The domain of a function may be restricted algebraically, graphically, or by the practical situation modeled by a function. Discontinuous domains and ranges include those with removable (holes) and nonremovable (asymptotes) discontinuities. A function can be described on an interval as increasing, decreasing, or constant over a specified interval or over the entire domain of the function. A function, f(x), is increasing over an interval if the values of f(x) consistently increase over the interval as the x values increase. A function, f(x), is decreasing over an interval if the values of f(x) consistently decrease over the interval as the x values increase. A function, f(x), is constant over an interval if the values of f(x) remain constant over the interval as the x values increase. For any x in the domain of f, determine f(x). (f) Represent relations and functions using verbal descriptions, tables, equations, and graphs. Given one representation, represent the relation in another form. (g) Describe the end behavior of a function. (h) Determine the equations of vertical and horizontal asymptotes of functions (rational, exponential, and logarithmic). (i) Determine the inverse of a function (linear, quadratic, cubic, square root, and cube root). (j) Graph the inverse of a function as a reflection over the line y = x. (j) Determine the composition of two functions algebraically and graphically. (k) Investigate and analyze characteristics and multiple representations of functions with a graphing utility. (a, b, c, d, e, f, g, h, i, j, k)
the data fall within two standard deviations of the mean, and approximately 99.7 percent of the data fall within three standard deviations of the mean. This is often referred to as the Empirical Rule or the 68-95-99.7 rule. NOTE: This chart illustrates percentages that correspond to subdivisions in one standard deviation increments. Percentages for other subdivisions require the table of Standard Normal Probabilities or a graphing utility. The mean and standard deviation of a normal distribution affect the location and shape of the curve. The vertical line of symmetry of the normal distribution falls at the mean. The greater the standard deviation, the wider (“flatter” or “less peaked”) the distribution of the data. A derived from a particular data value tells how many standard deviations that data value falls above or below the mean of the data set. It is positive if the data value lies above the mean and negative if the data value lies below the mean. A standard normal distribution is the set of all z-scores. The mean of the data in a standard normal distribution is 0 and the standard deviation is 1. This allows for the comparison of unlike normal data. The table of Standard Normal Probabilities and graphing utilities may be used to determine normal distribution probabilities. a z-score (z), the table of Standard Normal Probabilities (z-table) shows the area under the curve to the left of z. This area represents the proportion of observations with a z-score less than the one specified. Table rows show the z-score’s whole number and tenths place. Table columns show the hundredths place.
AII.12 The student will compute and distinguish between permutations and combinations. The Fundamental Counting Principle states that if one decision can be made n ways and another can be made m ways, then the two decisions can be made nm ways. A permutation is the number of possible ways to arrange a group of objects without repetition and when order matters (e.g., the outcome 1, 2, 3 is different from the outcome 3, 2, 1 when order matters; therefore, both arrangements would be included in the possible outcomes). A combination is the number of possible ways to select or arrange objects when is no repetition and order does not matter (e.g., the outcome 1, 2, 3 is the same as the outcome 3, 2, 1 when order does not matter; therefore, both arrangements would not be included in the possible outcomes). The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Compare and contrast permutations and combinations. Calculate the number of permutations of n objects taken r at a time. Calculate the number of combinations of n objects r at a time. Use permutations and combinations as counting techniques to solve practical problems. Calculate and verify permutations and combinations using a graphing utility.