Algebra 1 Notes
parallel lines
Inconsistent-independent
When graphing an inequality on a number line
Open circle for < and > Closed circle for ≤ and ≥ Shading to the right for "greater than" Shading to the left for "less than"
PEMDAS
Parentheses, Exponents,Multiplication and Division(reversible), Addition and Subtraction (reversible)
Integers
Positive and negative whole numbers including zeros. This category does not include any fractions or decimals
Quotient of Powers Property
To divide two powers with the same base, subtract the exponents.
Power of a Power Property
To find a power of a power, multiply the exponents.
Power of a Product Property
To find a power of a product, find the power of each factor and then multiply.
Power of a Quotient Property
To find a power of a quotient, find the power of each part of the quotient, and then divide by canceling common factors.
Product of Powers Property
To multiply two powers with the same base, add the exponents.
Use ________to simplify a radical.
Use the greatest perfect square method or prime factorization method to simplify a radical.
Units in formulas
Use units that appropriately model the formula. Use units that match with the quantities measured by the formulas. Remember that units follow algebraic operations. (1 ft • 1 ft = 1 ft^2)
Selecting units
When selecting units, be mindful of the situation you are modeling. Make sure the unit is not too large or too small to represent the measurement.
Unequal Bases
When working with unequal bases, it may be necessary to rewrite them with common bases.
average rate of change
[F(b) - F(a)]/ b-a
Addition Property
a = b, then a + c = b + c.
Subtraction Property
a = b, then a - c = b - c.
Multiplication Property
a = b, then a • c = b • c.
Division Property
a = b, then a/c = b/c.
Substitution property
a = b, then b can replace a in any expression without changing the value of the expression.
a system of equations
a collection of two or more equations
Linear function
a function that is defined by a linear equation.
Distributive Property
any number multiplied to a sum or difference of two or more numbers is equal to the sum or difference of the product
Transitive Property
any real numbers a, b, and c if a = b, and b = c, then a = c.
Reflexive Property
anything is equal to itself
y- intercept
can be written as an ordered pair where x is zero.
x- intercept
can be written as an ordered pair where y is zero.
explicit formula
can generate any term in a sequence just by inputting the term number. This is the same as evaluating any other function you learned about.
relation
describes a relationship and pairs input values with output values
subtraction terms
difference of minus less than decreased by subtracted from
Elimination Method
eliminating one of the variables when combining the two equations
rational exponent
exponent expressed as a fraction. These expressions will convert to radical expressions.
Exponential functions
f(x) = a (b)^x where a is the y-intercept and b is the base of the exponential expression. f(x) = P (1+r)^x where P is the principal amount and r is the rate of change in decimal form.
arithmetic explicit formula
finds each term based on the first term and number of terms. f(n) = f(1) + d(n - 1), where n > 0
geometric explicit formula
finds each term based on the first term and number of terms. f(n) = f(1) • rn-1, where n > 0
arithmetic recursive formula
finds each term based on the previous term. f(n) = f(n - 1) + d, where n > 0
geometric recursive formula
finds each term based on the previous term. f(n) = f(n - 1) • r, where n > 0
recursive formula
generate terms one at a time by relating the term to one or more previous terms
graphically solving a system of equations
graphing both equations and finding the point of intersection
Associative Property
grouping symbols does not affect the outcome
common difference of an arithmetic sequence
same as the slope of the corresponding linear function.
Function notation
sets an expression equal to f(x)
Addition terms
sum of plus more than increased by total
Coefficients
the numbers that come before the variable telling you how many times the variable has been added
Commutative property
the order in which you perform an operation does not affect the outcome.
Symmetric Property
the order on either side of the equal sign does not matter.
solution
the pair of values that make both equations true.
exponent
the small number to the right of the value and tells you how many times the value must be multiplied by itself
Conjunctions
two inequalities connected by the word "and."
Disjunctions
two inequalities connected by the word "or."
