Algebra 2 End of Course Review

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2 Natural numbers are not just whole numbers. Integers are all whole numbers both positive and negative. Rational or fractional numbers are all fractional types and are considered ratios because they divide into one another. i is an imaginary number. i 2 = -1 The absolute value of a number is always a positive number since absolute value measures distance. That is, the distance the particular number is from 0 on the number line. Thus, (-3) = 30 +(-3) = 27. Logarithms: 10 4 = 10,000. Then what is the log 10 10,000? 4. The logarithm of 10,000 with base 10 is 4. 4 is the exponent to which 10 must be raised to produce 10, = 10,000 is called the exponential form. log 10 10,000 = 4 is called the logarithmic form. Here is the definition: log b x = n means b n = x. That base, b, with that exponent, n, produces x. Simplify: 65i + 2i. The solution is to add the coefficients, = 67. the answer is 67i. Recall that the area of a rectangle is length times width. Is that area more than or less than 500 square feet? The identify property of multiplication is a number times its reciprocal = one If x, the side of a square, is 36 inches, the area of the square could not be >10 square feet. Number logic: (a) If a = b then a and b have the same values. (b) If you add two odd numbers the answer is always an even number (c) If you add two even numbers the answer is always an even number (d) If you add an odd and an even number or any 2 consecutive numbers the answer is always an odd number. (e) If you multiply two odd numbers the answer is always odd A rubber ball bounces at 96 inches per second. Convert this to a rate of feet per second. Convert 96 inches to feet. There are 12 inches in one foot. Therefore 96 in./12 in. = 8 feet. The diameter of a circle is twice the radius. d = 2 r. The circumference of a circle is π d or 22 7 d The measure of an inscribed angle is half the measure of the intercepted arc. 2

3 Example: A rubber ball rolls along a glass table at 30 inches per second. If the ball were rolled for 60 inches, how much time would it take? Solution: T = D / R = 60 / 30 = 2 seconds Example: Floor plans must often be prepared on graph paper using an appropriate scale. A room on graph paper was 20 inches by 24 inches. If the scale was 1 inch = 8 feet, what was the actual size of the room in square feet? 1 / 8 = 20 / Y Y = /8 = 24 / y Y = times 192 = 30,720 sq.ft. (Obviously this is a banquet hall!) Example: A map showed a scale of 10 miles to 1 cm. What was the length of a highway measuring 100 cm on the map? 1 cm / 10 miles = 100 cm / Y Y= 10 X 100 = 1000 miles How many 12 inch tiles would it take to fit a kitchen 18 feet wide by 20 feet long? Solution: first find the area of the kitchen. A = l x w. A = 18 ft x 20 ft = 360 sq.ft. The tiles are each 1 sq. ft. in area. Therefore it would take 360 sq.ft. / 1 sq.ft. = 360 tiles. The shadow cast by a pole 12 feet tall was 24 feet. How long is the distance from the far end of the shadow to the top of the pole? Solution: Use the Pythagorean Theorem to solve. Review the Pythagorean Theorem. The sides or legs of right triangles have a special relationship. One side, the hypotenuse, is always longer than the other two. That side is opposite the largest angle, the right angle. The square of side A plus the square of side B is equal to the square of side C. A 2 + B 2 = C 2 In our examples, let side A = 12 feet, side b = 24 feet = C = C = C = C = Problem: One bookcase has a height of 6 feet and a width of 2 feet. The ratio of height to width is 3:1. The height of a similar bookcase is 12 feet. What is the width of the second bookcase. The properties of similarity and proportionality are inherent in geometric objects and are applied in an integrated manner in an algebra context. Objects that are similar are in proportion to one another or have some specific ratio. If an object is twice the size of another, then its corresponding parts have a 2:1 ratio. Objects may be enlarged or reduced. The perimeter, area, and volume have specific ratios as well. In this example given 6 ft. / 2 ft = 12 ft,/ Y ft. Use cross multiplication to get 6 Y = 24 and dividing both sides of the equation by 6 to get 4 ft. as the width of the second bookcase. Note that the ratio of the dimensions of the second bookcase, 12:4,is the same as the ratio of the first bookcase. Congruency means equal in all respects. Congruent shapes have all sides, angles, perimeter, area, volume, and ratios equal. Similar shapes are smaller or larger than their originals; the ratio of their perimeters is the same but the ratio of their areas is proportional. Dilations are enlargements (reductions) and are similar in shape and proportional in ratio The distance from the point of tangency to the center of a circle is the diameter or radius. 3

