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Today in Pre-Calculus

Study key concepts including domain, square roots, increasing/decreasing trends, extrema, symmetry, continuity, and graph sketching. Practice with examples of domain analysis, boundedness, extrema identification, symmetry recognition, and graph sketching.

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Today in Pre-Calculus

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  1. Today in Pre-Calculus • Review Chapter 1 • Go over quiz • Make ups due by: Friday, May 22

  2. Domain • Look for square roots and denominators • Square roots set radicand ≥0 (numerator) or >0 (denominator). Solve for x. If x2 or higher, test. • Denominators, if not under radical, set ≠0, and solve. These solutions must be excluded from domain. • ( or ) point not included • [ or ] point included

  3. Domain - examples

  4. Increasing/Decreasing • Read from left to right, is graph going up (increasing), down (decreasing) or constant. • Think in terms of slope (for curves tangent lines to the curves). • State intervals using x values.

  5. Bounded • Bounded Above (graph does not go above a particular level) B= • Bounded Below (graph does not go below a particular level) b= • Bounded (bounded above & below) B= and b= • Unbounded (none of the above) • B and b are y values

  6. Extrema • Local (relative) Minima and Maxima • Absolute Minima and Maxima • State as “local minimum of y-value at x =___” • Note: the x values should match all of the intervals in increasing/decreasing.

  7. Example Using the graph: state on what intervals the function is increasing, decreasing , and/or constant. State the boundedness of the function. State any local or absolute extrema

  8. Symmetry • Graph can be symmetry to x-axis, y-axis (even functions) or origin (odd functions). • For origin symmetry parts in quadrant 1 have mirrors in quadrant 3, quadrant 2 mirrors are in quadrant 4.

  9. Continuity • Is graph continuous? (Can you draw the entire graph without picking up your pencil? • Discontinuity: • Removable (just a hole) • Jump • Infinite (do pieces on either side of graph at the point of discontinuity go to infinity –positive or negative)

  10. Continuity

  11. Asymptotes • Vertical asymptotes – occur where function DNE – check domain of function (term does not divide out) • Horizontal asymptotes – from end behavior • Slant asymptotes – degree in numerator must be one more than degree in denominator, use polynomial long division

  12. Intercepts • x – intercept: set numerator = 0 and solve for x • y – intercept: substitute 0 for x and simplify

  13. Sketching Graph

  14. Homework • Pg 102: 10, 13, 15 25-28 (also state boundedness) 47-54 (just with graph only) 55-62 (also find slant asymptotes) 63-66 Know the graphs of the 10 basic functions

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