Algebra A - Notes
Chapter 1 – Expressions, Equations and Functions
1.1 Variables and Expressions
Algebraic Expressions – An expression consisting of one or more numbers and variables along with one or more arithmetic operations.
Example:
Variable – Symbols that are used to represent unspecified numbers. For example: (m times s)
Term – A number, a variable, or a product or quotient of numbers and variables. For example: x, , 4y
Factors – In a multiplication expression, the quantities being multiplied are called factors.
Product – The result of multiplication
Ways in which multiplication is shown in algebra…
This manner is usually avoided because the multiplication symbol appears like a the variable “x”
Power – An expression of the form is known as a power.
Exponent – In the expression , the exponent is
So…
Base
Base – In the expression, the base is
Example: Read as six cubed or six to the third power! (6 x 6 x 6 = 216)
Translating Verbal Expressions into Algebraic Expressions and Vice Versa
OperationSignVerbal PhraseExample
Addition+more than, sum, plus,The sum of a and b
increased by, added to
Subtraction-less than, subtracted from,The difference of a and b
difference, decreased by,Or, a decreased by b
minus
Multiplicationxproduct of, multiplied by,The product of 8 and x
times, of
Divisionquotient of, divided byThe quotient of x and 2
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1.2 Order of Operations
Evaluate – To find the value of an expression when the values of the variables are known. For example: Evaluate (3 x 3 x 3 x 3 = 81)
Order of Operations – The rule that lets you know which operation to perform first.
P1) Simplify the expressions inside grouping symbols, such as parenthesis, brackets, and braces, and as indicated by fraction bars.
E2) Evaluate all Powers/Exponents
M3) Do all multiplication/division from left to right
D
A4) Do all additions and subtractions from left to right.
S
Which one is correct?
Example: or
191735
Using PEMDAS gives you the correct steps to the answer. The correct answer is 191.
Example: The correct answer is 41.
Example: The correct answer is 18.
s+3
s
Perimeter
Find the perimeter of the rectangle above
when: s = 5
First, use the formula for perimeter:
P = 2L + 2W
P = 2(s+3) + 2s
Then substitute and solve using
PEMDAS!
P = 2(5+3) + 2(5)
P = 16 + 10
Area
Find the area of the rectangle
when: s = 5
First us the formula for area:
A = LW
A = (s+3)s
Again, substitute and solve!
A = (5+3)5
A = (8)5
A = 40
P = 26
The fraction bar is another grouping symbol. It means the numerator and denominator should each be treated as a single value.
Evaluate:
when x = 4 and y = 2
= = = 6
Evaluate: when a=4/5 x=2 and y=3
= = = = = = .972
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1.3 Properties of Numbers
Equivalent Expression – Expressions that represent the same number. (3x + 8x and 11x)
Reflexive Property of Equality – Any quantity is equal to itself. (For any number , =)
Symmetric Property of Equality – If one quantity equals a second quantity, then the second quantity equals the first. (For any numbers and , if =, then =)
Transitive Property of Equality – If one quantity equals a second quantity and the second quantity equals a third quantity, then the first quantity equals the third quantity. (For any numbers, , and , if = and =, then =)
Substitution Property of Equality – A quantity may be substituted for its equal in any expression. (If =, then may be replaced by in any expression.)
Additive Identity – For any number , the sum of and 0 is . ()
Additive Inverse – A number and its opposite are additive inverses of each other. ( + - = 0)
Multiplicative Identity – For any number , the product of and 1 is . ()
Multiplicative Property of Zero – For any number , the product of and 0 is 0. ()
Multiplicative Inverse Reciprocals – For every nonzero number , where , ≠ 0,there isexactly one number such that
Commutative Property – For any numbers and ,
+ = +
and
=
Associative Property – For any numbers , and ,
(+)+ = +(+)
and
() = ()
Proof – A logical argument in which each statement you make is supported by a statement that is accepted as true.
Closure Property – The sum of any two whole numbers is always a whole number. So the set of whole numbers is said to be closed under addition.