4 Solution: Use the Pythagorean Theorem to solve. Review the Pythagorean Theorem. The sides or legs of right triangles have a special relationship. One side, the hypotenuse, is always longer than the other two. That side is opposite the largest angle, the right angle. The square of side A plus the square of side B is equal to the square of side C. A 2 + B 2 = C 2. The slant side is the hypotenuse of the triangle. Solution: v = a (h). First, find the area of the top or bottom of the cylinder. The area is equal to π r 2. Given the diameter divide it in half to get the radius. Using a = π r 2 and then multiplying by the height the volume of the cylinder can be determined. The area of a cone = 1/3 b h. What is the area of a cone whose base = 31 cm and whose height is 80 cm? 31 times 80 = 2,480 /3 = cm 2 Fractals are representation of similar objects. The distributive property cannot be used to find the perimeter or area of a square from tangents to a circle. A glide transformation will have symmetry, congruency, and similarity. A radius is perpendicular to the tangent of a circle. Use the Elimination Method. Example 1: Solve x + y = 5 x y = 1 x + y = 5 line 1 x y = 1 line 2; what can we do the get rid of one of the variables? (add) Add lines 1 and 2: x y = 1 x + y = 5 2x = 6; x = 3. Substitute x = 3 in any of the equations to get the value of y. y = 2. Solve this quadratic equation by factoring. x 2 + 5x 6 = 0 (x + 6)(x 1) = 0 x + 6 = 0 or x 1 = 0 x = 6 or x = 1 Remember: A "quadratic" is a polynomial that looks like "ax 2 + bx + c", where "a", "b", and "c" are just numbers. For the easy case, you will find two numbers that multiply to the constant term "c", and add to "b", the coefficient on the x-term. The minimum number of points that are necessary to derive an equation is 2.: Using the slope-intercept form where Y = m x + b, and m the coefficient of x is the slope of the equation. Example: in the equation Y = 5 X + 7, 5/1 is the slope and 7 is the y intercept. Use the slope-intercept form where y = m x + b, and m the coefficient of x is the slope of the equation and b is he y intercept. Example: y = -7x + 4. The slope is -7 (and the y intercept is 4.) 4

5 Example: Two concentric circles have a ratio of 3 to 1. If the circumference of the smaller is 40 cm, mentally find the circumference of the larger circle. If the circumference of the smaller circle is 40 cm and the ratio of the larger to smaller is 3 to 1, then the circumference of the larger circle is 3 times that of the smaller or 120 cm. Example: The sum of the ages of 2 brothers was 25. Twice the age of the younger and 3 times the age of the older equaled 55. How old were the brothers? Solution: Let x be one brother s age; let y be the other brother s age. X + y = 25; 2x + 3y = 55; multiply equation 1 by 2 = 2x + 2y = 50 (line 3); subtract line 3 from line 2. y = 5; x = 20. To check your answers substitute the values in both equations: = 25; = 55. If a vertical line intersects a graph at two points the graph is not a function. A line graph has two coordinates of (4, 3) and (8, -3). The slope is therefore negative. If the Y intercept of a function is zero, the X intercept can only be zero We use measures of central tendency or measures of dispersion to interpret and analyze data. The measures of central tendency are the mean, the mode, and the median. A total of 3.. any of these can be used to describe data. The mode is a data item that appears more than once. If no data element appears more often than any other then there is no mode in the set of data. The outlier is a measure of dispersion; it describes how far a value is from the mean. A box and whisker plot uses the median to divide the data. A scatter plot compares similar data. Order is important in a permutation; it is not in a combination. Extrapolating a graph will determine the trend. C (6,5) means to select 5 choices from 6 offerings. Statistical data may be expressed in tables, graphs, charts, or probability. 5

6 Glossary of New Terms NEW TERMS DEFINITION/EXAMPLE Natural numbers 1, 2, 3, 4, 5, 6 Whole numbers All natural numbers and zero: 0, 1, 2, 3 Integers All whole positive and negative numbers: -2, -1, 0, 1, 2 Rational numbers All fractional terminating types (ratios) like ½,.65, 10%, 1:4 Irrational numbers All fractional numbers that are non-terminating or repeating like pi, 1/3, 1/6, An irrational decimal becomes rational by rounding. Real numbers All natural, whole, integers, rational, and irrational numbers Imaginary numbers All numbers that are not real containing a small i. I is the square root of negative 1 or -1; since we cannot find the root of a negative number we say i 2 = 1. Complex numbers Complex numbers (C) contain real and imaginary numbers and are often within an expression: 5 3i Scientific notation A form of expression used to abbreviate large numbers containing many zeros. Scientific notation is expressed as a product of a power of 10. Exponents The number of times the same factor is multiplied: = 2 3 Radicals The square root sign: Absolute value The distance on a number line between a value and zero. Absolute value is always positive. Polynomials An algebraic phenomenon containing terms and expressions; polynomials may have variables, positive exponents, and constants. Polynomials do not contain negative exponents, radicals, or variables in their denominators. Logarithms Inverse exponents that answer the question: To what power is a number raised to obtain a certain value? Two to what power is 2 raised to get 8? 6

