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Definitions, Rules, and Formulas The Real Numbers Set-Builder Notation and Graphs Natural Numbers: {1, 2, 3, …} {xa � x � b} a b Whole Numbers: {0, 1, 2, 3, …} {xa � x � b} Integers: {… , -3, -2, -1, 0, 1, 2, 3, …} a b Rational Numbers: a a and b are integers, b 0 {xa � x � b} b a b Irrational Numbers: {x|x is real and not rational} 5 0 6 {xx � b} a b {xx � a} a b Basic Rules of Algebra Commutative: a + b = b + a; ab = ba Associative: (a + b) + c = a + (b + c); Slope Formula (ab)c = a(bc) Distributive: aa((bb +- cc)) == aabb +- aacc; slope(m) = CChhaannggee iinn yx = yx22 -- xy11, x2 - x1 0 # Identity: a + 0 = a; a 1 = a 1. If m 7 0, the line rises from left to right. If m 6 0, the Inverse: a + (-a) = 0; a#1a = 1(a 0) line falls from left to right. Multiplication Properties: (-1)a = -a; 2. The slope of a horizontal line is 0. The slope of a (-1)(-a) = a; a#0 = 0; (-a)(b) = (a)(-b) = -ab; vertical line is undefined. (-a)(-b) = ab 3. Parallel lines have equal slopes. Order of Operations Equations of Lines 1. Perform operations above and below any fraction 1. Standard form: Ax + By = C bar, following steps (2) through (5). 2. Slope-intercept form: y = mx + b 2. Perform operations inside grouping symbols, innermost m is the line’s slope and b is its y-intercept. grouping symbols first, following steps (3) through (5). 3. Simplify exponential expressions. 3. Point-slope form: y - y1 = m(x - x1) m is the line’s slope and (x , y ) is a fixed point on the 1 1 4. Do multiplication and division as they occur, working line. from left to right. 4. Horizontal line parallel to the x-axis: y = b 5. Do addition and subtraction as they occur, working 5. Vertical line parallel to the y-axis: x = a from left to right. Definitions, Rules, and Formulas (continued) Properties of Exponents Rational Exponents 1. bm#bn = bm+n 1. an1 = n a 2. (bm)n = bmn 2. amn = 2n a m = n am 3. (ab)m = ambm 2 2 m 1 1 1 4. bm = bm-n 3. a-n =1amn =2 n a m = n am bn 2 2 a m am 1 2 5. b = bm The Quadratic Formula 6. ba0 =b 1, where b 0 The solutions of ax2 + bx + c = 0 with a 0 are 7. b-n = b1n and b1-n = bn,where b 0 x = -b { b2 - 4ac . 22a Special Factorizations Imaginary and Complex Numbers 1. Difference of two squares: A2 - B2 = A + B A - B 1. i = -1 and i 2 = -1 1 2. The set of numbers in the form a + bi is the set of 2. Perfect square trinomial1s: 21 2 complex numbers. If b = 0, the complex number a + bi A2 + 2AB + B2 = A + B 2 is a real number. If b Z 0, the complex number a + bi A2 - 2AB + B2 = A - B 2 is an imaginary number. 1 2 3. Sum of two cubes: 1 2 A3 + B3 = A + B A2 - AB + B2 The Graph of y ax2 bx c 4. Difference of two1 cubes: 21 2 1. The graph of y = ax2 + bx + c is called a parabola, A3 - B3 = A - B A2 + AB + B2 shaped like a bowl. If a 7 0, the parabola opens upward, and if a 6 0, the parabola opens downward. 1 21 2 The turning point of the parabola is the vertex. Variation 2. Graph y = ax2 + bx + c by finding any x-intercepts English Statement Equation (replace y with 0), the y-intercept (replace x with 0), y varies directly as x. y = kx the vertex, and additional points near the vertex and k y varies inversely as x. y = x intercepts. The x-coordinate of the vertex is -b . 2a -b Properties of Radicals The y-coordinate is found by substituting for x in 2a y = ax2 + bx + c and solving for y. All roots represent real numbers. 