a review of the precalculus
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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) - PowerPoint PPT PresentationTRANSCRIPT
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A REVIEW OF THE PRECALCULUS
By: Will Puckett
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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”
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Parent Functions and their Graphs
http://learn.uci.edu/oo/getOCWPage.php?course=OC0111113&lesson=004&topic=13&page=1
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http://www.wkbradford.com/posters/geomforms.html
These formulas can be used to find the volume of a three dimensional solid.
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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
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The Quadratic Formula
http://blogs.discovermagazine.com/loom/2008/05/04/quadratic-vertebrae/
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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
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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.
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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 1A^0=1
A negative exponent is equal to the multiplicative inverse of the functionA^-b=1/a^b
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Solve the following expressions
1. (x^8) / (x^3) =2. (x^2 +2) (x^2 – 2) =3. 468x^0 =4. 2x^(-3) =
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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
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Tests for symmetry1. The graph of an equation is symmetric with
respect to the y-axis if replacing x with –x yields and equivalent equation
2. The graph of an equation is symmetric with respect to the x-axis if replacing y with –y yields an equivalent equation
3. 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
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Check the following equations for symmetry wrt both axes and the origin
1. x – y^2 = 02. Xy = 4
3. Y = x^4 – x^2 + 3
1. X-axis2. Origin3. Y-axis
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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
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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 asymptotesIf 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
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Graph the functionF (x) = 2(x^2 – 9)
(x^2 – 4)
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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
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Determine whether A, B, C, and D are relative maximums or relative Minimums
http://image.wistatutor.com/content/feed/tvcs/relative20maximum20help20graph20of20function.JPG
A- Relative MinimumB- Relative MaximumC- Relative MinimumD- Relative Maximum
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Transformations of GraphsThere are four different types of
transformations that can change the appearance of a graph.
Rigid transformationsTranslationReflection
Non-rigid transformationsStretchShrink
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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^2Y=(x^2)+2
The graph was shifted up two units on the y-axis
Graphs produced using mathgv
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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.
The graph was flipped, or reflected, about the x-axis
Y=x^2 Y=-(x^2)
Graphs produced using mathgv
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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 stretchedOr 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
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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
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http://www.dsusd.k12.ca.us/users/bobho/Alg/parabola.htm
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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
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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
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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
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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
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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
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Properties of Logarithms
http://www.apl.jhu.edu/Classes/Notes/Felikson/courses/605202/lectures/L2/L2.html
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Properties of Natural Log
http://www.tutorvista.com/math/natural-logarithm-exponential
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Properties of the Exponential Function
Domain: All Real numbers
Range: y>0 Always increasing
Lne^x = xe^(lnx) = xA^x = e^(xlna)
Inverse of the natural logarithmic function
http://www.craigsmaths.com/number/graphs-of-exponential-functions/
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Exponential Growth and Decay
A = Ce^kt A: amount at a given
timeC: Initial amountK: rate of growth or
decay (growth when positive, decay when negative)
T: time
http://www.tutornext.com/help/exponential-growth-function
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The population P of a city isP = 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
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Trigonometry
http://www.tutorvista.com/math/trigonometric-functions-chart
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Graphs of Trig Functions
http://www.xpmath.com/careers/topicsresult.php?subjectID=4&topicID=14
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F (x) = asin(2π/b) (x – c) + dabsA = amplitudeabsB = periodabsC = horizontal shiftabsD = vertical shift
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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
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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
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The Unit Circle
http://etc.usf.edu/clipart/43200/43215/unit-circle7_43215.htm
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1979 AB 1Given the function f defined by f (x)
= 2x^3 – 3x^2 – 12x + 20a) Find the zeros of fb) Write an equation of the line normal to
the graph of f at x = 0c) Find the x- and y- coordinates of all
absolute maximum and minimum points on the graph of f. Justify your answers
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Will Puckett2011