Graphs

1
Graphs

• Chapter 1

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Rectangular CoordinatesGraphing Utilities

• Section 1.1

3
Rectangular Coordinate System
4
Rectangular Coordinate System

• Example.
• Problem Plot the points (0,7), (6,0), (6,4) and
  (3,5)
• Answer

5
Rectangular Coordinate System

• The points on the axes are not considered to be
  in any quadrant

Quadrant I x gt 0, y gt 0
Quadrant II x lt 0, y gt 0
Quadrant III x lt 0, y lt 0
Quadrant IVx gt 0, y lt 0
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Distance Formula

• Theorem Distance Formula The distance between
  two points P1 (x1, y1) and P2 (x2, y2),
  denoted by d(P1, P2), is

7
Distance Formula

• Example.
• Problem Find the distance between the points
  (6,4) and (3,5).
• Answer

8
Midpoint Formula

• Theorem Midpoint Formula The midpoint M (x,y)
  of the line segment from P1 (x1, y1) to P2
  (x2, y2) is

9
Midpoint Formula

• Example.
• Problem Find the midpoint of the line segment
  between the points (6,4) and (3,5)
• Answer

10
Key Points

• Rectangular Coordinate System
• Distance Formula
• Midpoint Formula

11
Graphs of Equations in Two Variables

• Section 1.2

12
Solutions of Equations

• Solutions of an equation Points that make the
  equation true when we substitute the appropriate
  numbers for x and y
• Example.
• Problem Do either of the points (3,10) or
  (2,4) satisfy the equation y 3x 1?
• Answer

13
Graphs of Equations

• Graph of an equation Set of points in plane
  whose coordinates (x, y) satisfy the equation
• To plot a graph
• List some solutions
• Connect the points
• More sophisticated methods seen later

14
Graphs of Equations

• Example.
• Problem Graph the equation y 3x1
• Answer

15
Graphs of Equations

• Example.
• Problem Graph the equation y2 x
• Answer

16
Intercepts

• Intercepts Points where a graph crosses or
  touches the axes, if any
• x-intercepts x-coordinates of intercepts
• y-intercepts y-coordinates of intercepts
• May be any number of x- or y-intercepts

17
Intercepts

• Example.
• Problem Find all intercepts of the graph
• Answer

18
Intercepts

• Finding intercepts from an equation
• To find the x-intercepts of an equation, set y0
  and solve for x
• To find the y-intercepts of an equation, set x0
  and solve for y

19
Intercepts

• Example.
• Problem Find the intercepts of the equation 4x2
  25y2 100
• Answer

20
Symmetry

• Symmetry with respect to the x-axis If (x,y) is
  on the graph, then so is (x, y)
• Symmetry with respect to the y-axis If (x,y) is
  on the graph, then so is (x, y)
• Symmetry with respect to the origin If (x,y) is
  on the graph, then so is (x, y)

21
Symmetry and Graphs

• x-axis symmetry means that the portion of the
  graph below the x-axis is a reflection of the
  portion above it

22
Symmetry and Graphs

• y-axis symmetry means that the portion of the
  graph to the left of the y-axis is a reflection
  of the portion to the right of it

23
Symmetry and Graphs

• Origin symmetry
• Reflection across one axis, then the other
• Projection along a line through origin so that
  distances from the origin are equal
• Rotation of 180 about the origin

24
Symmetry and Equations

• To test an equation for
• x-axis symmetry Replace y by y
• y-axis symmetry Replace x by x
• origin symmetry Replace x by x and y by y
• In each case, if an equivalent equation results,
  the graph has the appropriate symmetry

25
Symmetry and Equations

• Example.
• Problem Test the equation x2 4x y2 5 0
  for symmetry
• Answer

26
Important Equations

• y x2
• x-intercept x 0
• y-intercept y 0
• Symmetry y-axis only

27
Important Equations

• x y2
• x-intercept x 0
• y-intercept y 0
• Symmetry x-axis only

28
Important Equations

• x-intercept x 0
• y-intercept y 0
• Symmetry None

29
Important Equations

• yx3
• x-intercept x 0
• y-intercept y 0
• Symmetry Origin only

30
Important Equations

• y
• x-intercept None
• y-intercept None
• Symmetry Origin only

31
Key Points

• Solutions of Equations
• Graphs of Equations
• Intercepts
• Symmetry
• Symmetry and Graphs
• Symmetry and Equations
• Important Equations

32
Solving Equations in One Variable Using a
Graphing Utility

• Section 1.3

33
Using Zero or Root to Approximate Solutions

• Example.
• Problem Find the solutions to the equation x3
  6x 3 0. Approximate to two decimal places.
• Answer

34
Use Intersect to Solve Equations

• Example.
• Problem Find the solutions to the equation x4
  3x3 2x2 2x 1. Approximate to two decimal
  places.
• Answer

35
Key Points

• Using Zero or Root to Approximate Solutions
• Use Intersect to Solve Equations

36
Lines

• Section 1.4

37
Slope of a Line

• P (x1, y1) and Q (x2,y2) two distinct points
• P and Q define a unique line L
• If x1 ? x2, L is nonvertical. Its slope is
  defined as
• x1 x2, L is vertical. Slope is undefined.