1. The product rule: 2. The quotient rule: 3. The vertex of y = ax2 + bx + c is a minimum point n # n n n when a 7 0 and a maximum point when a 6 0. a b = ab a n a = , b 0 2 2 2 2n b b 2 B Triangles 3. The Pythagorean Theorem 1. The sum of the measures of the interior angles of a In any right triangle with triangle is 180°. B hypotenuse of length c and legs 2. Similar triangles have corresponding angles with of length a and b, c2 = a2 + b2. the same measure and corresponding sides that are c a proportional. Two triangles are similar if two angles Hypotenuse Leg of one are equal in measure to two corresponding angles of the other. A b C Leg Common Formulas for Area, Perimeter, and Volume Square Rectangle Circle Triangle Trapezoid A = s2 A = lw A = pr2 A = qbh A = qh(a + b) P = 4s P = 2l + 2w C = 2pr b s w r h h s l b a Rectangular Circular Cube Solid Cylinder Sphere Cone V = s3 V = lwh V = pr2h V = dpr3 V = apr2h r s h h r h w r l s s Other Formulas 1. Temperature 5 C = (F - 32) (Fahrenheit to Celsius) 9 9 F = C + 32 (Celsius to Fahrenheit) 5 2. Percent A = PB (A is P percent of B.) 3. Simple Interest I = Prt (Interest is principal times rate times time.) 4. Distance d = rt (Distance is rate times time.) 5. Distance between Points The distance from (x1, y1) to (x2, y2) is (x2 - x1)2 + (y2 - y1)2 . 2 Definitions, Rules, and Formulas The Real Numbers 4. Horizontal line parallel to the x-axis: y = b Natural Numbers: {1, 2, 3, p} 5. Vertical line parallel to the y-axis: x = a Whole Numbers: {0, 1, 2, 3, p} Systems of Equations Integers: {p , -3, -2, -1, 0, 1, 2, 3, p} Rational Numbers: {a | a and b are integers, b Z 0} b y y y Irrational Numbers: {x | x is real and not rational} Basic Rules of Algebra x x x Commutative: a + b = b + a; ab = ba Associative: (a + b) + c = a + (b + c); (ab)c = a(bc) One solution: No solution: Infinitely many Distributive: a(b + c) = ab + ac; a(b - c) = ab - ac consistent inconsistent solutions: dependent and consistent # Identity: a + 0 = a; a 1 = a Inverse: a + (-a) = 0; a#1a = 1(a 0) A system of linear equations may be solved: Multiplication Properties: (-1)a = -a; (a) graphically, (b) by the substitution method, (c) by the # (-1)(-a) = a; a 0 = 0; (-a)(b) = (a)(-b) = -ab; addition or elimination method, (d) by matrices, or (e) by (-a)(-b) = ab determinants. a b Set-Builder Notation, Interval Notation, a1b1 = a1b2 - a2b1 and Graphs 2 2 (a, b) = {x | a 6 x 6 b} a b Cramer’s Rule: [a, b) = {x | a … x 6 b} 2 2 a b Given a system of a equations of the form (a, b] = {x | a 6 x … b} a b c b a c 1 1 1 1 [a, b] = {x | a … x … b} a b a1x + b1y = c1 , then x = c2b2 and y = a2c2 . (-, b) = {x | x 6 b} b a2x + b2y = c2 a1b1 a1b1 (-, b] = {x | x … b} a2b2 a2b2 b (a, ) = {x | x 7 a} 2 2 2 2 a Absolute Value [a, ) = {x | x Ú a} a 2 2 2 2 (-, ) = {x | x is a real number} = {x | xR} x if x Ú 0 1. x = -x if x 6 0 2. I0f |0x| = c, then x = c or x = -c. (c 7 0) Slope Formula 3. If |x| 6 c, then -c 6 x 6 c. (c 7 0) Change in y y2 - y1 4. If |x| 7bc, then x 6 -c or x 7 c. (c 7 0) slope (m) = Change in x = x2 - x1, x1 x2 Special Factorizations 1 2 Equations of Lines 1. Difference of two squares: 1. Slope-intercept form: y = mx + b A2 - B2 = (A + B)(A - B) m is the line’s slope and b is its y-intercept. 2. Perfect square trinomials: 2. Standard form: Ax + By = C 3. Point-slope form: y - y1 = m x - x1 A2 + 2AB + B2 = (A + B)2 m is the line’s slope and (x1, y1) is a fixed point on A2 - 2AB + B2 = (A - B)2 the line. 1 2 3. Sum of two cubes: 2. The complex numbers a + bi and a - bi are con- jugates. Conjugates can be multiplied using the A3 + B3 = (A + B)(A2 - AB + B2) formula 4. Difference of two cubes: (A + B)(A - B) = A2 - B2. A3 - B3 = (A - B)(A2 + AB + B2) The multiplication of conjugates results in a real number. Variation 3. To simplify powers of i, rewrite the expression in English Statement Equation terms of i2. Then replace i2 with -1 and simplify. y varies directly as x. y = kx y varies directly as xn. y = kxn Quadratic Equations and Functions k y varies inversely as x. y = 1. The solutions of a quadratic equation in standard form x k ax2 + bx + c = 0, a 0, y varies inversely as xn. y = xn are given by the quadratic formula y varies jointly as x and z. y = kxz -b { b2 - 4ac x = . 