
You can represent “half a number x” as x/2.
Then subtract 3 to get: x/2 – 3.
1-4 Properties of Real Numbers
You can use properties such as the ones below
to simplify and evaluate expressions.
Commutative Properties: -2 + 7 = 7 + (-2)
3 × 4 = 4×3
Associative Properties: 2× (14×3) = (2×14) × 3
3 + (12 + 2)= (3 + 12) + 2
Identity Properties: -6 + 0 = -6
21 × 1 = 21
Zero Property of Multiplication: -7 × 0 = 0
Multiplication Property of -1:
6 ∙ (-1) = -6
2-1 and 2-2 Solving One- and Two-Step
Equations
To solve an equation, get the variable by itself
on one side of the equation. YOU can use
properties of equality and inverse operations
to isolate the variable. For example, use
multiplication to undo its inverse, division.
2-1 and 2-2 Solving One- and Two-Step
Equations
What is the solution of _y_ + 5 = 8
2
_y_ + 5 – 5 = 8 – 5
Subtract to undo +
2
_y_ = 3
Simplify
2
2 * _y_ = 3*2
Multiply
2
y=6
Simplify
2-3 Solving Multi-Step Equations
To solve some equations, you may need to
combine like terms or use the Distributive
Property to clear fractions or decimals.
2-3 Solving Multi-Step Equations
You do!
What is the solution of 12 = 2x + _4_ – _2x_ ?
3
3
8=x
2-4 Solving Equations With Variables
on Both Sides
When an equation has variables on both
sides, you can use properties of equality to
isolate the variable on one side. An equation
has no solution if no value of the variable
makes it true. An equation is an identity if
every value of the variable makes it true.
2-4 Solving Equations With Variables
on Both Sides
What is the solution of 3x – 7 = 5x + 19 ?
3x – 7 – 3x = 5x + 19 – 3x
Subtract 3x
-7 = 2x + 19
Simplify
-7 – 19 = 2x + 19 – 19
Subtract 19
-26 = 2x
Simplify
-26 = 2x
Divide by 2
2 2
-13 = x
Simplify
2-5 Literal Equations and Formulas
A literal equation is an equation that involves
two or more variables. A formula is an
equation that states a relationship among
quantities. You can use properties of equality
to solve a literal equation for one variable in
terms of others.
2-5 Literal Equations and Formulas
You Do!
What is the width of a rectangle with area 91
ft2 and length 7 ft?
13 = w
3-1 Inequalities and Their Graphs
A solution of an inequality is any number that
makes the inequality true. You can indicate all
the solutions of an inequality on the graph a
closed or dot indicates that the midpoint is a
solution. An open dot indicates that the
midpoint is not a solution.
3-1 Inequalities and Their Graphs
What is the graph of x ≤ - 4?
-4
3-2 Solving Inequalities Using Addition
or Subtraction
You can use the addition and subtraction
properties of inequality to transform an
inequality into a simpler, equivalent inequality.
3-2 Solving Inequalities Using Addition
or Subtraction
What are the solutions of x + 4 ≤ 5 ?
x+4≤5
x+4–4≤5–4
Subtract 4
x≤1
Simplify
3-3 Solving Inequalities Using
Multiplication or Division
You can use the multiplication and division
properties of inequality to transform an
inequality. When you multiply or divide each
side of an inequality by a negative number
you have to reverse the inequality symbol.
3-3 Solving Inequalities Using
Multiplication or Division
What are the solutions of -3x > 12 ?
-3x > 12
-3x < 12
Divide each by -3
-3
-3
Reverse Inequality Symbol
x < -4
Simplify
3-4 Solving Multi-Step Inequalities
When you solve inequalities, sometimes you
need to use more than one step. You need to
gather the variable terms on one side of the
inequality and the constant terms on the
other side.
3-4 Solving Multi-Step Inequalites
You do!
What are the solutions of 3x + 5 > -1 ?
x > -2
3-5 Working With Sets
The complement of a set A (A’) is the set of all
elements in the universal set that are not in A.