38
Slope of a Line
39
Slope of a Line

• Interpretation of the slope of a nonvertical line
• Average rate of change of y with respect to x, as
  x changes from x1 to x2

40
Slope of a Line

• Any two distinct points serve to compute the
  slope
• The slope from P to Q is the same as the slope
  from Q to P

41
Slope of a Line

• Example.
• Problem Compute the slope of the line containing
  the points (7,3) and (2,2)
• Answer

42
Slope of a Line

• Move from left to right
• Line slants upward if the slope is positive
• Line slants downward if slope is negative
• Line is horizontal if the slope is 0
• Larger magnitudes correspond to steeper slopes

43
Slope of a Line
44
Slope of a Line

• Example.
• Problem Draw the graph of the line containing
  the point (1,5) with a slope of
• Solution

45
Equations of Lines

• Theorem Equation of a Vertical LineA vertical
  line is given by an equation of the form
• x a
• where a is the x-intercept

46
Equations of Lines

• Example.
• Problem Find an equation of the vertical line
  passing through the point (1, 2)
• Answer

47
Equations of Lines

• Theorem. Equation of a Horizontal LineA
  horizontal line is given by an equation of the
  form
• y b
• where b is the y-intercept

48
Equations of Lines

• Example.
• Problem Find an equation of the horizontal line
  passing through the point (1, 2)
• Answer

49
Point-Slope Form of a Line

• Theorem. Point-Slope Form of an Equation of a
  LineAn equation of a nonvertical line of slope
  m that contains the point (x1, y1) is
• y y1 m(x x1)

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Point-Slope Form of a Line

• Example.
• Problem Find an equation of the line with slope
  passing through the point (1, 2)
• Answer

51
Point-Slope Form of a Line

• Example.
• Problem Find an equation of the line containing
  the points (1, 2) and (5,3).
• Answer

52
Slope-Intercept Form of a Line

• Theorem. Slope-Intercept Form of an Equation of
  a LineAn equation of a nonvertical line L with
  of slope m and y-intercept b
• y mx b

53
Slope-Intercept Form of a Line

• Example.
• Problem Find the slope-intercept form of the
  line in the graph
• Answer

54
General Form of a Line

• General form of a line L
• Ax By C
• A, B and C are real numbers, A and B not both
  0.
• Any line, vertical or nonvertical, may be
  expressed in general form
• The general form is not unique
• Any equation which is equivalent to the general
  form of a line is called a linear equation

55
Parallel Lines

• Parallel Lines Two lines which do not intersect
• Theorem. Criterion for Parallel Lines Two
  nonvertical lines are parallel if and only if
  their slopes are equal and they have different
  y-intercepts.

56
Parallel Lines

• Example.
• Problem Find the line passing through the point
  (1, 2) which is parallel to the line y 3x 2
• Answer

57
Perpendicular Lines

• Perpendicular lines Two lines that intersect at
  a right angle

58
Perpendicular Lines

• Theorem. Criterion for Perpendicular Lines Two
  nonvertical lines are perpendicular if and only
  if the product of their slopes is 1.
• The slopes of perpendicular lines are negative
  reciprocals of each other

59
Perpendicular Lines

• Example.
• Problem Find the line passing through the point
  (1, 2) which is parallel to the line y 3x 2
• Answer

60
Key Points

• Slope of a Line
• Equations of Lines
• Point-Slope Form of a Line
• Slope-Intercept Form of a Line
• General Form of a Line
• Parallel Lines
• Perpendicular Lines

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Circles

• Section 1.5

62
Circles

• Circle Set of points in xy-plane that are a
  fixed distance r from a fixed point (h,k)
• r is the radius
• (h,k) is the center of the circle

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Standard Form of a Circle

• Standard form of an equation of a circle with
  radius r and center (h, k) is
• (xh)2 (yk)2 r2
• Standard form of an equation centered at the
  origin with radius r is
• x2 y2 r2

64
Standard Form of a Circle

• Example.
• Problem Graph the equation
• (x2)2 (y4)2 9
• Answer

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Unit Circle

• Unit Circle Radius r 1 centered at the origin
• Has equation x2 y2 1

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General Form of a Circle

• General form of the equation of a circle
• x2 y2 ax by c 0
• if this equation has a circle for a graph
• If given a general form, complete the square to
  put it in standard form

67
General Form of a Circle

• Example.
• Problem Find the center and radius of the circle
  with equation
• x2 y2 6x 2y 6 0
• Answer

68
Key Points

• Circles
• Standard Form of a Circle
• Unit Circle
• General Form of a Circle