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Comprehensive Review of Honors Analysis: Counting, Probability, Graphing, and Functions

This resource provides a thorough review of key concepts from Semester 1 Honors Analysis, including counting techniques such as the Fundamental Counting Principle, permutations, and combinations. It also covers introductory probability concepts, including the calculation of odds and the probability of events. The review extends into graphical representation of data, solving linear equations, and understanding piecewise functions. Additionally, it addresses methods for factoring and solving quadratics, ensuring a solid foundation for advanced mathematical study.

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Comprehensive Review of Honors Analysis: Counting, Probability, Graphing, and Functions

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  1. Semester 1 REview Honors Analysis

  2. 1.1: Counting Problems • Fundamental Counting Principle • Factorial Calculations (No Calculator!) • Permutation Calculation (No Calculator!) • Arrangement Problems (Permutations): n! • Circular Arrangements: (n – 1)! • Unique arrangements of letters in words 1.2 Combinations • Combinations Formula (by hand) • Combinations & Fundamental Counting Principle • Distinguish between perm/comb

  3. 1.3 Intro to probability • Calculate Probability/Odds • Create a sample space to determine prob. • Prob of Union of Events (OR Problems) • Remember: If events aren’t mutually exclusive, the intersection must be subtracted!! • Probability of Intersections (AND Problems) • Adjust for independent vs. dependent events (such as replacement) • Calculate probability of the complement • Reword probability scenarios using AND/OR

  4. 1.4 Extended Probability • Use combinations to calculate complex probabilities (specifically when order doesn’t matter) • Conditional Prob: P(A | B) (Probability of Event A given that Event B has occurred) • P(A | B) = 1.5 Geometric Probability • Know 30-60-90 & 45-45-90 triangle patterns • Find basic areas (circles, triangles, rectangles, trapezoids…) • Subtract areas of shapes from other regions to find partial areas

  5. 1.6 Mathematical Expectation (Expected Value) • (Calculated as sum of the products of the probability of each event and the gain/loss)

  6. Unit 2 Topics • Misleading graphs • Quantitative vs. Categorical Variables • Graph Types: • Bar graph vs. Histogram • Frequency table vs. Relative Frequency Table • Stem Plot • Pie Chart/Circle Graph • Comparative Bar Chart • Dot Plot

  7. Unit 2 Topics • Five Point Summary (quartiles, IQR) • Box Plot • Standard Deviation (By hand, calc) • Basic Normal Curve (given simple curve) • Z-Scores • Calculate probabilities using Z-scores

  8. Unit 3 topics • Calculate Slope • Graph linear equations using a table • Graph linear equations using x & y intercepts • Graph linear equations using slope-int form • Horizontal lines: y = k • Vertical lines: x = k • Perpendicular lines: Negative Recip. Slopes • Graph functions to find intersection point • Write equations of lines using pt-slope form:

  9. Unit 3 topics • Graph 2-variable data using a scatter plot • Approximate equation of line of best fit • Use graphing calculator (STAT/lists) to create linear regression line • Use linear equations to make predictions about data • Evaluate positive/negative correlation of data • Calculate midpoint:

  10. Unit 3 topics • Solve linear equations • Write linear equations based on application problems • Write linear equations involving supplements and complements • Write median equation (passes through triangle vertex and mdpt of opposite side) • Write equation of perpendicular bisector of side (passes through midpoint; perpendicular to slope of side) • Write equation of altitude of triangle (passes through vertex; slope perpendicular to base)

  11. UNIT 3 topics • Solve systems using substitution • Solve systems using elimination • Find intersection point of medians (centroid), altitudes (orthocenter), perpendicular bisectors (circumcenter) • Solve systems of three variables • Write equation of parabola using a system of three variables.

  12. Unit 4 Topics • Evaluate, analyze, and graph piecewise functions • Write the equation of piecewise functions • Determine domain and range of a function using the graph (or given a function such as • Determine values that make piecewise functions continuous • Evaluate Greatest Integer Function values • Modular Arithmetic

  13. Unit 4 Topics • Solve distance = rate * time word problems (use chart setup!) • Calculate average rate of change of a function from a table or function • Estimate instantaneous rate of change of a function • Estimate definite integrals by counting blocks on a graph (WATCH OUT FOR GRAPH SCALE!!) • Calculate definite integrals by calculating areas (constant functions, linear functions, etc.) • Estimate definite integrals (area under the curve) using the Trapezoidal Rule (may be given function OR a table of values – always best to draw a graph first!!) • Determine units for rate problems (y unit divided by x unit!) • Determine units for integral/area problems (x unit times y unit!)

  14. Unit 5 Topics • Factoring Methods: • Factor out the GCF • Difference of Squares • Trinomial (FOIL Pattern) • Grouping • Find vertex of a parabola by completing the square • Solve a quadratic using the quadratic formula • Solve a quadratic by factoring

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