Subtract 6 to both sides Divide by 2 on both sides Answer: x = -9 Cross Multiply. = 3 Distribute 2 to parenthesis Combine like terms Subtract 4x to both sides Subtract 10 from both sides x = -20 Subtract 2x from both sides Add 10 to both sides Divide by 3 to both sides x = 5 Divide both sides by lh = h Subtract by from both sides Divide by a to both sides = Combine Like Terms 7x 3 Distributive Property 6x + 10 Subtract 3 from both sides Divide by 2 to both sides x > 4 Distributive Property Combine like terms 9x 15
-1 0 1 2 3 4 5 Subtract 3 from both sides Divide by -2 to both sides flip the inequality symbol since you divided by a negative x < - 4 x > 3 Flip the inequality symbol when you flip the two sides -1 0 1 2 3 4 5 or < or > x or input independent variable y or output dependent variable Put -3 in all t places and simplify. C(-3) = 3(-3) + 6 C(-3)= -3 Put 3 in all x places and simplify. f(3) = - 2(3) + 3 f(3)= -3
x values don t repeat Put 11 in for f(x) and solve. 11 = - 2x + 3 x = - 4 No 0 gives us 0 or 3 Yes Yes passes vertical line test No Vertical line touches more than once y 5 4 3 2 1 5 4 3 2 1 1 2 3 4 5 x 1 2 3 4 5 y 5 4 3 2 1 5 4 3 2 1 1 2 3 4 5 x 1 2 3 4 5 D: all real numbers R: -1 x 1 D: x -4 R: all real numbers Yes (2, 4), (-2, 5), (3, -6), (-3, -5) No, 2 repeats (2, 4), (-2, 5), (3, -6), (2, -5)
No, x repeats x 2 2 2 2 y 3 4 5 6 Yes x 3 4 5 6 y 2 2 2 2 = = 1 2 Slope-intercept form y = mx + b Positive (2, 3) and (5, -6) = = 6 3 5 2 = 9 3 = 3 Undefined (0 on bottom) Negative Zero (zero on top) Standard Form Ax + By = C
Same or equal Point-slope Form = ( ) = Opposite and Reciprocal m = 2 m = - = = (2, 5) and (-2, 3) 3 5 2 2 = 2 4 = 1 2 x-intercept of 2 and a y-intercept of -3 Use y=mx+b to find b 5 = ½ (2) + b b = 4 y = x + 4 m = 2 (3, -4) = = y-int. = b= 1 y = x 1 y = mx+b -4 = 2(3) + b b=-10 y = 2x - 10
2x 4y = 4 x-int = 2 y-int = 1 y = 2x 1 Subtract 2x and divide by 5 2x + 5y = 10 = 2 5 + 2 Multiply by 2, add 6, subtract 2y y = ½ x 3 x 2y = 6 Distribute ½ and add 2 y-int. = b = -10 2 = 1 2 ( 4) y = 1 2 x + 0 y = x y = 0 Where the line crosses the x-axis and y = 0 Add 10, divide by 3 x = Substitution 3x + 4(2x) = -11 (-1, -2) Where the line crosses the y-axis, b value in slope-intercept form, and x = 0
Elimination by Multiplication First Multiply bottom equation by 2 (1, -2) Elimination by Addition (1, -2) h(x) = g(x) 3x = 4x 1 x = 1 f(x) = g(x) 2x + 9 = 7x 6 x = 3 Multiply Exponents (x 2 y 3 ) 5 = x 10 y 15 Add Exponents x 2 y 3 x 3 y 5 = x 5 y 8 Anything to the zero power is 1 (x 2 y 3 ) 0 = 1 Subtract Exponents = x3 y 1 = x 3 y ( )( ) ( ) = ( )( ) = x 2 y 3 x 3 y -5 = x 5 y -2 = Move negative exponents to bottom
= = Simplify (take out perfect squares) and combine like terms 75 + 2 27 3 5 3 + 2 3 3 3 10 3 Take out perfect squares 75 = 5 3 Keep Change Change (only when Subtr.) then Combine Like Terms (2x 2 + 3x 5) (3x 2 6x 2) (2x 2 + 3x 5) + (-3x 2 + 6x + 2) -x 2 + 9x 3 Combine Like Terms (2x 2 + 3x 5) + (3x 2 6x 2) 5x 2 3x 7 FOIL, Box Method, or Double Distributive Property (3x 5) (2x + 3) 6x 2 + 9x 10x 15 6x 2 x 15 Distributive Property 2x(2x 2 + 3x 5) 4x 3 + 6x 2 10x FOIL, Box Method, Double Distributive Property, or Difference of Two Squares (3x 4y) (3x + 4y) 9x 2 16y 2 FOIL, Box Method, Double Distributive Property, (3x + 4y) 2 (3x + 4y)(3x + 4y) 9x 2 + 12xy + 12xy + 16y 2 9x 2 + 24xy + 16y 2
Greatest Common Factor (GCF) 4x 2 y 3 + 8x 2 y 2 12xy 4xy (xy 2 + 2xy 3) Divide everything in the top by the bottom = + 2 8 Box Method 2x 2 x 6 = (2x + 3)(x 2) What two numbers multiply to be -4 and add to be 3? x 2 + 3x 4 = (x + 4)(x 1) Factor by remembering the Difference of Two Squares a 2 b 2 = (a b)(a+b) 4x 2 9y 2 (2x 3y) (2x + 3y) Take out a GCF first, then what two numbers multiply to be -4 and add to be 3? 2x 2 + 6x 8 2(x 2 + 3x 4) 2(x + 4)(x 1) a = -1, so a shape y= - x 2 + 6x + 2 Factor by remembering the special case a 2 + 2ab + b 2 = (a + b) 2 4x 2 + 12xy + 9y 2 (2x + 3y) 2 a = -1, so a shape maximum a = 1, so a U shape y = x 2 + 6x + 3
