Ch 1 – Functions and Their Graphs

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