22a Exponents 2. The discriminant, b2 - 4ac, of the quadratic equation Definitions of Rational Exponents ax2 + bx + c = 0 determines the number and type 1. an1 = n a 2. amn = n a m or n am of solutions. 21 2 2 3. a-mn = m 1 2 Discriminant Solutions an Positive perfect square 2 rational solutions Properties of Rational Exponents with a, b, and c rational numbers If m and n are rational exponents, and a and b are real Positive and not a perfect 2 irrational solutions numbers for which the following expressions are defined, square then Zero, with a, b, and c 1 rational solution bm rational numbers 1. bm#bn = bm+n 2. = bm-n bn Negative 2 imaginary solutions 3. (bm)n = bmn 4. (ab)n = anbn a n an 3. The graph of the quadratic function 5. = b bn f (x) = a(x - h)2 + k, a 0, a b is called a parabola. The vertex, or turning point, is Radicals (h, k). The graph opens upward if a is positive and downward if a negative. The axis of symmetry is a 1. If n is even, then n an = a . vertical line passing through the vertex. The graph 2. If n is odd, then n2an = a. can be obtained using the vertex, x-intercepts, if any, 0 0 3. The product rule2: n a# n b = n ab [set f (x) equal to zero], and the y-intercept (set x = 0). 2n a 2 a 2 4. A parabola whose equation is in the form n 4. The quotient rule: = 2n b b f (x) = ax2 + bx + c, a 0, 2 A Complex Numbers has its vertex at b b 1. The imaginary unit i is defined as - , f - . 2a 2a i = -1 , where i2 = -1. a a bb 1 If a 7 0, then f has a minimum that occurs at The set of numbers in the form a + bi is called the set b x = - . If a 6 0, then f has a maximum that occurs of complex numbers. If b = 0, the complex number 2a is a real number. If b 0 the complex number is an b imaginary number. at x = -2a. Definitions, Rules, and Formulas (continued) Exponential and Logarithmic Distance and Midpoint Formulas Functions 1. The distance from (x , y ) to (x , y ) is 1 1 2 2 1. Exponential Function: f (x) = bx, b 7 0, b 1 2 2 Graphs: (x2 - x1) + (y2 - y1) . 3 2. The midpoint of the line segment with endpoints (x , y ) and (x , y ) is y 1 1 2 2 x1 + x2 y1 + y2 , . 2 2 a b f(x) = bx f(x) = bx Conic Sections Circle 0 < b < 1 b > 1 y (0, 1) (x − h)2 + (y − k)2 = r2 x Center 2. Logarithmic Function: f (x) = logb x, b 7 0, b 1 (h, k) y = logbx is equivalent to x = by. Graphs: Radius: r Any point on the circle (x, y) y y y = x y = x x Ellipse y y Vertex (h, k + a) f(x) = bx f(x) = bx (0, 1) (x −a2 h)2+(y −b2 k)2= 1 x (0, 1) x (x −b2 h)2+(y −a2 k)2= 1 (1, 0) (1, 0) Major axis (h, k) (h, k) Vertex (h − a, k) Vertex (h + a, k) x f−1(x) = log x x b f−1(x) = logb x Vertex (h, k − a) Major axis Hyperbola b > 1 < b < 1 y2 x2 3. Properties of Logarithms xa22− by22= 1 y Ays y=mptbaotxe: a2− b2= 1 y Ays y=mptbaotxe: a. logb(MN) = logb M + logb N (0, a) (0, b) M b. logb N = logb M - logb N (−a, 0) (a, 0) (−b, 0) (b, 0) c. logb aMpb= plogb M x x log M ln M log M a d. logb M = log b = ln b = log b (0, −b) a (0, −a) ef.. lbologgbb bxx==x;x 1; 0lologg x10=x =x; exl;n l xn e=xx= x Ay s=ym −ptobatex: Ay s=ym −ptobatex:
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