3-5 Working With Sets
Suppose U = {1, 2,3,4,5,6} and Y = {2,4,6}.
What is Y’?
The elements in U that are not in Y are 1, 3, and
5. So Y’ = {1, 3, 5}
3-8 Unions and Intersections of Sets
The union of 2 or more sets is the set that
contains all elements of the sets. The
intersection of 2 or more sets is the set of
elements that are common to all the sets.
Disjoint sets have no elements in common.
4-4 Graphing a Function Rule
A continuous graph is a graph that is
unbroken. A discrete graph is composed of
distinct, isolated points. In real-world graph,
show only points that make sense.
4-4 Graphing a Function Rule
The total height h of a stack of cans is a
function of the number n of layers of 4.5-in.
cans used. This situation is represented by h =
4.5n. Graph the function.
4-5 Writing a Function Rule
To write a function rule describing a real-world
situation, it is often helpful to start with a
verbal model of the situation.
4-5 Writing a Function Rule
At a bicycle motocross (BMX) track, you pay $40
for a racing license plus $15 per race. What is
the function rule that represents your total
cost?
Total cost = license fee + fee per race ∙ # of races
C =
40
+
15
∙ r
A function rule is C = 40 + 15r
4-6 Formalizing Relations and
Functions
A relation pairs numbers in the domain with
numbers in the range. A relation may or may
not be a function.
4-6 Formalizing Relations and
Functions
Is the relation [(0,1), (3,3), (4,4), (0,0)] a
function?
The x-values of the ordered pairs form the
domain, and the y-values form the range. The
domain value 0 is paired with two range
values, 1 and 0. So the relation is not a
function.
4-7 Sequences and Functions
A sequence is an ordered list of numbers,
called terms, that often forms a pattern. In an
arithmetic sequence, there is a common
difference between consecutive terms.
4-7 Sequences and Functions
Tell whether the sequence is arithmetic.
5
2
-3
-1
-3
-4……
-3
The sequence has a common difference of -3, so
it is arithmetic
Summary
Our Objectives were that:
SWBAT make connections with content from
Chapters 1 – 4.
Homework
In TEXTbook
NC EOC Test Practice
Chapter 1 pg. 74 – 76
Chapter 2 pg. 158-160
Chapter 3 pg. 228 – 230
Chapter 4 pg. 286-288
Problems 1 – 20 even
Problems 1 – 18 even
Problems 1 – 20 even
Problems 1 – 14 even
Class Assignment
In the paper back NC Algebra 1 EOC Test
Workbook
Complete Problems:
1-9; 11-12; 14-22; 24-26; 28-29; 31 – 34
On pages 1-7
SHOW ALL WORK
If there is no computation to answer the
question, EXPLAIN your reasoning for getting
your answer choice.
YOU MAY write in the text book.
EARLY BIRDS
Review Released EOC test booklet and choose
questions from the booklet you need to go
over. Have these questions ready for
Thursday’s review.
Objective
SWBAT make connections with content from
Chapters 5 – 8.
Do Now!!
Factor each expression:
1) h2 + 8h + 16
2) d2 – 20d + 100
3) m2 + 18m + 81
5-1 Rate of Change and Slope
Rate of change shows the relationship between
two changing quantities. The slope of a line is
the ratio of the vertical change (the rise) to
the horizontal change (the run).
slope = rise = y2 – y1
run x2 – x1
The slope of a horizontal line is 0, and the slope
of a vertical line is undefined.
5-1 Rate of Change and Slope
What is the slope of the line that passes through
the points (1, 12) and (6, 22)?
Slope = y2-y1 = 22 – 12 = 10 = 2
x2-x1 6 – 1
5
5-2 Direct Variation
A function represent a direct variation a direct
variation if it has the form y = kx, where k ≠0.
The coefficient k is the constant of variation.
