Download College Algebra Quick Reference Cheat Sheet and more Cheat Sheet Algebra in PDF only on Docsity! College Algebra Quick Reference Sheet Set Notation Interval Notation Set-Builder Notation (a, b) { x | a < x < b } [a, b] { x | a ≤ x ≤ b } [a, b) { x | a ≤ x < b } (a, b] { x | a < x ≤ b } (a, ∞) { x | a < x } [a, ∞) { x | a ≤ x } (-∞, b) { x | x < b } (-∞, b] { x | x ≤ b } Set Operations Operation Elements Logic Union All OR Intersection Common AND Coordinate Plane Quadrants II I III IV Distance and Midpoint Formulas If P1=(x1,y1) and P2=(x2,y2) are two points, the distance between them is and the midpoint coordinates are Intercepts of an Equation x-intercepts Set y = 0; solve for x y-intercepts Set x = 0; solve for y Symmetry of the Graph of an Equation Type Mathematical Geometrical x-axis Unchanged when y replaced by -y Unchanged when reflected about x-axis y-axis Unchanged when x replaced by -x Unchanged when reflected about y-axis origin Unchanged when y replaced by –y & x replaced by -x Unchanged when rotated 180° about origin Function Notation y = f(x) Domain Set of all valid x Range Set of all valid y Function Arithmetic Transformations of Graphs of Functions Horizontal Vertical Shift (left) (right) (up) (down) Reflect (y-axis) (x-axis) Scale (compress) (expand) 1. Subtract h from each of the x-coordinates of the points on the graph of f. This results in a horizontal shift to the left if h > 0 (positive h) or right if h < 0 (negative h). 2. Divide the x-coordinates of the points on the graph obtained in Step 1 by b. This results in a horizontal scaling, but may also include a reflection about the y-axis if b < 0 (negative b). 3. Multiply the y-coordinates of the points on the graph obtained in Step 2 by a. This results in a vertical scaling, but may also include a reflection about the x-axis if a < 0 (negative a). 4. Add k to each of the y-coordinates of the points on the graph obtained in Step 3. This results in a vertical shift up if k > 0 (positive k) or down if k < 0 (negative k). Properties of Equality Properties of Inequalities Lines or Linear Functions Slope of Line through points (x1, y1) & (x2, y2) Slope-Intercept Form - slope m and point (0, b) Point-Slope Form - slope m and point (x1, y1) or Horizontal Line through point (0, b) Vertical Line through point (a, 0) Average Rate of Change The average rate of change m for function y=f(x) between x=a and x=b is Absolute Value Properties Absolute Value Function as a Piecewise-Defined Function Absolute Value Equations and Inequalities If c is a positive number: Parabolas or Quadratic Functions General Form The graph has a smile if a is positive and a frown if a is negative, and has a vertex at coordinates: Vertex Form The graph has a smile if a is positive and a frown if a is negative, and has a vertex at (h, k). Special Factoring Formulas Special Product Formulas Quadratic Formula Solve If , then 2 real unequal solutions If , then 2 real duplicate solutions If , then no real solutions Factored Form for real factors: Page 1