7 NEW TERMS DEFINITION/EXAMPLE Commutative property of addition Adding in any order does not affect the answer in an addition problem: a + b + c = c + b + a Commutative property of multiplication Multiplying out of order does nor affect the answer in a multiplication problem: a b = b a Associative property of addition and The answer is not affected when terms are multiplication grouped in a different order for problems that are all addition or all multiplication: (a + b) + c = (c + a) + b or a (b c) = b (a c) Distributive property This property applies a common factor to terms inside parentheses: (2 x 4) + (2 x 5) = 2 (4 + 5) Identity of addition/subtraction Adding or subtracting zero does not change the value of a number: a 0 = a; b + 0 = b Identity of multiplication/division The product of a number and its reciprocal = 1 Inverse property of addition/subtraction A number and its opposite equals zero. Inverse property of multiplication/division To divide by a fraction multiply by the reciprocal. Closure property Any math operation on real numbers results in real numbers. Equality property (a) Reflexive: a = a; (b) symmetric: if a = b, then b = a; (c) transitive: if a = b and b = c, then a = c; substitution: if a = b, then the value of a can replace b. Square numbers Numbers that can be arranged to form squares Triangular numbers Numbers that can be arranged to form triangles Recursive sequences Finding the next number in these means determining another sequence. Triangular or square numbers are recursive sequences. Fibonacci numbers Another recursive sequence found in the growth of plants and other aspects of nature. Pascal s Triangle An array of numbers that uses several sequences (triangular and counting numbers) within it. 7

8 NEW TERMS DEFINITION/EXAMPLE Distance formula D = RT; where d = distance, r = rate, and t = time Rate R = D/T Time T = D/R Scale The numbers on either the x or the y axis depicting a range of values. Scales have consistent distance between values called intervals. Interval The constant space distance between the numbers on any axis of a graph. Independent variable The variable that affects another in a function. Dependent variable The variable that changes because of changes in the independent variable. Function A graph in which some values change because of others. Circumference The distance around a circle. C = d π Diameter The distance across the circle through the center. D = 2r or C/π Radius One half of the diameter. R = ½ d or C π 2 Center The point inside the circle equidistant from the circumference. Each radius meets at the center. Arc A fraction of the circumference. Chord A line segment that joins any 2 points of the circumference. Central angle Angles formed at the center of the circle. Central angles have the same measure as their intercepted arc. Inscribed angle Angles formed at the circumference of the circle. Inscribed angles are ½ the measure of their intercepted arcs or the central angles. Similarity Objects that are proportional to each other. Hypotenuse The longest leg of a right triangle. Perpendicular A line segment at right angles to another. 8

9 NEW TERMS DEFINITION/EXAMPLE Parallel A line that is equidistant from another at every point along that segment Perpendicular slopes The product of these is 1. Parallel slopes If the slopes of two lines are the same, the lines are parallel. Congruency Equal in all respects. Transformations All forms of rotations slide or glide movements, reflections, and flips of geometric objects. Dilations Enlargements or reductions of a shape. Symmetry An equal division of a shape or picture with both parts exactly the same (congruent). Fractals Shapes that divide in the same proportion continuously. Vertices The point where 2 lines meet to form an angle. Popular fractals Sierpinski s Triangle, Koch s Snowflake, and Mandlebrot s Fractals Tangents Lines drawn outside of circles touching the circle at one point only. Mid-point The average distance between 2 points. Vertical line test A test to determine if a graph is a function. Measures of central tendency Statistical measures such as the mean, median and the mode used to describe data. Measures of dispersion Statistical measures such as the range, outlier, variance, and standard deviation used to describe data. Frequency table A table listing how often a range of data occurs. TAILS 5 elements present on graphs: Title, axis, interval, label, and scale. LE Lower extreme or smallest data in a box and whisker plot. 9

10 NEW TERMS UE M LQ UQ Outlier IQ Frequency of occurrence Trends Simple or classical probability Permutations Combinations Experimental probability DEFINITION/EXAMPLE Upper extreme or highest data in a box and whisker plot Median or middle number in a numerically organized data set. Lower quartile or median of the lower half of the data. Upper quartile or median of the upper half of the data. A piece of data that lies outside of the expected range of the data in a box and whisker plot. The expected range is between the 2 quartiles. An outlier should be 1.5 times the interquartile range. Interquartile range is the difference between the upper and lower quartiles. A common way of predicting outcomes based on how often an event occurs. The direction a graph might take. A simple way of calculating the likelihood of random events. An arrangement of objects in which order is important. An arrangement of objects in which order is not important. Determining probability by using an experiment 10

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