By: Will Puckett A REVIEW OF THE PRECALCULUS For those who dont 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 Parent Functions and their Graphs http://learn.uci.edu/oo/getOCWPage.php? These formulas can be used to find the volume of a three dimensional solid. http://www.wkbradford.com/posters/geomforms.html
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
The Quadratic Formula http://blogs.discovermagazine.com/loom/2008/05/04/quadratic-vertebrae/ 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
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. 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 Solve the following expressions 1. 2. 3. 4. (x^8) / (x^3) =
(x^2 +2) (x^2 2) = 468x^0 = 2x^(-3) = 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 Tests for symmetry 1. 2.
3. 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 Check the following equations for
symmetry wrt both axes and the origin x y^2 = 0 2. Xy = 4 Y = x^4 x^2 + 3 1. 3. 1. X-axis 2. Origin 3. Y-axis
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 yaxis 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 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 Graph the function F (x) = 2(x^2 9) (x^2 4) 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 yaxis 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
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 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 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
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 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 Complete the following Transformations: Shift the graph of Y=2x + 3
to the right two and down three Reflect the graph of y=x^2 + 2 about the x-axis http://www.dsusd.k12.ca.us/users/bobho/Alg/parabola.htm 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
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 Y^2 = 4px vertical axis
horizontal axis http://people.richland.edu/james/lecture/m116/conics/translate.html 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
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 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 Properties of Logarithms http://www.apl.jhu.edu/Classes/Notes/Felikson/courses/605202/lectures/L2/L2.html
Properties of Natural Log http://www.tutorvista.com/math/natural-logarithm-exponential 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/ 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 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 Trigonometry http://www.tutorvista.com/math/trigonometric-functions-chart Graphs of Trig Functions
http://www.xpmath.com/careers/topicsresult.php?subjectID=4&topicID=14 F (x) = asin(2/b) (x c) + d absA = amplitude absB = period absC = horizontal shift absD = vertical shift 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
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 The Unit Circle http://etc.usf.edu/clipart/43200/43215/unit-circle7_43215.htm 1979 AB 1 Given the function f defined by f (x) = 2x^3 3x^2 12x + 20 a) b) c)
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 Will Puckett 2011