5-2 Direct Variation
Suppose y varies directly with x, and y = 15 when
x = 5. Write a direct variation equation that
relates x and y. What is the value of y when x =
9?
y = kx
15 = k(5)
3=k
y = 3x
The equation y = 3x relates x and y. When x = 9, y =
27
5-3, 5-4, and 5-5 Forms of Linear
Equations
The graph of a linear equation is a line. You can
write a linear equation in different forms.
The slope-intercept form of a linear equation is
y = mx + b, where m is the slope and b is the yintercept.
The point-slope form of a linear equation is
y – y1 = m(x – x1), where m is the slope and
(x1,y1) is a point on the line.
The standard form of a linear equation is Ax+By = C,
where A, B, and C are real numbers, and A and B
are not both zeros.
5-6 Parallel and Perpendicular Lines
Parallel lines are lines in the same plane that
never intersect. Two lines are perpendicular if
they intersect to form right angles.
5-6 Parallel and Perpendicular Lines
Are the graphs of y = 4/3 x + 5 and y = -3/4x + 2
parallel, perpendicular, or neither? Explain.
The slope of the graph of y = 4/3x + 5 is 4/3
The slope of the graph of y = -3/4x + 2 is -3/4
(4/3)∙(-3/4) = -1
The slopes are opposite reciprocals, so the graphs
are perpendicular.
What type of slopes do parallel lines have?
The same slope
5-7 Scatter Plots and Trend Lines
A scatter plot displays two sets of data as
ordered pairs. A trend line for a scatter plot
shows the correlation between the two sets of
data. The most accurate trend line is the line
of best fit. To estimate or predict values on a
scatter plot, you can use interpolation or
extrapolation.
6-1 Solving Systems by Graphing
One way to solve a system of linear equations is
by graphing each equation and finding the
intersection point of the graph, if one exists.
6-1 Solving Systems by Graphing
What is the solution of the system?
y = -2x + 2
y = -0.5x – 3
The solution is (2, -2)
6-2 Solving Systems Using Substitution
6-3 Solving Systems Using Elimination
• You can solve a system of equations by solving
one equation for one variable and then
substituting the expression for that variable
into the other equation.
• You can add or subtract equations in a system
to eliminate a variable. Before you add or
subtract, you may have to multiply one or
both equations by a constant to make
eliminating a variable possible.
6-5 and 6-6 Linear Inequalities and
Systems of Inequalities
A linear Inequality describes a region of the
coordinate plane with a boundary line. Two or
more inequalities form a system of inequality.
The system’s solutions lie where the graphs of
the inequalities overlap.
6-5 and 6-6 Linear Inequalities and
Systems of Inequalities
What is the graph of the system?
y > 2x – 4
y ≤ -x + 2
7-1 Zero and Negative Exponents
You can use zero and negative integers as
exponents. For every nonzero number a, a0 =
1. For every nonzero number a and any
integer n, a-n = 1/an. When you evaluate an
exponential expression, you, you can simplify
the expression before substitution values for
the variables.
7-3 and 7-4 Multiplication Properties
of Exponents
To multiply powers, with the same bases, add
the exponents am∙an = am+n, where a≠0 and m
and n are integers.
To raise a power to a power, multiply the
exponents. (am)n = amn, where a≠0 and m and
n are integers.
To raise a product to a power, raise each factor
in the product to the power.
(ab)n = anbn, where a≠0, b≠0, and n is an
integer
7-5 Division Property of Exponents
To divide powers with the same base, subtract
the exponents.
am = am-n, where a ≠ 0 and m and n are integers
an
To raise a quotient to a power, raise the
numerator and the denominator to the power.
_a_ n = _an where a≠0 & b≠0; n is an integer
b
bn
7-3 and 7-4 Multiplication Properties
of Exponents
7-5 Division Property of Exponents
You Do
a) 310∙34
b) (x5)7
c) (pq)8
d) 5x4 3
z2
7-6 Exponential Functions
An exponential function involves repeated
multiplication of an initial amount a by the
same positive number b. The general form of
an exponential function is y = a ∙ bx, where
a≠0, b>0, and b≠1.
