Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Fundamental theorem of algebra wikipedia, lookup

System of polynomial equations wikipedia, lookup

Signal-flow graph wikipedia, lookup

Elementary algebra wikipedia, lookup

System of linear equations wikipedia, lookup

Linear algebra wikipedia, lookup

History of algebra wikipedia, lookup

Quadratic equation wikipedia, lookup

Quartic function wikipedia, lookup

Cubic function wikipedia, lookup

Transcript

SOL Algebra 2 Properties Commutative (order), Associative (grouping), Additive Identity (+ 0), Additive Inverse, Multiplicative Identity (x 1), Distributive Property Reflexive, Symmetric, & Transitive Properties of Equality (Transitive – Inequality) If a > b and b > c, then a > c. Addition Property, Subtraction Property, Multiplication Prop. If a = b, then a + c = b + c. If x + 1 > y, then x + 1 + z > y + z. (for inequality) Simplifying Expressions Imaginary numbers: i, i2 = -1, i3 = -i, i4 = 1 pattern Factoring (top top, bottom bottom, cancel top & bottom) Basic Exponent Laws (add or subtract exponents, distribute) Factoring techniques: CMF: Factor out what is common among terms. DOTS: a2 – b2 = (a + b)(a - b) SOTC: a3 + b3 = (a + b)( - + ) DOTC: a3 – b3 = (a - b)( + + ) Easy T (factors of last term that add up to middle term) Perfect Square T: (x – 3)2 or (x + 5)2 not the same as (x2 + 52) Hard T (guess and check, or grouping) Convert: Rational Exponent to Radical (root) Expressions TI – 84: Dealing with imaginary numbers: Use (a + bi) MODE. Use the exponent (^) key and USE ( ) for fractions. Radical expressions (MATH menu: cube root, others) The STORE key may be helpful to check your answers. Parabola: y = a (x – h)2 + k Technique: Find the vertex (h, k) a > 0 (up) a < 0 (down) OR y – k = a (x – h)2 For horizontal parabolas: x = a (y – k)2 + h OR x – h = a (y – k)2 a > 0 (facing right) a < 0 (facing left) SOL Algebra 2 Polynomials x-intercepts zeros solutions (roots) Graph Function Equation If k is a zero, then f(k) = 0. When k is plugged into x, the answer is 0. If 5 is a zero, then (x – 5) is a factor. If (x + 3) is a factor, then –3 is a zero. Think opposite! If ½ is a zero, then (x – ½ ) is a factor. Using SLIDE, this becomes the factor (2x – 1). If -3/4 is a zero, then (x + ¾) is a factor. Using SLIDE, this becomes the factor (4x + 3). FOIL the factors to get the function: y = f(x) or equation: f(x) = 0. # of turns + 1 = degree of function (linear, quadratic, cubic, quartic) Double roots (graph merely touches the x-axis) no real zeros graph doesn’t touch or cross the x-axis a > 0 (upward) a < 0 (downward) TI-84: Y=…, Graph, 2nd Trace [Calculate] Use the STORE key to determine if a number is a zero or not. For easy input, try 2nd Enter [Entry]. Don’t Forget: Parentheses, Parentheses, Parentheses Functions Function value, domain & range, operations including composition TI-84: Y = …, GRAPH, TABLE, TABLESET (ASK)… Pick a point (x,y) from the graph. Plug in the x & y and see if the equation is true. Eliminate choices where equation is false. Pick another easy point (x,y) if necessary. The x- and y- intercepts are good choices. Transformations of Basic Graphs: effects of h & k Horizontal shift: h (left or right) Vertical shift: k (up or down) Example: y = /x/ (basic graph with vertex at origin) y = a /x – h/ + k (absolute value function) Or y – k = a /x – h/ Think opposite…(h,k) new vertex SOL Algebra 2 Solving Equations Use the STORE key. Check to see if there are 2 or more solutions. Use ( Type in the left side and/or right side of the equation carefully. ) with fractions, rational expressions, quantities inside a square root. If the discriminant b2 – 4ac for a quadratic equation is negative, then the solutions are imaginary (involves i). Rational Expressions: You can also use the LCD or crossmultiplication methods. Radical Expressions: You can square both sides of EQ at the right time (isolate radical first). Remember if the root is even, put + and -. Quadratic Expressions: You can factor the left side or use the quadratic formula. Get the standard form first: ax2 + bx + c = 0. Absolute Value Equations/Inequalities /2x + 1/ = 7 , , This branches out into two equations. 2x + 1 = 7 or 2x + 1 = -7. Solve for x. The graph consists usually of two points. endpoints have holes full endpoints (dots) /2x + 1/ > 7 (same with ) The graph consists of two separate parts. Solution has the word ‘or’. Try solving: 2x + 1 > 7. /2x + 1/ < 7 (same with ) The graph consists of one major part (sANDwich). Solution looks like: a x b. Solve: -7 < 2x + 1 < 7. TI-84 keys: Y= … MATH NUM ABS for absolute value symbol 2nd MATH [TEST] for inequality symbols GRAPH SOL Algebra 2 Scatterplots/Linear Regression/Best Fit Is there a correlation at all? Positive (rising line), Negative (falling line), None (scattered) Slope: m = rise / run Y-intercept b Is it a line that best fits the data points? Then think of y = mx + b. Also, draw a line that fits the data. Then read the coordinates. Is it a curve (parabola) that fits the data? Then think of y = ax2+bx+c Sometimes, you have to change the equation first. (Isolate y on the left.) a > 0 (parabola facing up) a < 0 (down) TI-84: Think of the right entries for L1 & L2. You may simplify entries for L1. (example: number of years since 1980) STAT EDIT …(input)… STAT CALC 4 [LinReg ax+b] y = ax + b gives the equation from which you can predict y Just plug number into x! L1 independent variable L2 variable to predict or find Systems of Equations/Inequalities Look for intersection points of 2 graphs. Or solve the system. (You usually solve for y first in terms of x.) Input: Y1= … Y2 = …. GRAPH. Look at the coordinates of intersection points. Or try 2nd TRACE [Calculate] Intersection Linear Programming: Get the objective function. Plug in x & y coordinates of vertices. Check what is asked for: maximum or minimum? solid line Use y = mx + b form instead of standard > or < dashed line > upper portion shaded < lower portion Sequences & Variation Sigma notation, arithmetic (d), geometric (r) patterns Problem solving, arithmetic & geometric means, rule for an Direct vs. Inverse Joint: z = kxy y = kx y=k/x (square, cube roots, etc.) x/y=x/y xy = xy Word Problems (direct: divide & inverse: product = product) REVIEW your Sequences Quiz and STATISTICS QUIZ!