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Chapter 3 Review. The nature of Graphs. Odd/ Even functions. O dd function: f(-x) = -f(x) Which means it has o rigin symmetry -it can be flipped diagonally across the origin (y=x line) *ex) y= x³ Even function: f(-x) = f(x) It has Y axis symmetry -can be flipped over y axis

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chapter 3 review

Chapter 3 Review

The nature of Graphs

odd even functions
Odd/ Even functions
  • Odd function: f(-x) = -f(x)

Which means it has origin symmetry

-it can be flipped diagonally across the origin (y=x line)

*ex) y= x³

  • Even function: f(-x) = f(x)

It has Y axis symmetry

-can be flipped over y axis

*ex) y = x⁴

families of graphs
Families of Graphs 
  • Constant function

Y remains the same

  • Linear Equation

(Straight line)

families of functions continued
Families of Functions continued 
  • Polynomial

(X to a power)

  • Square root function

(y= )

Y = x²

families of functions continued1
Families of Functions continued 
  • Absolute Value

(shape of a V)

* Greatest Integer function (step)

-y is the same for an entire integer

(ex from 1.01 to 1.99 y=1)

families of functions continued2
Families of Functions continued 
  • Rational Functions

(a.k.a. fractions,

and it has asymptotes)

Ex) y=

Y = ( - 1) +2

trig graphs
Trig Graphs 
  • Sine/ Cosecant

Sin/Csc

trig graphs continued
Trig Graphs Continued 
  • Cosine/ Secant

Cos/Sec

trig graphs continued1
Trig Graphs Continued 
  • Tangent

Tan

  • Cotangent

Cot

how to move a graph
How to move a graph

Reflections

Y = - f(x) is over the X axis

(How to remember: f(x) is the same as Y, so if the negative is outside, it does NOT affect the Y axis)

Y = f(-x) is over the Y axis

(How to remember: f(x) is the same as Y, so since the negative is inside, it DOES affect the Y axis)

how to move a graph1
How to move a graph

Translations

Y = f(x) +c is moving c units UP

(its adding height)

Y = f(x) –c is moving c units DOWN

(its subtracting height)

Y = f(x + c) is moving c units left

(if its inside the parenthesis, it will go in the opposite direction of the sign)

Y= f(x – c) is moving c units right

(if its inside the parenthesis, it will go in the opposite direction of the sign)

how to move a graph2
How to move a graph

Dilations

To expand a graph horizontally (wider):

Y = f(cx) and c is a fraction between 0 and 1

To compress a graph horizontally (skinnier):

Y = f(cx) and c is greater than 1

To expand a graph Vertically (taller):

Y = c·f(x) and c is greater than 1

To compress a graph Veritcally (shorter):

Y = c·f(x) and c is a fraction between 0 and 1

inverses
Inverses
  • To find an inverse:
  • Whether it’s an equation, graph, or a table, switch x and y
  • Then, solve for y if its an equation
  • Use the vertical line test on the inverse to figure out if the inverse is a function
  • Vertical line test is: if there is two different y values for one x (the vertical line hits the graph twice) then the inverse is NOT a fuction

Way to remember-

I:SSV

Inverses: switch, solve, vertical line test

I smell stinky vomit

continuity discontinuity
Continuity/ Discontinuity
  • A graph is continuous if it has no breaks and there is a y value for every x value in the given interval
  • Infinite discontinuity: y keeps increasing or decreasing as you approach the x value in question (like a graph right before an asymptote)
  • Jump discontinuity: the graph stops at a certain y value on the x axis, and continues at a different y value on the same x axis (like the step graph)
  • Point discontinuity: The graph is missing a point (function does not exist at that point, but if the point were inserted the graph would be continuous)
end behavior
End Behavior

End behavior = what the y values of the graph do as X goes to ± infinity

Ex) *as x approaches infinity, y increases

*as x approaches negative infinity, y also increases

Y = x²

critical points
Critical Points
  • Maximum (when the function is increasing to the left of x=c and decreasing to the right of x=c, then the maximum is x=c)
  • Minimum (when the function is decreasing to the left of x=c and increasing to the right of x=c, then the minimum is x=c)
  • Point of inflection (graph changes curvature/concavity, a.k.a. curving up or down)
  • Absolute Max (the point at which the highest value of the function occurs. x=m)
  • Absolute Min (the point at which the lowest value of the function occurs. x=m)
  • Relative Max (not the highest point in the function, but the highest point on some interval of the domain x=m )
  • Relative Min (not the lowest point in the function, but the lowest point on some interval of the domain x=m)
rational functions
Rational functions
  • F(x) =

Where G(x) cannot equal zero (undefined)

Asymptote: Horizontal = Y = = = Y

Vertical = use G(x) = 0 to get the

X value of the vertical asymptote

variation
Variation
  • Direct:

Y = k ·x^na.k.a. Y= kx

  • Inverse:

Y = a.k.a Y=

  • Joint:

Y = k ·x^n· z^n

bibliography
Bibliography
  • http://de.wikipedia.org/w/index.php?title=Datei:Ygleichxhoch3.png&filetimestamp=20110719011320
  • http://www.squarecirclez.com/blog/how-to-reflect-a-graph-through-the-x-axis-y-axis-or-origin/6255
  • http://www.northstarmath.com/sitemap/ConstantFunction.html
  • http://upload.wikimedia.org/wikibooks/en/5/55/LinearSystems3.jpg
  • http://www.math.utah.edu/online/1010/parabolas/
  • http://math.tutorvista.com/calculus/real-function-graphs.html
  • http://en.wikipedia.org/wiki/Absolute_value
  • http://www.icoachmath.com/math_dictionary/Greatest_Integer_Function.html
  • http://www.analyzemath.com/RationalGraphTest/RationalGraphTutorial.html
  • http://jwilson.coe.uga.edu/emt668/EMAT6680.2000/Umberger/EMAT6690smu/Day6/Day6.html
  • http://regentsprep.org/Regents/math/algtrig/ATT7/othergraphs.htm