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A Review of the PreCalculus

By: Will Puckett. A Review of the PreCalculus. For those who don’t already know… What is Calculus?. Definition of CALCULUS a : a method of computation or calculation in a special notation (as of logic or symbolic logic)

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A Review of the PreCalculus

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  1. By: Will Puckett A Review of the PreCalculus

  2. For those who don’t already know…What is Calculus? Definition of CALCULUS • a: a method of computation or calculation in a special notation (as of logic or symbolic logic) • b: the mathematical methods comprising differential and integral calculus —often used with “the”

  3. Parent Functions and their Graphs http://learn.uci.edu/oo/getOCWPage.php?course=OC0111113&lesson=004&topic=13&page=1

  4. These formulas can be used to find the volume of a three dimensional solid. http://www.wkbradford.com/posters/geomforms.html

  5. These formulas can be used to find the surface area of a three dimensional solid, which is equal to the sum of the areas of all sides of the figure added together http://www.wkbradford.com/posters/geomforms.html

  6. The Quadratic Formula http://blogs.discovermagazine.com/loom/2008/05/04/quadratic-vertebrae/

  7. Discriminant and its Implications The discriminant of a function is shown as • If the discriminant is… • <0, the function has no real solutions • =0, the function has one real solution • >0, the function has two real solutions

  8. Try it out • Determine the number of solutions of the equation y= x^2 + 7x + 33, and solve using the quadratic formula to find those solutions.

  9. Exponents • When dividing two powers with the same base, subtract the exponents • (a^b)/(a^c)=a^(b-c) • When multiplying two powers with the same base, add the exponents • (a^b)(a^c)=a^(b+c) • Any number raised to the power of zero equals 1 • A^0=1 • A negative exponent is equal to the multiplicative inverse of the function • A^-b=1/a^b

  10. Solve the following expressions • (x^8) / (x^3) = • (x^2 +2) (x^2 – 2) = • 468x^0 = • 2x^(-3) =

  11. Symmetry of a graph • A graph is symmetrical with respect to the x-axis if, whenever (x, y) is on the graph, (x,-y) is also on the graph • A graph is symmetrical with respect to the y-axis if, whenever (x, y) is on the graph, (-x,y) is also on the graph • A graph is symmetrical with respect to the origin if, whenever (x, y) is on the graph, (-x, -y) is also on the graph

  12. Tests for symmetry • The graph of an equation is symmetric with respect to the y-axis if replacing x with –x yields and equivalent equation • The graph of an equation is symmetric with respect to the x-axis if replacing y with –y yields an equivalent equation • The graph of an equation is symmetric with respect to the origin if replacing x with –x and y with –y yields an equivalent equation

  13. Check the following equations for symmetry wrt both axes and the origin • x – y^2 = 0 • Xy= 4 • Y = x^4 – x^2 + 3 X-axis Origin Y-axis

  14. Even and Odd Functions • A function is even if for every x in the domain, -x is also in the domain, and f(-x)=f(x) • A function can be even if and only if it is symmetrical to the y-axis • Example of an even function: • Y=x^2 • A function is odd if for every x in the domain, -x is also in the domain, and f(-x)=-f(x) • A function can be odd if and only if it is symmetrical to the origin. • Example of an odd function: • Y=x^3

  15. Asymptotes of graphs • Horizontal asymptotes • If the power of the denominator is… • >power of numerator: y=0 is a horizontal asymptote • =power of numerator: y=ratio of the coefficients is a horizontal asymptote • Vertical asymptotes • Vertical asymptotes are found by finding the zeroes of the denominator • Oblique asymptotes • If the power of the numerator is larger than the power of the denominator, you must use the long division method in order to find the asymptote of the graph

  16. Graph the function F (x) = 2(x^2 – 9) (x^2 – 4)

  17. Relative Extrema • Relative extrema are also commonly known as local extrema, or relative maximums and minimums. • A relative minimum is the lowest point on the y-axis that a function reaches between two points of inflection when concave up • A relative maximum is the highest point on the y-axis that a function reaches between two points of inflection when concave down • A polynomial of degree “n” can have a maximum of “n-1” relative extrema

  18. Determine whether A, B, C, and D are relative maximums or relative Minimums A- Relative Minimum B- Relative Maximum C- Relative Minimum D- Relative Maximum http://image.wistatutor.com/content/feed/tvcs/relative20maximum20help20graph20of20function.JPG

  19. Transformations of Graphs There are four different types of transformations that can change the appearance of a graph. • Rigid transformations • Translation • Reflection • Non-rigid transformations • Stretch • Shrink

  20. Translation • A transformation in which the graph of a geometric figure is shifted up, down, or diagonally from its original location without any change in size or orientation Y=x^2 Y=(x^2)+2 The graph was shifted up two units on the y-axis Graphs produced using mathgv

  21. Reflection • A transformation in which the graph of a function is reflected about an axis of reflection, such as the x-axis or a line such as y=2, creating a symmetrical figure with its original graph. Y=-(x^2) Y=x^2 The graph was flipped, or reflected, about the x-axis Graphs produced using mathgv

