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) Square root function (y= ) Y = x²

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

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: 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 = = = Y Vertical = use G(x) = 0 to get the X value of the vertical asymptote

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 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