Factoring
1) Factor out the Greatest Common Factor (GCF) if possible.
2) Determine if it is a Binomial, Trinomial or a Polynomial with 4 Terms.
3) If it is a Binomial:
A) The only way to factor is if it is a Difference of Two Squares.
1) Can you take the square root of the first and last terms?
2) Is there a minus sign between the two terms?
3) a² - b² = ( a + b ) ( a - b )
4) If it is a Trinomial:
A) The AC Test. 1) Multiply a and c . 2) Then find the factors that will add or subtract to be the middle term. A) Use the rules for the signs. 3) Write each factor as a fraction by putting “a” in the denominator. 4) Reduce each fraction. 5) Write your answer as two binomials
C) Perfect Square Trinomial.
1) Can you take the square root of the First and Last terms?
2) The signs have to be + +
or - +
3) a² + 2ab + b² = ( a + b )²
a² - 2ab + b² = ( a - b )²
4) Do the test 2 * a * b Does it = the middle term? D) Polynomial with 4 Terms
1) Group the first two terms in parentheses and the last two terms in parentheses.
A) Don’t forget to leave the sign in the middle of the parentheses.
2) If the sign in between the parentheses is negative, change the sign of the 2nd term in the last set of parentheses.
3) Factor out the GCF from each set of parentheses.
4) State your answer as two binomials.
5) Rules for the Signs: ax² + bx + c → ( + )( + ) ax² - bx + c → ( - )( - ) ax² + bx – c → ( + )( - ) or ( - )( + ) ax² - bx – c → ( + )( - ) or ( - )( + ) Formulas
Area of a Rectangle:
A = L * w L = Length w = width A = Area
Perimeter of a Rectangle:
P = 2 L + 2 w L = Length w = width P = Perimeter
Perimeter of a Triangle:
P = S1 + S2 + S3 P = Perimeter S1 = side 1 S2 = side 2 S3 = side 3
Direct Variation: y = k x k = constant of variation = slope Forms a linear function that passes through the origin. k = y/x k is called the constant of variation=slope
To Find the Missing Coordinate of a Direct Variation: use the proportion below and solve.
Y1 = Y2 X1 X2
Inverse Variation: x * y = k k = constant of variation Forms a curve.
Probability Formula:
P ( event ) = number of favorable outcomes number of possible outcomes
Positive Correlation: Both sets of data increase or decrease together.
Negative Correlation: One set of data increases as the other set decreases. No Correlation: The data sets are NOT related…they do NOT have an effect on each other.
Probability of Two Events: A) Independent Events (with replacement )
P ( A and B ) = P ( A ) * P ( B )
B) Dependent Events ( without replacement )
P ( A and B ) = P ( A ) * P ( B after A )
Complement of an Event:
P ( Complement of event ) = 1 - P ( event )
Slope ( m ) :
m = y2 - y1 slope = constant of variation of a direct variation = rate x2 - x1 of change
Slope - Intercept Form of a Linear Equation:
y = m x + b m = slope b = y - intercept
Point - Slope Form: it is a formula to find the equation of a line given the slope and an ordered pair. y - y1 = m ( x - x1 )
Standard Form of a Linear Equation:
A x + B y = C Clear out any fractions by multiplying every term by the LCD. Then write in standard form.
Domain: x-coordinates or numbers substituted in for a variable or Input values.
Range: y-coordinates or the answer after substitution…Output Values. Quadrants of a Coordinate Plane:
Quadratic Formula: used to solve a quadratic equation (find the x - intercepts, roots, solutions, answers.) The Function MUST be written in standard form ax ² + bx + c = 0
Discriminant: used to find the number of solutions, x - intercepts, roots, answers, of a Quadratic Equation. The Function MUST be written in standard form ax² + bx + c = 0 b² - 4ac
If b² - 4ac > 0 → Two solutions If b² - 4ac = 0 → One solution If b² - 4ac < 0 → No solutions
Pythagorean Theorem: used to find the missing length of a right triangle. a² + b² = c² Distance Formula: used to find the distance between two ordered pairs.
( x1 , y1 ) ( x2 , y2 )
d =
Midpoint Formula: used to find the ordered pair half-way between two points.