  22. Stretch or Compress • A transformation in which the graph of a function is either compressed or stretched horizontally, changing the shape of the graph. Y=abs(x) Y=3abs(x) Y=1/3abs(x) The two red curves represent the transformations in which the graph was stretched Or compressed. When multiplied by three, it was compressed towards the y-axis. When divided by three, it was stretched away from the y-axis. Graphs produced using mathgv

  23. Complete the following Transformations: Reflect the graph of y=x^2 + 2 about the x-axis Shift the graph of Y=2x + 3 to the right two and down three

  24. http://www.dsusd.k12.ca.us/users/bobho/Alg/parabola.htm

  25. Types of Conic Sections • Parabola- the set of all points (x,y) that are equidistant from a fixed line (directrix) and a fixed point (focus) not on the line • Ellipses- set of all points (x,y) the sum of whose distances from two distinct fixed points (foci) is constant • Hyperbola- A hyperbola is the set of all points (x,y) the difference of whose distances from two distinct fixed points (foci) is a positive constant

  26. Parabola • Standard form of equation with vertex at (h, k) • (x - h)^2 = 4p(y - k), p cannot equal 0 • Vertical axis, directrix: y = k - p • (y - k)^2 = 4p(x - h), p cannot equal 0 • Horizontal axis, directrix: x = h – p • The focus lies on the axis p units from the vertex. If the vertex is at the origin… • X^2 = 4py vertical axis • Y^2 = 4px horizontal axis http://people.richland.edu/james/lecture/m116/conics/translate.html

  27. Ellipse • Standard form of equation with center (h, k) and major and minor axes of lengths 2a and 2b, where 0 < b < a • (x – h)^2 + (y – k)^2 = 1 a^2 b^2 • (x – h)^2 + (y – k)^2 = 1 b^2 a^2 • The Foci lie on major axis, c units from the center, with c^2 = a^2 + b^2. http://www.tutorvista.com/math/solving-major-axis-of-an-ellipse

  28. Hyperbola • Standard form of equation with center at (h,k) • (x – h)^2 -(y – k)^2 = 1 a^2 b^2 • (x – h)^2 -(y – k)^2 = 1 b^2 a^2 • Vertices are a units from the center, and the foci are c units from the center. C^2 = a^2 + b^2 http://people.richland.edu/james/lecture/m116/conics/translate.html

  29. Circle • Round shape with all points equidistant r units from center at (h, k) where r is the radius. (x – h)^2 + (y – k)^2 = r^2 http://www.mathsisfun.com/algebra/circle-equations.html

  30. Properties of Logarithms http://www.apl.jhu.edu/Classes/Notes/Felikson/courses/605202/lectures/L2/L2.html

  31. Properties of Natural Log http://www.tutorvista.com/math/natural-logarithm-exponential

  32. Properties of the Exponential Function • Domain: All Real numbers • Range: y>0 • Always increasing • Lne^x = x • e^(lnx) = x • A^x = e^(xlna) • Inverse of the natural logarithmic function http://www.craigsmaths.com/number/graphs-of-exponential-functions/

  33. Exponential Growth and Decay A = Ce^kt A: amount at a given time C: Initial amount K: rate of growth or decay (growth when positive, decay when negative) T: time http://www.tutornext.com/help/exponential-growth-function

  34. The population P of a city is P = 140,500e^(kt) Where t = 0 represents the year 2000. IN 1960, the population was 100,250. Find the value of k, and use this result to predict the population in the year 2020. K=.0084; P=166,203

  35. Trigonometry http://www.tutorvista.com/math/trigonometric-functions-chart

  36. Graphs of Trig Functions http://www.xpmath.com/careers/topicsresult.php?subjectID=4&topicID=14

  37. F (x) = asin(2π/b) (x – c) + d absA = amplitude absB = period absC = horizontal shift absD = vertical shift

  38. Amplitude of a Graph • Describes how high or low the graph of the function goes on the y-axis. Changing the amplitude transforms the graph by stretching or compressing it vertically The graph shows the difference in amplitude between f (x)= sin(x) and f (x) = 3sin(x). Notice the vertical stretch made by multiplying the function by three. Graph produced by mathgv

  39. Period of a Graph • Describes the distance or time it takes for the graph of the function to repeat itself, or distance from crest to crest. Changing the period transforms the graph by stretching or compressing it horizontally The graph shows the difference in period between f (x) = sin(x) and F (x) = sin(x/2).Notice the horizontal stretch, and how the period of the modified function in red is double that of the original Graph created by mathgv

  40. The Unit Circle http://etc.usf.edu/clipart/43200/43215/unit-circle7_43215.htm

  41. 1979 AB 1 Given the function f defined by f (x) = 2x^3 – 3x^2 – 12x + 20 • Find the zeros of f • Write an equation of the line normal to the graph of f at x = 0 • Find the x- and y- coordinates of all absolute maximum and minimum points on the graph of f. Justify your answers

  42. Will Puckett 2011

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