MATH 117, Study Guide #1
MATH 117, Study Guide #1 Math 117
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This 4 page Study Guide was uploaded by CJ Wechsler on Sunday September 25, 2016. The Study Guide belongs to Math 117 at University of Southern California taught by Matthew Thomas Hogancamp in Fall 2016. Since its upload, it has received 97 views. For similar materials see Introduction to Mathematics for Business and Economics in Math at University of Southern California.
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Date Created: 09/25/16
Study Guide #1 1.4: Quadratic Equations 2 ax + bx + c = 0 Zero Factor Property: If ab = 0, then a = 0 or b = 0 2 Square Root Property: If x = k, then x = √k or x = -√k Completing the Square: 1) If a ≠ 1, divide by a 2) Move constant to own side of equation 2 3) Add (b/2) 4) Factor 5) Use square root property and solve 2 Quadratic Formula: -b ± √b -4ac / 2a discriminant: the solution to b -4ac to reveal the amount of solutions to the equation - if discriminant < 0 è no solutions - if discriminant = 0 è one solution - if discriminant > 0 è two solutions 1.7 Inequalities Notation: - [ ] are inclusive - ( ) are exclusive General Strategy: 1) Solve corresponding equation 2) Draw dots on number line 3) Test points on each side to determine interval **If rational inequality, find all values of x for which a side of the inequality is undefined (denominator is 0). Remove values from number line and use to make intervals 4) Shade appropriately 1.8 Absolute Values definition: distance from given number to zero Equation / Inequality How to Solve |a| = b a = ±b |a| < b -b < a < b |a| > b a < -b or a > b |a| = |b| a = ±b ** b ≥ 0 2.1 Rectangular Coordinates & Graphs Distance Formula: √(x – 1 ) +2(y - y 1 2 2 Midpoint Formula: (x + 1 / 22, y + y1/ 2)2 x-intercept: where curve of equation hits x-axis (y = 0) y-intercept: where curve of equation hits y-axis (x = 0) Drawing Graphs: 1) Find intercepts 2) Find other solutions 3) Plot and connect the dots 2.2 Circles 2 2 2 Center-Radius Form: (x-h) + (y-k) = r center: (h , k) radius: r General Form: x + y + Dx + Ey + F = 0 Convert center-radius form to general: FOIL Convert general form to center-radius: Complete the Square 2.3 Functions function: rule for assigning elements of one set to another set - machine with inputs and outputs Vertical Line Test: graph is a function if each x-value has 0 or 1 y-value - vertical line hits curve no more than once Increasing Interval: when x < x1impli2s f(x ) < f(1 ) 2 - uphill Decreasing Interval: when when x < x i1plies2f(x ) > f(x )1 2 - downhill Constant Interval: when f(x ) =1f(x ) 2 - flat ** intervals are always open (parentheses) because a specific point cannot be increasing or decreasing 2.5 Linear Models Point-Slope Form: m = y – y / x 1 x 1 Slope-Intercept Form: y = mx + b - lines are parallel if they have the same slope - lines are perpendicular if their slopes are opposite reciprocals 2.6 Graphs of Basic Functions continuous: interval where graph is connected; can be drawn without lifting pen discontinuous: interval where graph is broken piecewise function: defined by different rules over different intervals of its domain 2 3 Squaring function: f(x) = x Cube Root function: f(x) = √x Cubing function: f(x) = x 3 Absolute Value Function: f(x) = |x| Square Root function: f(x) = √x 2.7 Graphing Techniques y = a • f(x) Stretches or shrinks graph of f(x) vertically - If a > 1, stretch by factor of a - If 0 < a < 1, shrink by factor of a - If a < 0, first reflect across x-axis y = f(a•x) Stretches or shrinks graph of f(x) horizontally - If a > 1, shrink by factor of a - If 0 < a < 1, stretch by factor of a - If a < 0, first reflect across y-axis y = f(x) + a Translate (shift) graph of f(x) vertically y = f(x+a) Translate (shift) graph of f(x) horizontally
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