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Memory Aid Help “c” b2 = c2 - a2 a2 = c2 - b2 must be the hypotenuse. In a right triangle that has 30o and 60o angles, the longest side ( the hypotenuse) is always twice the length of the shortest side. Natural number: positive integers and no zero. Whole number: natural + zero. Example: -45, -39, -8, 0, 123, 29874, 30000000 Rational: number can be written as a ratio (fraction) of two integers. (in decimal form are terminating or repeating. Example: 0,1,2,3.....76....3456.....282763.... Integer: whole numbers and their opposites (no decimal) Example: 1,2,3,4,....89,.....756,.....1000000 Example: ½ , 5.2222..., 0.19, -11/3, 2, -4.5, √25 Terminating decimal numbers: 5/2 = 2.5, 5/8 = 0.625 Repeating decimal numbers: 1/9 = 0.1111111...... or 0.1 Irrational: number that cannot be written as a fraction of integers and whose decimal numbers are infinite and nonperiodic (does not repeat). Example: √2, √5, ∏ Reverse x and y to get an inverse function If x increases, y decreases and vice versa When the product of each variables’ values is a constant you get an inverse variation function. a relation is a function when each value of the x-axis (abscissa) has one y-axis (ordinate) associated with it. x-axis (abscissa) = independent variable y-axis (ordinate) = dependent variable [included] ]excluded[ Intervals with infinity: infinity is never included. [-4, +∞[ = from -4 to positive infinity. ]- ∞, -1[ = negative infinity up to but excluding -1. Domain (X): all x values from left to right. Range (Y): all y values from down to up Variation (X): it can increase, decrease or remain constant. Extrema (Y): The minimum: smallest value of y. The maximum: largest value of y. Sign (X): above x-axis is positive and below is negative. X-intercept (zero) & y-intercept (initial value). Domain: ]-∞,+ ∞[ Range: ]- ∞,8] Variation Increasing: ]- ∞,-4] U [-1,3] Decreasing: [-4,-1] U [3, + ∞[ Constant: none Extrema Min: - ∞ Max: 8 Sign Positive: [-6,-2] U [1,5] Negative: ]- ∞,-6] U [-2,1] U [5,+ ∞[ Zero: -6, -2, 1, 5 Initial value: -2 Variables are qualitative (words) or quantitative (numbers). Discrete quantitative (counting numbers) E.g. Dolls on a shelf Continuous quantitative (all values included within an interval – can be decimal points) E.g. Height 1. simple random: by chance (from a hat) 2. systematic: regular intervals from a list of the whole population (every 10th member) 3. cluster: A random selection of clusters is chosen to represent the whole. Every individual within a selected cluster is selected. 4. stratified: taking representative samples from each group. Percentage: 10% of 254 = 10/100 x 254 = 25.4 Sources of bias are different reasons that could lead researchers or survey people to draw the wrong conclusion from a survey or census. There are 6 different sources of bias: A non-representative sample of the population A poorly formulated question The attitude of the person doing the survey Inadequate representation of the results Large part of the sample is rejected A processing error that occurs when compiling the data. Median: is the number in the middle when values are placed in order. Mode: the number that occurs most often in a distribution (list of numbers). Mean: average of all numbers (sum of all values divided by the number of values). Range: highest value – lowest value Table of condensed data: mostly used when data values are repeated. Table with data grouped into classes: data is grouped into intervals [a,b[ (included, excluded) – very few repeated values. Need to determine the number of groups and how much data each one can carry (amplitude). Amplitude = range/number of classes. Amplitude of each interval must be the same! A) mode: class with highest frequency is called the modal class. Middle of modal class ≈ mode B) median: the class that includes the median is called the median class. Middle of median class ≈ median C) mean: sum of midpoints of each class multiplied by its frequency divided by the number of data values. D) range is a measure of dispersion In condensed data: Highest value – lowest value In grouped data: upper bound of highest group or class – lower bound of smallest group or data. Relative frequency is a percentage of a group within the total (how many red pens in a box full of colored pens) Relative frequency Independent = x values Dependent = y values ______y______ Before depends on ____x________. starting a slope type word problem, figure out which variable is x and which is y. 1. locate two ordered pairs (table or graph) 2. find the rate of change (y2-y1)/(x2-x1) 3. using the a you just found, substitute the variables of an ordered pair from your graph or table of values. 4. solve for b. 5. put a and b in the generic rule. 6. y=ax+b 1. using the a you are given, substitute the variables of an ordered pair from your graph, table of values or description. 2. solve for b. 3. put a and b in the generic rule.