( x1 , y1 ) ( x2 , y2 )
(X1 + X2) , (y1 + y2) 2 2
Trigonometric Ratios: used to find the missing length of a right triangle given the measurement of an angle and the measurement of a side. The Calculator MUST be in DEGREE MODE
Use Sin-1, Cos-1, & Tan-1 to find the angle measurements of a right triangle.
Permutation: Arrangement ( order matters )
n P r n = number of objects r = number of selections to make
5! = 5 * 4 * 3 * 2 * 1 Multiplication Counting Principal - If there are m ways to select the 1st item, and n ways to select a 2nd item, then there are m n ways to make both selections.
Combinations: order does NOT matter.
n C r n = number of objects r = number chosen
Rules for Exponents:
A) a0 = 1
B) a-2 = 1 1 = a3 a2 a-3
C) ( a2 ) ( a3 ) = a2+3 = a5
D) a 5 = a2 a 4 = 1 a3 a7 a3
E) ( a b2 )3 = a3 b6
Percent of Change:
(Largest Number - Smallest Number) * 100 Original Number
Percent Proportion:
% = is 100 of Axis of Symmetry & the x - coordinate of the vertex: used to find the Axis of Symmetry & the x - coordinate of the vertex. ax² + bx + c = 0 x = - b 2a To find the y-coordinate of the vertex: substitute x into the quadratic equation and solve.
Greatest Possible Error: It is one-half of the measuring unit.
Examples: 10 cm - Greatest Possible Error is .5 cm 2.3 in - Greatest Possible Error is .05 in 23.12 mm - Greatest Possible Error is .005 mm
Percent Error:
Percent Error = Greatest Possible Error * 100 Measurement Given
Arithmetic Sequence: adding a fixed number to each term. a + ( n - 1 ) * d a is the 1st term of the sequence. n is the term number you are finding. d is the common difference (the number you are adding) If the common difference is positive, d will be positive. If the common difference is negative, d will be negative.
Geometric Sequence: multiplying a fixed number to each term. a * r (n-1) a is the 1st term of the sequence. n is the term number you are finding. r is the common ratio (the number you multiplying). Distance Formula: rate * time = distance R * T = D time must be in the form of hours.
Profit Formula:
Profit = Revenue - Expenses Consecutive Integers: x, x+1, x+2, x+3, x+4 etc
Consecutive Even or Odd Integers: x, x+2, x+4, x+6, x+8 etc.
Interest Formula: I = p * r * t 100 I = Interest p = principal r = rate, divide it by 100 first. t = time.
Solving Inequalities: RULE: Switch the direction of the Inequality symbol if you multiply or divide both sides of the inequality by a NEGATIVE NUMBR.
To Find the X-Intercept: Substitute 0 in for y and solve the equation.
To Find the Y-Intercept: Substitute 0 in for x and solve the equation.
Compound Inequalities: Inequalities and Absolute Value Inequalities.
A) Conjunction - answers must work for both of the inequalities. 1) Both inequalities are joined together. 2) The word “and” is between the two inequalities. 3) Absolute Value Inequalities have < , ≤ B) Disjunction - answers will work for one of the inequalities, or both inequalities. 1) The word “or” is between the two inequalities. 2) Absolute Value Inequalities have > , ≥
Absolute Value Equations: Will have 2 answers if the absolute value equals a positive number. Will have one answer if the absolute value equals zero. Will have “No Solution” if the absolute value equals a negative number.
Vertical Line Test: used to determine if a graph is a function.
If a vertical line intersects a graph once, it is a function. If a vertical line intersects a graph 2 or more times, it is NOT a function.
Function: If the x - coordinates do NOT repeat, it is a function. If the x - coordinates repeat it is NOT a function.
Parallel Lines: Have the same slope BUT different y - intercepts.
Perpendicular Lines: The slopes of perpendicular lines are the opposite & the reciprocal of each other…NEGATIVE RECIPROCALS.
