Real Analysis

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 8 July 2014

Lecture 5

Graphing Techniques

  1. Basic graphs
  2. Shifting
  3. Scaling
  4. Flipping
  5. Limits
  6. Asymptotes
  7. Slopes: Increasing/Decreasing
  8. Concave up/Concave down points of inflection.

Basic Graphs

1 1 x y y=x x y -1 1 1 y=x2 x y -1 1 -1 1 x2+y2=1 x y 1 -1 x2-y2=1 y 1 -1 x -3π -5π2 -2π -3π2 -π -π2 π2 π 3π2 2π 5π2 3π y=sinx y 1 -1 x -3π -5π2 -2π -3π2 -π -π2 π2 π 3π2 2π 5π2 3π y=cosx x y 1 y=ex

Shifting

Graph (x-3)2+(y-2)2=1. 37.790960 x 1 2 3 y 1 2 (3,2) 1 Notes:

(a) x2+y2=1 is the basic graph, a circle of radius 1.
(b) Center is shifted by 3 to the right in the x-direction and 2 upwards in the y-direction.

Scaling

Graph 2y=sin3x. y 1 -1 x -π -5π6 -3π3 -π2 -π3 -π6 π6 π3 π2 2π3 5π6 π Notes:

(a) y=sinx is the basic graph.
(b) The x axis is scaled (squished) by 3.
(c) The y axis is scaled (squished) by 2.

Flipping

Graph y=-e-x. x y -1 Notes:

(a) y=ex is the basic graph.
(b) y=-e-x is the same as -y=e-x.
(c) The x-axis is flipped. The y-axis is flipped.

Graph y=sin(1x). x y 1 -1 -1π -12π 12π 1π 2π Notes:

(a) y=sinx is the basic graph.
(b) Positive x axis is flipped in x=1. Negative x axis is flipped in x=1.
(c) As x, sin(1x)0+. As x-, sin(1x)0-.
(d) As x0+, sin(1x) goes between +1 and -1.

Graph y=sin-1x. x -1 1 y 3π2 π π2 -π2 -π -3π2 Notes:

(a) y=sinx is the basic graph.
(b) y=sin-1x is siny=x so x and y are switched from the y=sinx graph.

Example f(x)= { 1-cosxx2, ifx0, 1, ifx=0. y 1 -1 x -3π -5π2 -2π -3π2 -π -π2 π2 π 3π2 2π 5π2 3π y=cosx y 1 -1 x -3π -5π2 -2π -3π2 -π -π2 π2 π 3π2 2π 5π2 3π y=-cosx x y 2 1 -3π -5π2 -2π -3π2 -π -π2 π2 π 3π2 2π 5π2 3π y=1-cosx x y -1 1 1 y=1x2 x y 12 1 y=2x2 y = f(x) = { 1-cosx x2 , x0 , 1 , x=0 , Notes:

(a) As x0 limx0 1-cosxx2 = limx0 1-(1-x22!+x44!-x66!+) x2 = limx0 x22!- x44!+ x66!- x2 = limx0 12!- x24!+ x66!- =12. (if x=.0001 then x2=.0000001 and x4=.0000000000001).
(b) At x=0, f(x)=1.
(c) At the peaks of 1-cosx, 1-cosxx2=2x2.

A function is continuous at x=a if it doesn't jump at x=a.

A function is continuous at x=a if limxaf(x) =f(a).

Notes and References

These are notes from a 2010 course on Real Analysis 620-295. This page comes from 100310Lect5.pdf and was given on 10 March 2010.

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