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Ch 1 – Functions and Their Graphs

Ch 1 – Functions and Their Graphs. Different Equations for Lines Domain/Range and how to find them Increasing/Decreasing/Constant Function/Not a Function Transformations Shifts Stretches/Shrinks Reflections Combinations of Functions Inverse Functions. Ch 1 – Functions and Their Graphs.

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Ch 1 – Functions and Their Graphs

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  1. Ch 1 – Functions and Their Graphs • Different Equations for Lines • Domain/Range and how to find them • Increasing/Decreasing/Constant • Function/Not a Function • Transformations • Shifts • Stretches/Shrinks • Reflections • Combinations of Functions • Inverse Functions

  2. Ch 1 – Functions and Their Graphs 1.1 Formulas for lines slope vertical line point- horizontal slope line slope- parallel intercept slopes general perpendicular form slopes

  3. 1.2 Functions domain (input) range (output)

  4. 1.2 Functions domain (input) range (output)

  5. 1.2 Functions Increasing/decreasing/constant on x-axis only (from left to right)

  6. 1.2 and 1.3 Functions Functions Not functions

  7. 1.2 and 1.3 Functions Function or Not a Function? Domain? Range? y-intercepts? x-intercepts? increasing? decreasing?

  8. 1.2 and 1.3 Functions Finding domain from a given function. Domain = except: x in the denominator x in radical Can’t divide by zero Can’t root negative

  9. 1.4 Shifts (rigid) horizontal shift vertical shift

  10. 1.4 Stretches and Shrinks (non-rigid) vertical horizontal stretch shrink shrink stretch

  11. 1.4 Reflections In the x-axis In the y-axis If negative can be move to other side, flipped on x-axis. If can’t, flipped on y-axis.

  12. 1.5 Combination of Functions

  13. 1.5 Combination of Functions

  14. 1.6 Inverse Functions

  15. Ch 2 – Polynomials and Rational Functions • Quadratic in Standard Form • Completing the Square • AOS and Vertex • Leading Coefficient Test • Zeros, Solutions, Factors and x-intercepts • Given Zeros, give polynomial function • Given Function, find zeros • Intermediate Value Theorem, IVT • Remainder Theorem • Rational Zeros Test • Descartes’s Rule • Complex Numbers • Fundamental Theorem of Algebra • Finding Asymptotes

  16. Ch 2 – Polynomials and Rational Functions 2.1 Finding the vertex of a Quadratic Function 1. By writing in standard form (completing the square) 2. By using the AOS formula

  17. 2.1 Writing Equation of Parabola in Standard Form

  18. 2.2 Leading Coefficient Test Leading Coefficient a Positive Negative Leading exponentn Odd Even

  19. 2.2 Zeros, solutions, factors, x-intercepts There are 3 zero (or roots), solutions, factors, and x-intercepts.

  20. 2.2 Zeros, solutions, factors, x-intercepts Find the polynomial functions with the following zeros (roots). If the above are zeros, then the factors are: Can be rewritten as

  21. 2.2 Zeros, solutions, factors, x-intercepts Find the polynomial functions with the following zeros (roots). Writing the zeros as factors: Simplifying.

  22. 2.2 Intermediate Value Theorem (IVT) IVT states that when y goes from positive to negative, There must be an x-intercept.

  23. 2.3 Using Division to find factors Long Division Synthetic Division

  24. 2.3 Remainder Theorem

  25. 2.3 Rational Zeros Test Possible

  26. 2.3 Descartes’s Rule Count number of sign changes of f(–x) for number of positive zeros + – + – 1 2 3 = 3 or 1 positive zeros Count number of sign changes of f(–x) for number of negative zeros. – – – – 0 negative zeros (+) (–) (i)

  27. 2.3 Complex Numbers Complex number = Real number + imaginary number Treat as difference of squares.

  28. 2.3 Complex Numbers

  29. 2.5 Fundamental Theorem of Algebra A polynomial of nth degree has exactly n zeros. has exactly 4 zeros.

  30. 2.5 Finding all zeros 1. Start with Descartes’s Rule 2. Rational Zeros Test (p/q) 3. Test a PRZ (or look at graph on calculator).

  31. 2.6 Finding Asymptotes Vertical Asymptotes Where f is undefined. Set denominator = 0 Horizontal Asymptotes BOBO Degree larger in D, y = 0. BOTN Degree larger in N, no h asymptotes. Degrees same in N and D, take ratio of coefficients.

  32. Ch 3 – Exponential and Log Functions • Exponential Functions • Logarithmic Functions • Graphs (transformations) • Compound Interest (by period/continuous) • Log Notation • Change of Base • Expanding/Condensing Log Expressions • Solving Log Equations • Extraneous Solutions

  33. Ch 3 – Exponential and Log Functions 3.1 Exponential Functions Same transformation as If negative can be move to other side, flipped on x-axis. If can’t, flipped on y-axis. Shifted 1 to right, 2 down. Flipped on x-axis. Flipped on y-axis.

  34. 3.1 Compounded Interest Compound by Period Compound Continuously

  35. 3.1 Compounded Interest A total of $12,000 is invested at an annual interest rate of 3%. Find the balance after 5 years if the interest is compounded (a) quarterly and (b) continuously.

  36. 3.2 Logarithms Used to solve exponential problems (when x is an exponent).

  37. 3.2 Logarithms Used to solve exponential problems (when x is an exponent). Change of base

  38. 3.3 Logarithms Expanding Log Expressions Condensing Log Expressions

  39. 3.4 Solving Logarithmic Equations Solve the Log Equation x in the exponent, use logs

  40. 3.4 Solving Logarithmic Equations Solve the Log Equation

  41. 3.4 Solving Logarithmic Equations Solve the Log Equation

  42. 3.4 Solving Logarithmic Equations Solve the Log Equation

  43. 3.4 Solving Logarithmic Equations Solve the Log Equation

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