To determine if a Table of Values is Linear: 1) Subtract the 1st x-coordinate FROM the 2nd x-coordinate for all of the x-coordinates. 2) Subtract the 1st y-coordinate FROM the 2nd y-coordinate for all of the y-coordinates. 3) Then divide each y-value by it’s coresponding x-value. 4) If they are all the same, then it is Linear. There are 3 ways to solve a system of Linear Equations: 1) By Graphing 2) By Substitution 3) By Elimination
There are 5 ways to solve a Quadratic Equation: 1) By hand if you are missing the bx term using Square Roots. 2) By Graphing 3) By Factoring 4) By Completing the Square 5) By the Quadratic Formula
Completing the Square: 1) If the first term has a coefficient > 1, divide every term by that coefficient. 2) Write the equation in the form ax² + bx = c . 3) Multiply ½ * b . 4) Square the answer. 5) Add it to both sides of the equation. 6) Factor the left side of the equation as a perfect square trinomial. 7) Take the square root of both sides of the equation. 8) Don’t forget the ± on the right side. 9) Write two equations. 10) Solve both equations.
To Find the Domain of a Radical Function: Set the Radicand > 0 and solve.
Quantitative Data: Data that has units of measure and can be numerically compared.
Qualitative Data: Data that describes a category and CANNOT be measured or numerically campared.
Univariate Data: Data that uses only one variable.
Bivariate Data: Data that uses 2 variable, Properties
Identity Property: a + 0 = a Additive Identity a * 1 = a Multiplicative Identity
Commutative Property: a + b = b + a of addition a * b = b * a of multiplication
Associative Property:
( a + b ) + c = a + ( b + c ) of addition
( a * b ) * c = a * ( b * c ) of multiplication
Property of Opposites: ( Additive Inverse )
a + ( - a ) = 0
Property of Reciprocals: ( Multiplicative Inverse ) a * 1 = 1 a
Distributive Property: a ( b + c ) = ab + ac a ( b - c ) = ab - ac
Multiplication Property of Zero: n * 0 = 0 Multiplication Property of -1:
-1 * n = -n -1 * -n = n
Properties of Equality:
Addition: If a = b then a + c = b + c
Subtraction: If a = b then a - c = b - c
Multiplication: If a = b then ac = bc
Division: If a = b then a = b c c
Reflexive: a = a
Symmetric:
If a = b, then b = a
Transitive:
If a = b and b = c, then a = c TI-83 Plus Graphic Calculator Procedures
Writing the Equation of a Line Given:
1) Two Coordinates: STAT - EDIT - ENTER - enter the x-coordinates under the L1 column and y-coordinates under the L2 column - STAT - CALC - #4 LinReg - ENTER - ENTER. a = slope, b = y-intercept for y = mx + b.
2) A Table of Values: SAME AS ABOVE.
Writing the Equation of a Trend Line and Finding the Correlation: 2ND - 0 CATALOG - Scroll down to DIAGNOSTIC ON - ENTER - ENTER. STAT - EDIT - ENTER - enter the x-coordinates under the L1 and y-coordinates under the L2 - STAT - CALC - #4 LinReg - ENTER - ENTER.
a = slope, b = y-intercept for y = mx + b.
r = correlation
-1------0------1
Strong Correlation 0.5 to 1 & - 0.5 to - 1
No Correlation 0.49 to - 0.49
To Find the X-Intercepts of a Quadratic Function: 1) Graph the Quadratic Function on the Calculator. 2) Then use one of the following methods. A) Go to the Table of Values (2nd Graph) and find where the y-coordinates are zero. The corresponding x-coordinates are the solutions. This method only works if the answers are Integers. OR B) Press the 2nd button then the Trace button. Select 2 (Zero). Follow the “Left Bound”, “Right Bound” listed below. Put the cursor over the x-intercept when it says “guess”. Press Enter. Repeat the process for the other x-intercept.
To Solve a System of Linear Equations: Graph both equations on the calculator. Press 2nd, then Trace. Choose # 5 Intersect. When it says “1st curve”, Press Enter. When it says “2nd curve”, Press Enter. When it says “Guess”, Put the cursor over the point of intersection and Press Enter.
For Degree Mode: for trigonometric ratios. Press the Mode button. Move the cursor to Degree on the 3rd line. Press Enter. Press Clear.
For Scientific Notation Mode: Press the Mode button. Move the Cursor to Sci on the 1st line. Press Enter. Press Clear.
