Chapter 3 Review
The nature of Graphs
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 
• Constant function
Y remains the same
• Linear Equation
(Straight line)
Families of Functions continued 
• Polynomial
(X to a power)
Y = x²
• Square root function
(y= )
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 continued 
• Rational Functions
(a.k.a. fractions,
and it has asymptotes)
Ex) y=
Y = ( - 1) +2
Trig Graphs 
• Sine/ Cosecant
Sin/Csc
Trig Graphs Continued 
• Cosine/ Secant
Cos/Sec
Trig Graphs Continued 
• Tangent
Tan
• Cotangent
Cot
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 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 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
• To find an inverse:
1)
2)
3)
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
• 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 = 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
• 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
• F(x) =
Where G(x) cannot equal zero (undefined)
Asymptote: Horizontal = Y =
=
Vertical = use G(x) = 0 to get the
X value of the vertical asymptote
=Y
Variation
• Direct:
Y = k · x^n
a.k.a. Y= kx
• Inverse:
Y=
a.k.a Y=
• Joint:
Y = k · x^n · z^n
Bibliography
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http://de.wikipedia.org/w/index.php?title=Datei:Ygleichxhoch3.png&filetimestamp=2011071
9011320
http://www.squarecirclez.com/blog/how-to-reflect-a-graph-through-the-x-axis-y-axis-ororigin/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.h
tml
http://regentsprep.org/Regents/math/algtrig/ATT7/othergraphs.htm