7-6 Exponential Functions
What is the graph of y = ½ ∙5x ?
Make a table of values. Graph the ordered the
pairs.
7-7 Exponential Growth and Decay
When a > 0 and b >1, the function y = a ∙ bx
models exponential growth. The base b is
called the growth factor. When a >0 and
0<b<1, the function y = a ∙ bx models
exponential decay. In this case the base b is
called the decay factor.
7-7 Exponential Growth and Decay
You Do!
The population of a city is 25, 000 and decreases
1% each year. Predict the population after 6
years.
The population will be about 23,537 after 6
years.
Rule for an Arithmetic Sequence
The nth term of an arithmetic sequence with
first term A(1) and common difference d is
given by:
A(n) = A(1) + (n – 1)d
Where n = nth term
A(1) = the first term
n = term number
d= common difference
Arithmetic Sequence and Recursive
Formula
Arithmetic Sequence:
– Sequence with a constant difference between
terms.
Recursive Formula:
– Formula where each term is based on the term
before it
Recursive Formula for an Arithmetic Sequence:
a1

an  an 1  d , n  2
Arithmetic Sequence and Recursive
Formula
• If you buy a new car, you might be advised to
have an oil change after driving 1000 miles
and every 3000 miles thereafter. Then the
following sequence gives the mileage when oil
changes are required:
1000 4000 7000 10000 13000 16000
a1  1000

an  an 1  3000; n  2
Arithmetic Sequence and Recursive
Formula
You DO
• Briana borrowed $870 from her parents for
airfare to Europe. She will pay them back at
the rate of $60.00 per month. Let an be the
amount she still owes after n months. Find a
recursive formula for this sequence.
a1  870

an  an 1  60, n  2
8-1 Adding and Subtracting
Polynomials
A monomial is a number, a variable, or a
product of a number and one or more
variables. A polynomial is a monomial or the
sum of two or more monomials. The degree of
a polynomial in one variable is the same as
the degree of the monomial with the greatest
exponent. To add 2 polynomials, add the like
terms of the polynomials. To subtract a
polynomial, add the opposite of the
polynomial.
8-1 Adding and Subtracting
Polynomials
You Do!
What is the difference of 3x3 – 7x2 + 5 and
2x2 – 9x – 1?
3x3 – 9x2 + 9x + 6
8-2 Multiplying and Factoring
You can multiply a monomial and a
polynomial using the Distributive Property.
You can factor a polynomial by finding the
greatest common factor (GCF) of the terms of
the polynomial.
8-3 and 8-4 Multiplying Binomials
You can use algebraic tiles, tables, or
Distributive Property to multiply polynomials.
The FOIL method (First, Outer, Inner, Last) can
be used to multiply two binomials. You can
also use rules to multiply special case
binomials.
8-3 and 8-4 Multiplying Binomials
What is the simplified form of (4x + 3)(3x + 2)?
= 12x2 + 17x + 6
8-5 and 8-6 Factoring Quadratic
Trinomials
You can write some quadratic trinomials as
the product of two binomials factors. When
you factor a polynomial, be sure to factor out
the GCF first.
8-5 and 8-6 Factoring Quadratic
Trinomials
What is the factored form x2 + 7x + 12?
List the pairs of factors of 12. Identify the pair
with a sum of 7. Factors of 12 Sum of Factors
1, 12
13
2, 6
8
3,4
7
x2 + 7x + 12 = (x+3)(x+4)
8-7 Factoring Special Cases
When you factor a perfect-square trinomial, the
two binomial factors are the same.
a2 + 2ab + b2 = (a+b)(a+b) = (a+b)2
a2 – 2ab + b2 = (a-b)(a-b) = (a-b)2
When you factor a difference of squares of 2
terms, the 2 binomial factors are the sum and
the difference of the two terms.
a2 – b2 = (a+b)(a-b)
8-7 Factoring Special Cases
What is the factored form of 81t2 – 90t + 25 ?
