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Chapter 3 Review

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

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  1. Chapter 3 Review The nature of Graphs

  2. 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

  3. Families of Graphs  • Constant function Y remains the same • Linear Equation (Straight line)

  4. Families of Functions continued  • Polynomial (X to a power) • Square root function (y= ) Y = x²

  5. 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)

  6. Families of Functions continued  • Rational Functions (a.k.a. fractions, and it has asymptotes) Ex) y= Y = ( - 1) +2

  7. Trig Graphs  • Sine/ Cosecant Sin/Csc

  8. Trig Graphs Continued  • Cosine/ Secant Cos/Sec

  9. Trig Graphs Continued  • Tangent Tan • Cotangent Cot

  10. 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)

  11. 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)

  12. 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

  13. 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

  14. 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)

  15. 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

  16. 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)

  17. 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

  18. Variation • Direct: Y = k ·x^na.k.a. Y= kx • Inverse: Y = a.k.a Y= • Joint: Y = k ·x^n· z^n

  19. 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

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