1. Cheryl Corbett, Ivy Cook, and Erin Koch 5 th Period

3. http://itech.pjc.edu/falzone/handouts/parent_functions.pdf

5. Asymptotes Definition: A line such that the distance between the curve and the line approaches zero as they tend to infinity. Types: Horizontal, Vertical, and Oblique www.mathnstuff.com/.../300/fx/library/lines.htmwww.sparknotes.com/.../section2.rhtml

6. Horizontal Asymptotes 1.higher exponent on top—no horizontal asymptote 2.higher exponent on bottom—y=0 3.Exponents same— ratio of coefficients webgraphing.com/algebraictricksoftrade.jsp Example: y= x/(x 2 +1) 1/2

7. Vertical Asymptotes Definition: a vertical line (perpendicular the to x-axis) near which the function rose without bound. How to find it: The zeros of the function are the vertical asymptotes. www.sparknotes.com/.../section2.rhtml Example: f(x)= 1 / (x-5) X-5=0 X=5

8. Oblique Asymptotes Definition: are diagonal lines so that the difference between the curve and the line approaches zero as x tends positive or negative infinity. How to find it: If the power of the numerator is bigger than the denominator then you must do long division to find the asymptote. www.mathwords.com/a/asymptote.htm

9. Symmetry y-axis f(-x)=f(x) x-axis f(-x) = - f(x) origin f(-x,-y)=f(x,y); origin symmetry contains x and y symmetry Example: f(x)=x 2 Solution: f(-x)= (-x) 2 or f(-x)=x2 This problem has y-axis symmetry

10. Quadratic Formula Example: y=x 2 +5x+2 5 ± √17 2 5 ± √17 2

11. Trig Identities tanx=sinx/cosx cotx=cosx/sinx secx=1/cosx cscx=1/sinx sin 2 x+cos 2 x=1 tan 2 x+1=sec 2 x 1+cot 2 x=csc 2 x

12. Example: cotx*secx*sinx=1 Step 1:cosx * 1 * sinx = 1 sinx cosx 1 Step 2:cosx * 1 * sinx = 1 sinx cosx 1 Step 3: 1 * sinx = 1 sinx 1 Step 4: 1 * sinx = 1 sinx 1 Answer: 1 = 1 1

13. Unit Circle

14. Example of Unit Circle Problems 1.sin(∏/3)= 2.cos(5∏/4)= 3.cos(∏/2)= 4.sin(11∏/6)= 5.sin(7∏/4)= 6.cos(∏/6)= 7.sin(∏)= 1.√ (3)/2 2.-√(2)/2 3.0 4.-1/2 5.-√(2)/2 6.√ (3)/2 7.-1

15. Logarithmic Rules Logarithmic Rule 1: Logarithmic Rule 2: Logarithmic Rule 3:

16. Expand the following: log 2 (8x) Simplify the following: log 4 (3) - log 4 (x) Solve the following: log 3 (x) 5 log 2 (8x) = log 2 (8) + log 2 (x) log 4 (3) – log 4 (x) = log 4 (3/x) log 3 (x) 5 = 5log 3 (x)

17. Conic Sections Conic Sections are defined as the intersection of a plane and a cone (and, at a simple level, we just consider the four shapes obtained when the plane does not pass thru the vertex of the cone). http://www.mathwords.com/c/conic_sections.htm

18. Types of conic sections: – Circle – Parabola – Hyperbola – Ellipse Standard Equations for each conic section: – Circle: x 2 + y 2 = r 2 – Parabola: y 2 = 4ax – Hyperbola: x 2 - y 2 = 1 a 2 b 2 – Ellipse: x 2 + y 2 = 1 a 2 b 2

19. Circles Standard form: (x – h) 2 + (y – k) 2 = r 2 Center: (h, k) Radius: r http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_conics_circle.xml

20. Graph a circle with the following formula: The vertex is (2,-1) The radius is √9=3

21. Ellipses

22. Graph the ellipse: a=5 b=3 Center: (0,0) Vertices: (5,0), (-5,0) Foci: (4,0), (-4,0) Length of Major axis: 5 Length of Minor axis: 3

23. Parabolas

24. Graph the following parabola: x 2 =8y p=2 x=0 Vertex: (0,0)Focus: (0,2) Directrix: y=-2 Opens: up Focus Directrix

25. Hyperbolas

26. Graph the following: Center: (0,0) Foci: (2,0), (-2,0) Asymptotes: y=7/2x, y=-7/2x Foci

27. Change of Base Formula log b (x) = log d (x) log d (b)

28. Change of Base Examples Example 1: log3(6) = ln(6) = 1.791759 = 1.63093 ln(3) 1.098612

29. Limits What is a limit? – Holes, jump discontinuity, removable discontinuity Ways to find a limit… – Plug in h value and see if you get a y value – L’Hospital’s Rule

30. Limit Example –plugging in method Limx²-4= Lim (x-2)(x+2) = x→2x+2 x→2 (x-2) Lim(x+2) = Lim(2+2)=4 x→2 x→2

31. L’Hospitals Rule Part one: – If Lim f(x) = 0, then lim f¹(x) = x→ag(x) 0 x→a g¹(x) – Can be used as many times as needed until you get an answer Part two: – lim 1 = 0 x→∞ x – Find the highest power and divide every term by that number or use rules for finding asymptotes

32. Limit Example-L’hospital’s Rule lim 4x⁵ - 7x⁴ + ∏x³ + ex² - 1000 = 0 x→∞ 6x¹⁷ - 8x¹² + (1/x)

33. 1975 AB 1 Given the function f defined by f(x) = ln(x²-9). a.Describe the Symmetry of the graph of f. b.Find the domain of f. c.Find all values of x such that f(x) = 0 d.Write a formula for f¯¹(x), the inverse function of f, for x>3

34. FRQ a. f(-x) = ln((-x)²-9) f(-x) = ln(x²-9) – This function has y-axis symmetry. b. x² -9 ≥ 0 x² ≥ 9 IxI ≥ 3 c. 0 = ln(x²-9) 1 = x²-9 10 = x² ±√10 = x d. x = ln(x²-9) e x = y²-9 e x -9 = y² ±√(e x -9) = y f¯¹(x) = √(e x -9) ANSWER

35. Citations http://itech.pjc.edu/falzone/handouts/parent _functions.pdf http://itech.pjc.edu/falzone/handouts/parent _functions.pdf http://www.mathwords.com/c/conic_sections.htm http://www.mathwords.com/c/conic_sections.htm ©Cheryl Corbett, Erin Koch, and Ivy Cook. 02/18/10