(9t – 5)2
Summary
Our objective was:
SWBAT make connections with content from
Chapters 5 – 8.
Homework
Chapter 5 EOC pg. 354 – 356 Problems 1- 15
Chapter 6 EOC pg. 408 – 410 Problems 1 – 17
Chapter 7 EOC pg. 468 – 470 Problems 1- 19 odd
Chapter 8 EOC pg. 528 – 530 Problems 1–21 odd
Assignment
Resume yesterday’s assignment:
In the paper back NC Algebra 1 EOC Test Workbook
Complete Problems:
1-9; 11-12; 14-22; 24-26; 28-29; 31 – 34
On pages 1-7
SHOW ALL WORK
If there is no computation to answer the question,
EXPLAIN your reasoning for getting your answer
choice.
YOU MAY write in the text book.
Do Now
Find the length of the hypotenuse. Write your
answer in simplified radical form.
3√2
Objective
SWBAT make connections with the content from
Chapters 9 - 12
9-1 and 9-2 Graphing Quadratic
Functions
A function of the form y = ax2 + bx + c, where
a ≠0, is a quadratic function. Its graph is a
parabola. The axis of symmetry of a parabola
divides it into two matching halves. The vertex
of a parabola is the point at which the
parabola line intersects the axis of symmetry.
9-1 and 9-2 Graphing Quadratic
Functions
What is the vertex of the graph of
y = x2 + 6x – 2?
The vertex is (-3, -11)
9-3 and 9-4 Solving Quadratic
Equations
The standard form of a quadratic equation is
ax2 + bx + c = 0, where a ≠ 0. Quadratic equations
can have two, one, or no real-number solutions.
You can solve a quadratic equations by graphing
the related function and finding the x-intercepts .
Some quadratic equations can also be solved
using square roots. If the left side of ax2+bx+c = 0
can be factored, you can use the Zero-Product
Property to solve the equation.
9-3 and 9-4 Solving Quadratic
Equations
What are the solutions of 2x2 – 72 = 0 ?
x=+6
9-5 Completing the Square
You can solve any quadratic equation by
writing it in the form x2 + bx = c, completing
the square, and finding the square roots of
each side of the equations.
9-5 Completing the Square
What are the solutions of x2 + 8x = 513
x = 19 or x = -27
9-6 The Quadratic Formula and the
Discriminant
You can solve the quadratic equation ax2+bx+c =
0 where a ≠ 0, by using the quadratic formula
x = -b + √(b2 – 4ac)
2a
Discriminant: is the expression under the radical
sign in the quadratic formula, it tells how
many solutions the equation has.
x = -b + √(b2 – 4ac)
The
discriminant
2a
9-6 The Quadratic Formula and the
Discriminant
• How many real-number solutions does the
equation x2 + 3 = 2x have?
Because the discriminant is negative, the
equation has no real-number solutions.
9-8 Systems of Linear and Quadratic
Equations
Systems of linear and quadratic equations can
have 2 solutions, one solution, or no solution.
These systems can be solved graphically or
algebraically.
9-8 Systems of Linear and Quadratic
Equations
What are the solutions of the system?
y = x2 – 7x – 40
y = -3x + 37
(11, 4) and (-7, 58)
10-1 The Pythagorean Theorem
Given the lengths of 2 sides of a right triangle,
you can use the Pythagorean Theorem to find
the length of the third side. Given the lengths
of all 3 sides of a triangle, you can determine
whether it is a right triangle.
10-2 Simplifying Radicals
A radical expression is simplified if the following
statements are true:
• The radicand has no perfect-square factors
other than 1
• The radicand contains no fractions
• No radicals appear in the denominator of a
fraction
10-2 Simplifying Radicals
What is the simplified for of √(3x) ?