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Extend 1.3 Algebra Lab: Accuracy
Accuracy– Refers to how close a measured value comes to the actual or desired value. For example, a fraction is more accurate than a rounded decimal.
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1.4 The Distributive Property
Distributive Property – For any numbers , , and :
(+) = +and (+) = +
(-) = -and (-) = -
Like Terms: Terms that contain the same variables, with corresponding variables with the same power. For example:
Simplest Form – An expression is in simplest form when it is replaced by an equivalent expression having no like terms and no parenthesis.
Coefficient – The numerical factor in a term
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1.5 Equations
Open Sentence – Mathematical statements that contains algebraic expressions and symbols.
Equation – A mathematical sentence that involves an equal sign (=).
Solving an Open Sentence – Finding a replacement for the variable in an open sentence that results in a true sentence.
Solution – A replacement for the variable in an open sentence that results in a true sentence.
Replacement Set – A set of numbers from which replacements for a variable may be chosen
Set – A collection of objects or numbers. Sets are usually given capital letter names. For example: A={1,3,5} B={2,4,5}
Element – A member of a set.
Solution Set – The set of all replacements for the variable in an open sentence that results in a true sentence.
Identity – An equation that is true for every value of the variable.
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1.6 Relations
Coordinate System – The intersection of two number lines, the horizontal axis and the vertical axis.
Coordinate plane – The plane containing the x- and y- axes.
Y-Axis (Vertical Axis) – The vertical line in a graph that represents the dependent variable.
(y-axis—dependent variables—RANGE)
X-Axis (Horizontal Axis) – The horizontal line in a graph that represents the independent variable. (x-axis—independent variables—DOMAIN)
Origin – The point of intersection of the two axes in the coordinate plane.
Ordered Pairs – Pairs of numbers used to locate points in the coordinate plane.
X-Coordinate – Represents the horizontal placement of the point
Y-Coordinates – Represents the vertical placement of the point
Relation – A relation is a set of ordered pairs.
Mapping – Illustrates how each element of the domain is paired with an element in the range.
-2
-14
06
18
2
Domain – The set of all first coordinates from the ordered pairs in a relation.
Range – The set of all second coordinates from the ordered pairs in the relation.
Independent Variable – The variable in a function whose value is subject to choice is the independent variable. The independent variable affects the value of the dependent variable.
Dependent Variable – The variable in a function whose value is determined by the independent variable.
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1.7 Functions
Function – A relationship between input and output in which the output depends on the input.
Discrete Function – A graph that consists of points that are not connected
Continuous Function – A function graphed with a line or smooth curve
Vertical line test – A test for functionality of a graphed line. If a vertical line intersects the graph more than once, then the graph is not a function. Otherwise, the relation is a function.
Function Notation – Equation y = 3x-8 in function notation it is written f(x) = 3x-8. In a function, x represents the elements in the domain and f(x) represents the elements in the range.
Nonlinear function – A function with a graph that is not a straight line.
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Extend 1.7 Graphing Technology Lab: Representing Functions
No new vocabulary terms.
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1.8 Interpreting Graph of Functions
Intercept – The points where a graph intersects an axis.
X-Intercept – The y-coordinate of the point at which the graph intersects the y-axis.
Y-Intercept –The x-coordinate of the point at which the graph intersects the x-axis.
Line Symmetry –A graph possesses line symmetry in the y-axis, or some other vertical line, if each half of the graph on either side of the line matches exactly.
Positive – A function is positive on a portion of its domain where its graph lies above the x-axis.
Negative –A function is negative on a portion of its domain where its graph lies below the x-axis.
Increasing –The graph of a function goes up on a portion of its domain when viewed from left to right.
Decreasing – The graph of a function goes down on a portion of its domain when viewed from left to right.
Extrema –Extremely high or low compared to the other points
Relative Maximum – When no other nearby points have a lesser y-coordinate.
Relative Minimum – When no other nearby points have a greater y-coordinate.
End Behavior – Describes the values of a function at the positive and negative extremes in its domain.
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End