For Absolute Value Symbols: Press the MATH button. Scroll Right to NUM. Select # 1 abs( Press Enter. To Find the Minimum Value of a Quadratic Function: Graph the Quadratic Function. Press 2nd then Trace. Select # 3 minimum. Follow the “Left Bound”, “Right Bound” listed below. Place the cursor on the lowest point when it says “guess”. Press Enter. To Find the Maximum Value of a Quadratic Function: Graph the Quadratic Function. Press 2nd then Trace. Select # 4 maximum. Follow the “Left Bound”, “Right Bound” listed below. Place the cursor on the highest point when it says “guess”. Press Enter.
Standard Deviation: STAT - EDIT - ENTER - enter the DATA under the L1 column. STAT - CALC - # 1 VAR STATS - ENTER - ENTER. x = mean ( average ). σ x = standard deviation
To Solve Linear Equations in Standard Form: 2nd Button – X-1 Button (MATRIX) Scroll right to highlight EDIT at the top of the screen – ENTER MATRIX [A] 2 X 3 Enter the Coefficients into the Matrix. 2nd Button – Mode Button (QUIT) 2nd Button – X-1 Button (MATRIX) Scroll right to highlight MATH at the top of the screen. Scroll down the menu to letter B rref – ENTER. 2nd Button – X-1 Button (MATRIX) – ENTER – ENTER
To Graph Linear Inequalities: Press the Y= Button. Enter the Equation to the right of Y1=. Scroll to the left of Y1. Press the Enter Button 2 times for a Greater Than symbol >. Press the Enter Button 3 times for a Less Than Symbol <. ≤ or ≥ draw a SOLID line. < or > draw a DOTTED line. Section 8-8 Exponential Growth.
Objectives: To model exponential growth. To model exponential decay.
Exponential Growth: the function y = a * b for a > 0 and b > 1 . a → original amount (principal). b → growth factor. x → # of payments or # of years.
Growth Factor: b in the function y = a * b . To find the growth factor:
1) Divide the percent by 100. 2) Then divide by the number of payments in a year. 3) Then add one.
Compound Interest: when a bank pays interest on both the principal AND the interest an account has already earned.
Interest Period: is the length of time over which interest is calculated.
Exponential Decay: the function y = a * b where a > 0 and 0 < b < 1 . a → original amount. b → decay factor. x → # of years.
Decay Factor: the b in the function y = a * b . To find the decay factor: 1) 100% - the percent given. 2) Divide by 100. Functions
1) Linear Functions – highest power of x is 1. They form a straight line on the graph. The equations are in the form of y = mx + b. The equation of a horizontal line is in the form of y = the y-intercept. The equation a vertical is in the form of x = the x – intercept. A VERTICAL LINE is NOT a Function.
2) Quadratic Functions – the highest power of x is 2. They form a parabola that opens upward or downward. The equations are in the form of: y = ax² y = ax² + c y = ax² + bx + c 3) Absolute Value Functions - the variable “x” is contained inside the Absolute Value Symbols. The graph forms a V – shape that opens upward or downward.
4) Exponential Functions – are in the form of y = a · bx . The graph forms a curve upward or downward.
5) Square Root Functions – otherwise known Radical Functions – the variable “x” is contained in the radicand. The graph forms a curve to the right. 6) Rational Functions – contains the variable “x” in the denominator. The graph forms 2 curves and there are asymptotes that guide the graph.
To Find the Horizontal Asymptote: y = the number to the right of the fraction.
To find the Vertical Asymptote: Set the denominator equal to zero and solve the equation for x.
Quadratic Functions: Standard form is ax2 + bx + c = 0 If a is POSITIVE, the parabola opens upward. If a is NEGATIVE, the parabola opens downward. The LARGER the absolute value of a, the NARROWER the Parabola. c is the y-intercept. Translation
Addition: Subtraction: More Than Difference Plus Minus Sum Decreased by Increased by Less Than Switch the order Added To Subtracted From Switch the order
Multiplication: Division: Multiplied by Quotient Product Divided by Of Twice = multiply by 2 Triple=multiply by 3 Each Per
Parentheses: Equals: Quantity Is