y-inter = c = 2 y = - x 2 + 6x + 2 a = 1, so a U shape minimum = 2 y-int = c = 3 y = x 2 + 6x + 3 The maximum/minimum is the vertex Use the axis of symmetry equation to find x Plug into equation to find y (x, y) = ± 4 2 Vertex Form Vertex of y = 2(x 6) 2 + 3 is (6, 3) y = a (x h) 2 + k Vertex at (h, k) Factor, set each factor = 0, solve for x 0 = x 2 + 4x + 3 0 = (x + 1)(x + 3) x+1 = 0 x + 3 = 0 x = -1 x = -3 Quadratic Equation (cannot factor) 0 = x 2 + 6x + 3-0.55 and -5.45 (rounded to nearest hundredth)
Quadratic Formula (cannot factor) 0 = x 2 + 7x + 5-0.81 and -6.19 (rounded to nearest hundredth) Factor, set each factor = 0, solve 0 = x 2 + 4x 12 0 = (x + 6)(x 2) 0 = x + 6 0 = x 2 x = -6 x= 2 Set = 0, Factor, set each factor = 0, solve -5x = x 2 + 6 0 = x 2 + 5x + 6 0= (x + 2)(x + 3) 0 = x + 2 0 = x + 3 x = -2 x = - 3 Set = 0, Quadratic Formula -6 = x 2 + 6x 0 = x 2 + 6x + 6-4.73 and -1.27 (rounded to nearest hundredth) y = x 2 10x 6 y + 6 + 25 = x 2 10x +25 y + 31 = (x 5) 2 y = (x 5) 2 31 Vertex (5, -31) x 2 + 8x + 5 = 0 x 2 + 8x = -5 x 2 + 8x + 16 = -5 + 16 (x+4) 2 = 11 x = -4 ± 11 Create a table or Find vertex and x or y-intercepts Create a table or Find vertex and x or y-intercepts k f(x) The graph stretches vertically (shorter) (k<1) or shrinks vertically (taller) (k>1) It flips if k is negative (y is negative) f(x) + k Shifts graph up if k is positive Shifts graph down if k is negative
f(k x) The graph stretches vertically (wide) (k<1) or shrinks vertically (narrow) (k>1) It flips if k is negative (x is negative) f(x + k) Shifts graph to right if negative Shifts graph to the left if positive BACKWARDS Communicative Property of Multiplication Communicative Property of Addition Associative Property of Multiplication Associative Property of Addition Identity Property of Addition Distributive Property Addition Property of Equality Identity Property of Multiplication
Divide and convert decimal to percentage 0.4 = 40% Divide by 100 or move the decimal two places 0.06 An irrational number cannot be written as a fraction 2 = 1.41421356. A rational number can be written as a fraction (terminal or repeating decimal) ex. 0.25 = Rational number Rational number Irrational number Irrational number Create a table use at least one negative number Create a table use at least one negative number
= 1 + = 1300 1 + 0.05 12 = 1300(1.125) = 1 + A = Total Amount P = Principal r = interest rate n = compoundings per year t = number of years ( ) = 0.85 Decay, base is less than one so the numbers are getting smaller as x increases ( ) = 2 Growth, base is more than one so the numbers are getting bigger as x increases Decay Growth Explicit: can find any number = Recursive: using the last number = Explicit = = 3 2 Recursive = = 3 = 2
IQR = Q3 Q1 A number that is far away from the rest of the data Center mean and median When you have a normal distribution, standard deviation is a way to measure how spread out the data is. 68% 1 standard deviation away 95% 2 standard deviations away 99.7% 3 standard deviations away Skewed to the right Spread standard deviation and IQR Skewed to the left Normal distribution Median is the middle number 5 1 2 3 4 5 6 7 8
Mode is the most often 2 and 5 Mean is the average 4 Upper Quartile is the middle of the second half Q3 = 6 Lower Quartile is the middle of the first half Q1 = 2 Strong Positive Correlation r 0.9 1 2 3 4 5 6 7 8 No Correlation r 0 Weak Positive Correlation r 0.4 Strong Negative Correlation r -0.9 Weak Negative Correlation r -0.4
Causation is when one variable causes another.