Factors
values you can multiply together to get a product
point- slope form
y − y1 = m(x − x1)
Accuracy
A description of how close a measurement is to the true value of the quantity measured
Zero Exponent Property
Any number (except 0) with an exponent of 0 equals
Negative Exponent Property
Any number raised to a negative power is equivalent to the reciprocal of the positive exponent of the number.
the same line
Consistent-dependent
Intersecting lines
Consistent-independent
Steps to a consecutive integer problem
Identify the variable. Write the equation. Solve the equation. Check your solution.
slope formula
(y₂- y₁) / (x₂- x₁)
Volume
1 cup = 8 fluid ounces 1 pint = 2 cups 1 quart = 2 pints 1 gallon = 4 quarts 1 gallon = 3.785 liters 1 gallon = 128 fluid ounces 1 liter = 0.264 gallons
Distance
1 inch = 2.54 centimeters 1 meter = 39.37 inches 1 mile = 5,280 feet 1 mile = 1,760 yards 1 mile = 1.609 kilometers 1 kilometer = 0.62 mile
Mass/Weight
1 pound = 16 ounces 1 pound = 0.454 kilograms 1 kilogram = 2.2 pounds 1 ton = 2,000 pounds
Effects on the Graph of a Function
A graph will experience a vertical shift when f(x) + k = P(1 + r)x + k. A graph will experience a horizontal shift when f(x + h) = P(1 + r)x + h. A graph will be reflected over the y-axis when f(-x) = P(1 + r)-x
Arithmetic Sequences
A list of numbers, called terms, which share a common difference.
Geometric Sequences
A list of numbers, called terms, which share a common ratio.
Rational
A number that can be expressed as a ratio of two numbers (fraction). This includes decimals that terminate or repeat
Irrational Numbers
A number that cannot be expressed as a ratio of two numbers (fraction). This includes decimals that never stop.
Real Numbers
A number that is rational or irrational. This is the highest set of numbers and includes all non-imaginary numbers
Whole Numbers
All positive whole numbers including zero. This category does not include any negative whole numbers
Natural Numbers
All positive whole numbers, not including zero
slope
ratio of the vertical change to the horizontal change between two points.
Precision
Exactness of a measurement
Exponential Growth and Decay
Exponential growth is when the graph is increasing from left to right. Exponential decay is when the graph is decreasing from left to right.
To write the equation of a line parallel to a given line through a given point
Find the slope of the given line. Use that slope and the point on the new line to write the equation of the new line.
To write the equation of a line perpendicular to a given line through a given point
Find the slope of the given line. Use the opposite reciprocal of that slope and the point on the new line to write the equation of the new line.
Rational Exponent Property
Fractional powers, where a number is raised to a fraction, can be converted to a radical. The numerator becomes the exponent, and the denominator becomes the index of the radical.
Steps for Solving Systems of Inequalities by Graphing
Graph both inequalities on the same coordinate plane. Remember to determine if the line should be solid or dashed and which side of the line to shade. Test an ordered pair from the overlapping shaded region to be sure it makes both inequalities in the system true.
Five step problem solving process
Read and understand the situation within. Identify and pull out important information from the problem. Assign variables to unknown values. Set up and solve the equation. Check that your answer makes sense within the context of the problem.
Steps for graphing inequalities
Rearrange the inequality into slope-intercept form. Plot the y-intercept, and use the slope to find a second point. Connect the points with a dashed or solid line.
Steps of solving an equation
Step 1: Simplify each side of the equation. Step 2: Get the variable on one side of the equation Step 3: Get the variable by itself. Step 4: Check your solution.
Signifigant figures
The amount of significant figures in a number that can be found by counting
Rational and Irrational Numbers
The sums and products of two rational numbers are always rational. The sum of a rational number and an irrational number is always irrational. The product of a nonzero rational number and an irrational number is always irrational. The sums and products of two irrational numbers are either rational or irrational.
Substitution Method
isolating one variable in one of the equations and substituting it into the other equation
function
none of the same input values special relationship in that each input value is paired with only one output value
domain
possible x values or the input
range
possible y values or the output
Equivalent systems
produced when any algebraic property is applied to either or both equations in a system. Equivalent systems have the same solutions as the original equations.
Multiplication terms
product of times percent of per twice doubles triples
Division terms
quotient of ratio of half of third of
