Commutative
Associative
Distributive
Identity
Inverse
Zero
Square Root Property
If , then / Quadratic Equations
To solve:
1)Factor: set equation to 0; factor; set each factor to 0; solve; check
2)Quadratic Formula:
3)Completing the Square
- Divide by leading coefficient
- Move constant to other side
- Divide b value by 2 and square it, then add it to both sides
- Factor the perfect square trinomial as a binomial squared
- Use the square root property by taking the square root of both sides and solve
/ Factoring
Look for GCF first!!
GCF: Difference Of Perfect Squares:
Trinomial:
Polynomials
Add – combine like terms
Subtract – add the opposite
Multiply- distribute
Divide – divide each term
Variation
Direct Variation – x & y increase/decrease
together
Inverse Variation – as x increases, y decreases
Solving Equations
- Distribute to get rid of parentheses.
- Combine like terms on the same side of the =.
- Get all variables on one side and the constants on the other.
- Divide or multiply to solve.
- Solve like an equation.
- Graph on a # line (open circle <,>; closed circle )
- Test critical points or put in calc.
- State Solution with and /or.
Real Number System
Exponents
Absolute Value Equations
1)Isolate the absolute value. /
2)Separate into 2 cases “+”&”-“
3)Solve
4)Check /
/
Interval Notation
(1,5]
Use ( ) for < or >; [ ] for
Pythagorean Theorem
To find a missing side of a right triangle:
Where c is the hypotenuse
Solve System of Equations
Substitution – solve one of the equation for x or y and sub into the other.
Elimination/Addition – create opposites by mult equation and add equations / Degree
Monomial – sum of exponents
Polynomial – highest exponent
Functions
Relation – a set of ordered pairs
Function – a relation in which every x-element has only 1 y-element (no repeating x’s; passes vertical line test)
Vertical Line Test – every vertical line drawn will pass through
the graph only one time
Domain – x-values
Range – y-values / Features of Function Graphs
Intercepts
x-intercept – point where the graph passes through the x-axis
find the x-intercept by setting y to 0 and solving
(also called the roots, solution, and zeros)
y-intercept – point where the graph passes through the y-axis
find the y-intercept by setting x to 0 and solving
Positive/Negative Regions Example
f(x) is negative
when x<-2 and 0<x<2
f(x) is positive
when -2<x<0 and x>2
neither positive or negative on the x-axis
Increasing/Decreasing
Increasing –going up left to right
Decreasing – going down left to right
Constant – horizontal line
Restricted Domains Domain is all real numbers EXCEPT:
- Fractions Domain:
- Square Root Domain:
- Both: Square Root in the denominator
set value under radical and solve
Evaluate Function
Substitute if
End Behavior - appearance of the graph as it follows x in either direction / Minimum/Maximum
Highest or lowest point on the graph
/ Symmetry
Horizontal Vertical About the origin
Transformations
over x-axis shift left shift up vertical stretch/compress
over y-axis shift right shift dow horizontal stretch/compress / Average Rate of Change
Types of Functions
Linear where m=slope & b=y-intercept
D: All Real #s
R: All Real #s
x-intercept = (x,0)
y-intercept = (0,y)
slope =
positive negative zero undefined
Collinear – on the same line
Parallel lines have the same slope
Perpendicular lines have negative reciprocal slopes
(flip and change the sign) / Cubic
D: All Real #s
R: All Real #s / Square Root
D:
R: / Cube Root
D: All Real #s
R: All Real #s / Absolute Value
D: All Real #s
R: / Rational
D:
R:
Step
/ Piecewise
/ Rational Expressions
Add /Subtract: Get a common
denominator
Multiply:
Divide: Multiply by the reciprocal
Simplify: Factor and Divide Common
Terms
Quadratic or
D: All Real #s
R: depends on the vertex
x-intercept(s) =
roots=solution=zeros
y-intercept = c value
when a>0 parabola faces up;
a<0 parabola faces down
axis of symmetry:
turning point = vertex=highest or lowest point
(axis of symmetry, y) substitute axis of symmetry
value into equation and solve for y
Exponential where a = starting value; b = 1 + rate (growth) ; x = time; y = ending value
(principal) 1 – rate (decay)
D: All Real #s
R: y > 0
Statistics
Data
Quantitative – dealing with #s
Qualitative- dealing with descriptions
Univariate – one variable; no relationships
Bivariate – two variables; relationships; causes
Causal – cause/effect relationship
Population Data –all members of the group
Sample Data – a subset of the group / Shapes of Distributions
Histograms / Dot Plots / Box & Whisker / Curve
Symmetric (Bell Shaped) / / /
Symmetric (U Shaped) / / /
Skewed Left
/ / /
Skewed Right
/ / /
Uniform
/ /
/
Correlation Coefficient: indicates the strength of the relationship where
-1 = perfect negative correlation; 1 = perfect positive correlation; 0 = no correlation
Graphs
Histograms – data grouped in intervals
Dot Plots– shows all data
Box & Whisker – data grouped in quartiles
More Statistics
Measures of Central Tendency
Mean = average (calc = )
Median= middle term
Mode = most frequent / Measures of Dispersion/Spread
Range = maximum value – minimum value
Interquartile Range – Quartile 3 – Quartile 1
Standard Deviation – how much variation from the mean
STAT CALC 1VAR STATS σx
Also =
Variance – how far the data is spread out
(standard deviation)2
Outliers – values that are far away from the rest of the data;
affects mean; if outliers exist, it is best to use median / Regressions
STAT EDIT enter into list1 and list2
STAT CALC choose the correct regression
5 Statistical Summary
Minimum, Q1, median, Q3, maximum
Quartiles divide the data into 4 equal parts.
Percentiles divide the data into 100 equal parts.
Percentile Rank =
/ Residuals
-the difference between measured data and predicted
data (actual – predicted)
Contingency Table – a 2-way frequency table used to organize 2 categorical variables into 1 display
Frequency – using the actual counts in each cell
Cell – the intersection of 2 categories
Relative frequency – using the percent instead of the count in each cell
Marginal Totals –the totals for each row and column in a contingency table
Relative marginal distribution – the percent for each row and column
Conditional relative distribution – the percent you get in each row or column using just that row or column as the total (instead of the actual total)
Perimeter – the sum of all the sides
Area
Triangle =
Rectangle =
Square =
Parallelogram =
Sequences
Arithmetic – adding a common difference
Explicit equation:
Where is the first element in the sequence
And dis the common difference
Equation Type: Linear
Ex: {3,7,11,15,…} adding 4
/ Geometric – multiplying by a common ratio
Explicit equation:
Where is the first element in the sequence
And r is the common ratio
Equation Type: Exponential
Ex: {9,27,81,…} multiplying by 3
/ Recursive–the next term is found using the previous term; the first term must be given as part of the definition of the function
Ex: