<p> Factoring</p><p>1) Factor out the Greatest Common Factor (GCF) if possible.</p><p>2) Determine if it is a Binomial, Trinomial or a Polynomial with 4 Terms.</p><p>3) If it is a Binomial:</p><p>A) The only way to factor is if it is a Difference of Two Squares.</p><p>1) Can you take the square root of the first and last terms?</p><p>2) Is there a minus sign between the two terms?</p><p>3) a² - b² = ( a + b ) ( a - b )</p><p>4) If it is a Trinomial:</p><p>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</p><p>C) Perfect Square Trinomial.</p><p>1) Can you take the square root of the First and Last terms?</p><p>2) The signs have to be + +</p><p> or - +</p><p>3) a² + 2ab + b² = ( a + b )²</p><p> a² - 2ab + b² = ( a - b )²</p><p>4) Do the test 2 * a * b Does it = the middle term? D) Polynomial with 4 Terms</p><p>1) Group the first two terms in parentheses and the last two terms in parentheses.</p><p>A) Don’t forget to leave the sign in the middle of the parentheses.</p><p>2) If the sign in between the parentheses is negative, change the sign of the 2nd term in the last set of parentheses.</p><p>3) Factor out the GCF from each set of parentheses.</p><p>4) State your answer as two binomials.</p><p>5) Rules for the Signs: ax² + bx + c → ( + )( + ) ax² - bx + c → ( - )( - ) ax² + bx – c → ( + )( - ) or ( - )( + ) ax² - bx – c → ( + )( - ) or ( - )( + ) Formulas</p><p>Area of a Rectangle:</p><p>A = L * w L = Length w = width A = Area</p><p>Perimeter of a Rectangle:</p><p>P = 2 L + 2 w L = Length w = width P = Perimeter</p><p>Perimeter of a Triangle:</p><p>P = S1 + S2 + S3 P = Perimeter S1 = side 1 S2 = side 2 S3 = side 3</p><p>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</p><p>To Find the Missing Coordinate of a Direct Variation: use the proportion below and solve.</p><p>Y1 = Y2 X1 X2 </p><p>Inverse Variation: x * y = k k = constant of variation Forms a curve.</p><p>Probability Formula:</p><p>P ( event ) = number of favorable outcomes number of possible outcomes</p><p>Positive Correlation: Both sets of data increase or decrease together.</p><p>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.</p><p>Probability of Two Events: A) Independent Events (with replacement )</p><p>P ( A and B ) = P ( A ) * P ( B )</p><p>B) Dependent Events ( without replacement )</p><p>P ( A and B ) = P ( A ) * P ( B after A )</p><p>Complement of an Event:</p><p>P ( Complement of event ) = 1 - P ( event )</p><p>Slope ( m ) :</p><p> m = y2 - y1 slope = constant of variation of a direct variation = rate x2 - x1 of change</p><p>Slope - Intercept Form of a Linear Equation:</p><p> y = m x + b m = slope b = y - intercept</p><p>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 )</p><p>Standard Form of a Linear Equation:</p><p>A x + B y = C Clear out any fractions by multiplying every term by the LCD. Then write in standard form.</p><p>Domain: x-coordinates or numbers substituted in for a variable or Input values.</p><p>Range: y-coordinates or the answer after substitution…Output Values. Quadrants of a Coordinate Plane:</p><p>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</p><p>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</p><p>If b² - 4ac > 0 → Two solutions If b² - 4ac = 0 → One solution If b² - 4ac < 0 → No solutions</p><p>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.</p><p>( x1 , y1 ) ( x2 , y2 )</p><p> d = </p><p>Midpoint Formula: used to find the ordered pair half-way between two points.</p><p>( x1 , y1 ) ( x2 , y2 )</p><p>(X1 + X2) , (y1 + y2) 2 2</p><p>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</p><p>Use Sin-1, Cos-1, & Tan-1 to find the angle measurements of a right triangle.</p><p>Permutation: Arrangement ( order matters )</p><p> n P r n = number of objects r = number of selections to make</p><p>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.</p><p>Combinations: order does NOT matter.</p><p> n C r n = number of objects r = number chosen</p><p>Rules for Exponents:</p><p>A) a0 = 1</p><p>B) a-2 = 1 1 = a3 a2 a-3 </p><p>C) ( a2 ) ( a3 ) = a2+3 = a5 </p><p>D) a 5 = a2 a 4 = 1 a3 a7 a3</p><p>E) ( a b2 )3 = a3 b6 </p><p>Percent of Change:</p><p>(Largest Number - Smallest Number) * 100 Original Number</p><p>Percent Proportion:</p><p>% = 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.</p><p>Greatest Possible Error: It is one-half of the measuring unit.</p><p>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</p><p>Percent Error:</p><p>Percent Error = Greatest Possible Error * 100 Measurement Given</p><p>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.</p><p>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.</p><p>Profit Formula:</p><p>Profit = Revenue - Expenses Consecutive Integers: x, x+1, x+2, x+3, x+4 etc</p><p>Consecutive Even or Odd Integers: x, x+2, x+4, x+6, x+8 etc.</p><p>Interest Formula: I = p * r * t 100 I = Interest p = principal r = rate, divide it by 100 first. t = time.</p><p>Solving Inequalities: RULE: Switch the direction of the Inequality symbol if you multiply or divide both sides of the inequality by a NEGATIVE NUMBR.</p><p>To Find the X-Intercept: Substitute 0 in for y and solve the equation.</p><p>To Find the Y-Intercept: Substitute 0 in for x and solve the equation.</p><p>Compound Inequalities: Inequalities and Absolute Value Inequalities.</p><p>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 > , ≥</p><p>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.</p><p>Vertical Line Test: used to determine if a graph is a function.</p><p>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.</p><p>Function: If the x - coordinates do NOT repeat, it is a function. If the x - coordinates repeat it is NOT a function.</p><p>Parallel Lines: Have the same slope BUT different y - intercepts.</p><p>Perpendicular Lines: The slopes of perpendicular lines are the opposite & the reciprocal of each other…NEGATIVE RECIPROCALS.</p><p>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</p><p>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</p><p>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.</p><p>To Find the Domain of a Radical Function: Set the Radicand > 0 and solve.</p><p>Quantitative Data: Data that has units of measure and can be numerically compared.</p><p>Qualitative Data: Data that describes a category and CANNOT be measured or numerically campared.</p><p>Univariate Data: Data that uses only one variable.</p><p>Bivariate Data: Data that uses 2 variable, Properties</p><p>Identity Property: a + 0 = a Additive Identity a * 1 = a Multiplicative Identity</p><p>Commutative Property: a + b = b + a of addition a * b = b * a of multiplication</p><p>Associative Property:</p><p>( a + b ) + c = a + ( b + c ) of addition</p><p>( a * b ) * c = a * ( b * c ) of multiplication</p><p>Property of Opposites: ( Additive Inverse )</p><p> a + ( - a ) = 0</p><p>Property of Reciprocals: ( Multiplicative Inverse ) a * 1 = 1 a</p><p>Distributive Property: a ( b + c ) = ab + ac a ( b - c ) = ab - ac</p><p>Multiplication Property of Zero: n * 0 = 0 Multiplication Property of -1:</p><p>-1 * n = -n -1 * -n = n</p><p>Properties of Equality:</p><p>Addition: If a = b then a + c = b + c</p><p>Subtraction: If a = b then a - c = b - c</p><p>Multiplication: If a = b then ac = bc</p><p>Division: If a = b then a = b c c</p><p>Reflexive: a = a</p><p>Symmetric:</p><p>If a = b, then b = a</p><p>Transitive:</p><p>If a = b and b = c, then a = c TI-83 Plus Graphic Calculator Procedures</p><p>Writing the Equation of a Line Given: </p><p>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.</p><p>2) A Table of Values: SAME AS ABOVE.</p><p>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.</p><p> a = slope, b = y-intercept for y = mx + b.</p><p> r = correlation</p><p>-1------0------1</p><p>Strong Correlation 0.5 to 1 & - 0.5 to - 1</p><p>No Correlation 0.49 to - 0.49 </p><p>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.</p><p>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. </p><p>For Degree Mode: for trigonometric ratios. Press the Mode button. Move the cursor to Degree on the 3rd line. Press Enter. Press Clear.</p><p>For Scientific Notation Mode: Press the Mode button. Move the Cursor to Sci on the 1st line. Press Enter. Press Clear.</p><p>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.</p><p>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</p><p>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</p><p>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.</p><p>Objectives: To model exponential growth. To model exponential decay.</p><p>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.</p><p>Growth Factor: b in the function y = a * b . To find the growth factor:</p><p>1) Divide the percent by 100. 2) Then divide by the number of payments in a year. 3) Then add one.</p><p>Compound Interest: when a bank pays interest on both the principal AND the interest an account has already earned.</p><p>Interest Period: is the length of time over which interest is calculated.</p><p>Exponential Decay: the function y = a * b where a > 0 and 0 < b < 1 . a → original amount. b → decay factor. x → # of years.</p><p>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</p><p>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.</p><p>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.</p><p>4) Exponential Functions – are in the form of y = a · bx . The graph forms a curve upward or downward.</p><p>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.</p><p>To Find the Horizontal Asymptote: y = the number to the right of the fraction.</p><p>To find the Vertical Asymptote: Set the denominator equal to zero and solve the equation for x.</p><p>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</p><p>Addition: Subtraction: More Than Difference Plus Minus Sum Decreased by Increased by Less Than Switch the order Added To Subtracted From Switch the order</p><p>Multiplication: Division: Multiplied by Quotient Product Divided by Of Twice = multiply by 2 Triple=multiply by 3 Each Per</p><p>Parentheses: Equals: Quantity Is</p>