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Pre Calculus Review

Pre Calculus Review. By Kevin Young and Chris Haas. Parent Functions. Y=X Y= cosx Y =X^2 Y=1/X Y=X^3 Y= e^X Y= sinX Y= ln^X. Symmetry. Odd Symmetry Odd symmetry is when the function has symmetry with respect to the origin. Even Symmetry

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Pre Calculus Review

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  1. Pre Calculus Review By Kevin Young and Chris Haas

  2. Parent Functions • Y=X • Y=cosx • Y=X^2 • Y=1/X • Y=X^3 • Y=e^X • Y=sinX • Y=ln^X

  3. Symmetry Odd Symmetry • Odd symmetry is when the function has symmetry with respect to the origin Even Symmetry Even symmetry is when the function has symmetry with respect to the Y-axis .

  4. Asymptotes • An asymptote is a line that a function approaches but never intersects.

  5. Horizontal Asymptote • To find the Horizontal Asymptote take the highest exponential value of the numerator compared to the highest exponential value of the denominator • If Exponential value is greater on top there is no horizontal Asymptote • If Exponential value is greater on bottom the asymptote is Y=0 • If Exponential value is the same then take ratio the of coefficients

  6. Vertical Asymptote • To find the vertical Asymptote you have to find the zeros of the denominator.

  7. Oblique Asymptote • If the exponential power is higher on numerator then the denominator then divide the numerator by the denominator.

  8. Domain • The Domain is all the possible values that x can equal. The domain can be found by finding the zeros of the denominator. Domain-

  9. Range • The range is all the possible values that the function equals. Domain- Range-

  10. X and Y intercepts • The x-intercept is the x value when the function intersects the x-axis • The y-intercept is the y value when the function intersects the y-axis • To find the x-intercept set the y value equal to zero • To find the y-intercept set the x value equal to zero

  11. Exponential Growth and Decay Formula- A- The amount C- Initial Amount e- a number K- rate T-Time

  12. Unit Circle

  13. Trig values

  14. Trigonometry Graphs

  15. Trig identities Double Angle Pythagorean Identities

  16. Trig Identities Half Angle Sum and Difference

  17. Trig Properties

  18. Formulas • Quadratic Formula- • Slope-intercept form- • Point-Slope form- • Cone Volume- 1/3r2h • Cylinder Volume- r2h • Rectangular box- Length•Width•Height

  19. Formulas • Sphere Volume- 4/3r3 • Rectangular prism surface area-2ab + 2bc + 2ac • Sphere Surface area-4r 2 • Cylinder Surface area- 2r 2+2rh

  20. Natural Log Properties

  21. Log Properties • logx1 = 0 • logxx = 1 • logxxa = a • logxab = logxa + logxb • logx(a/b) = logx(a) – logx(b) • logx(ab) = b · logx(a)

  22. Change the Base formula

  23. Problem 1 f (x) = log2 (x + 2) • Find the domain of f and range of f • Find the vertical asymptote of the graph of f. • Find the x intercept of the graph of f if there are any.

  24. Problem 1 answer • the domain of f is the set of all x values such that  x + 2 > 0    Domain- x > -2 The range of f is the interval (-∞, +∞).

  25. Problem 1 answer • The vertical asymptote is obtained by solving  x + 2 = 0 which gives  x = -2 • To find the x intercept we need to solve the equation f(x)= 0 log2 (x + 2) = 0 Use properties of logarithmic and exponential functions to write the above equation as 2log2 (x + 2) = 20 Then simplify x + 2 = 1 x = -1

  26. Problem 2 Solve Trig Identities • Sin(2a)= • Cos(2a)= • 1-Sin²(x)= • 1+Tan²(x)= • 1+Cot ²(x)=

  27. Problem 2 Answer • Sin(2a)=2Sin(a)Cos(a) • Cos(2a)=Cos²(a)-Sin²(a) • 1-Sin²(x)=Cos ²(x) • 1+Tan²(x)=Sec ²(x) • 1+Cot ²(x)=Csc ²(x)

  28. Problem 3 f (x) = -3ln(x - 4) • Find the domain of f. • Find the vertical asymptote of the graph of f. • Find the x and y intercepts of the graph of f if there are any.

  29. Problem 3 Answer • The domain of f is the set of all x values such that  x - 4 > 0 Domain x > 4 • The vertical asymptote is obtained by solving x - 4 = 0    Vertical Asymptote-   x = 4

  30. Problem 3 Answer • To find the x intercept we need to solve the equation f(x) = 0 3ln(x - 4) = 0 • Divide both sides by -3 to obtain ln(x - 4) = 0 • Use properties of logarithmic and exponential functions to write the above equation as eln(x - 4) = e0 • Then simplify  x - 4 = 1 x-Int = 5

  31. Bibliography • http://www.google.com/imgres?imgurl=http://www.mathwords.com/t/t_assets/t80.gif&imgrefurl=http://www.mathwords.com/t/trig_values_of_special_angles.htm&usg=__-3XgQOrgRoJB7cDAAzWVc_auyz4=&h=159&w=309&sz=5&hl=en&start=0&zoom=1&tbnid=xOlmVubnsE31FM:&tbnh=93&tbnw=180&ei=RS50TfCaOMultweFuqiEDw&prev=/images%3Fq%3Dtrigometric%2Bvalues%26um%3D1%26hl%3Den%26safe%3Doff%26sa%3DX%26nfpr%3D1%26biw%3D1280%26bih%3D709%26tbs%3Disch:1&um=1&itbs=1&iact=hc&vpx=883&vpy=159&dur=245&hovh=127&hovw=247&tx=64&ty=71&oei=RS50TfCaOMultweFuqiEDw&page=1&ndsp=28&ved=1t:429,r:5,s:0 • http://www.freemathhelp.com/trig-double-angles.html • http://www.sciencedigest.org/unit%20circle.htm • http://math12.vln.dreamhosters.com/wiki/Sum_and_Difference_Identities • http://webgraphing.com/algebraictricksoftrade.jsp • http://www.elec-intro.com/even-function • http://www.sparknotes.com/math/algebra2/specialgraphs/section2.rhtml • http://www.biology.arizona.edu/biomath/tutorials/Rational/Asymptotes.html • http://www.analyzemath.com/Graphing/graphing_tangent_function.html • http://www.ltcconline.net/greenl/courses/105/Limits/INFLIM.HTM • http://www.purplemath.com/modules/asymtote3.htm • http://www.xpmath.com/careers/topicsresult.php?subjectID=4&topicID=14

  32. March 7,2011

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