√(2)
√(3x) = √(3x) ∙ √(2)
Multiply by √2
√(2)
√(2) √(2)
√2
= √(6x) = √(6x)
Simplify
√(4)
2
10-5 Graphing Square Root Functions
Graph a square root function by plotting points
or translating the parent square root function
y = √x.
The graph of y = √x + k and y = √x – k are vertical
translations of y = √x. The graphs of y = √(x+h)
and y = √(x – h) are horizontal translation of
y = √x
10-5 Graphing Square Root Functions
What is the graph of the square root function
y = √(x-2) ?
The graph of y = √(x – 2) is the graph of y = √x
shifted 2 units to the right.
11-5 Solving Rational Expressions
A rational expression is an expression that can
be written in the form: polynomial
polynomial
A rationale expression is in simplified form when
the numerator and denominator have no
common factors other than 1.
11-5 Solving Rational Expressions
What is the simplified form of x2 – 9
x2 – 2x -15
x2 – 9_ = (x-3)(x+3)
Factor numerator
x2 – 2x -15 (x+3)(x-5)
and denominator
(x-3)(x+3)
Divide out common
(x-5)(x+3)
factor
(x- 3)
Simplify
(x-5)
11-6 Inverse Variation
When the product of 2 variables is constant,
the variables form an inverse variation. You
can write an inverse variation in the form xy =
k or y = k/x, where k is the constant of
variation.
11-6 Inverse Variation
Suppose y varies inversely with x, and y = 8
when x = 6. What is an equation for the
inverse variation?
xy = k
General form of inverse variation
6(8) = k
Substitute
48 = k
Solve for k
xy = 48
Write equation
11-7 Graphing Rational Functions
A rational function can be written in the form
f(x) = polynomial . The graph of a rational
polynomial
function in the form y = _a_ + c has a vertical
x–b
asymptote at x = b and a horizontal asymptote at
y = c. A line is an asymptote of a graph if the graph
gets closer to the line as x or y gets larger in
absolute value.
12-2 Frequency and Histograms
The frequency of an interval is the number of
data values in that interval. A histogram is a
graph that groups data into intervals and
shows the frequency of values in each
interval.
12-3 Measures of Central Tendency
and Dispersion
The mean of a data set equals:
sum of data values
Total number of data values.
The median is the middle value in the data set
when the values are arranged in order. The
mode is the data item that occurs the most
times. The range of set of data is the
difference between the greatest and least
data values.
12-4 Box-and-Whisker Plots
A box-and-whisker plot organizes data values
into four groups using the minimum value, the
first quartile, the median, the third quartile,
and the maximum value.
1-7 Midpoint and Distance in the
Coordinate Plane
You can find the coordinates of the midpoint M
of AB with endpoints A(x1,y1) and B(x2,y2)
using the Midpoint Formula.
M( x1+x2 , y1+y2)
2
2
You can find the distance d between two points
A(x1,y1) and B(x2,y2) using the Distance
Formula.
d = √(x2-x1)2 + (y2-y1)2
1-8 Perimeter, Circumference, and
Area
Circles have a circumference C. The area A of a
polygon or a circle is the number of square
units it encloses.
Circle:
Circumference = ∏d or Circumference = 2∏r
Area = ∏r2
11-4, 11-5, 11-6 Volumes of Cylinders,
Pyramids, Cones, and Spheres
Cylinder: V = Bh or V = ∏r2h
Pyramids: V = 1/3Bh
Cones: V = 1/3Bh; or V = 1/3∏r2h
Spheres: V= 4/3∏r3
Summary
Our objective was:
SWBAT make connections with the content from
Chapters 9 - 12
Homework
Chapter 9 EOC Practice pg. 594 – 596 1 – 20 all
Chapter 10 EOC Practice pg. 646 – 648 1 – 17
odd
Chapter 11 EOC Practice pg. 708 – 710 1 – 21
odd
Assignment
In TEXTbook
Pg. 780 – 785 End-Of-Course Assessment
Complete Problems:
1 – 47 ALL
Do not do problems:
3, 27, 45