Graphs

Chapter 1

Rectangular Coordinates;Graphing Utilities

Section 1.1

Rectangular Coordinate System

Rectangular Coordinate System

Example. Problem: Plot the points (0,7),

({6,0), (6,4) and ({3,{5)Answer:

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Rectangular Coordinate System

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

Quadrant I x > 0, y > 0

Quadrant II x < 0, y > 0

Quadrant III x < 0, y < 0

Quadrant IVx > 0, y < 0

Distance Formula

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

Distance Formula

Example. Problem: Find the distance

between the points (6,4) and ({3,{5).

Answer:

Midpoint Formula

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

Midpoint Formula

Example. Problem: Find the midpoint of

the line segment between the points (6,4) and ({3,{5)

Answer:

Key Points

Rectangular Coordinate System

Distance FormulaMidpoint Formula

Graphs of Equations in Two Variables

Section 1.2

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:

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 solutionsConnect the pointsMore sophisticated methods

seen later

Graphs of Equations

Example.Problem: Graph the equation y

= 3x{1Answer:

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-5

5

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Graphs of Equations

Example.Problem: Graph the equation y2

= xAnswer:

2 4 6 8 10

-4

-2

2

4

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

-1 1 2 3 4

-2

-1

1

2

Intercepts

Example.Problem: Find all intercepts of

the graphAnswer:

Intercepts

Finding intercepts from an equationTo find the x-intercepts of an

equation, set y=0 and solve for x

To find the y-intercepts of an equation, set x=0 and solve for y

Intercepts

Example.Problem: Find the intercepts of

the equation 4x2 + 25y2 = 100Answer:

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

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)

-1 1 2 3 4

-2

-1

1

2

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

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

-2 -1 1 2

-1

1

2

3

4

Symmetry and Graphs

Origin symmetryReflection across one axis,

then the otherProjection along a line through

origin so that distances from the origin are equal

Rotation of 180± about the origin

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Symmetry and Equations

To test an equation forx-axis symmetry: Replace y by

{yy-axis symmetry: Replace x by

{xorigin symmetry: Replace x by

{x and y by {yIn each case, if an equivalent

equation results, the graph has the appropriate symmetry

Symmetry and Equations

Example.Problem: Test the equation

x2 {4x + y2 { 5 = 0 for symmetry

Answer:

Important Equations

y = x2

x-intercept: x = 0y-intercept: y = 0Symmetry: y-axis only

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Important Equations

x = y2

x-intercept: x = 0y-intercept: y = 0Symmetry: x-axis only

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Important Equations

x-intercept: x = 0y-intercept: y = 0Symmetry: None

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Important Equations

y=x3

x-intercept: x = 0y-intercept: y = 0Symmetry: Origin only

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Important Equations

y =

x-intercept: Noney-intercept: NoneSymmetry: Origin only

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Key Points

Solutions of EquationsGraphs of Equations InterceptsSymmetrySymmetry and GraphsSymmetry and Equations Important Equations

Solving Equations in One Variable Using a Graphing Utility Section 1.3

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:

Use Intersect to Solve Equations

Example.Problem: Find the solutions to

the equation {x4 + 3x3 + 2x2 = {2x + 1. Approximate to two decimal places.

Answer:

Key Points

Using Zero or Root to Approximate Solutions

Use Intersect to Solve Equations

Lines

Section 1.4

Slope of a Line

P = (x1, y1) and Q = (x2,y2) two

distinct pointsP 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.

Slope of a Line

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

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

Slope of a Line

Slope of a Line

Example.

Problem: Compute the slope of

the line containing the points

(7,3) and ({2,{2)

Answer:

Slope of a Line

Move from left to rightLine slants upward if the slope

is positiveLine slants downward if slope is

negativeLine is horizontal if the slope is

0Larger magnitudes

correspond to steeper slopes

Slope of a Line

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m1

m13

m0

m13

m1

m3

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Slope of a Line

Example.Problem: Draw the graph of the

line containing the point (1,5) with a slope of

Solution:

Equations of Lines

Theorem [Equation of a Vertical Line]A vertical line is given by an equation of the form

x = awhere a is the x-intercept

Equations of Lines

Example. Problem: Find an equation of

the vertical line passing through the point ({1, 2)

Answer:

Equations of Lines

Theorem. [Equation of a Horizontal Line]A horizontal line is given by an equation of the form

y = bwhere b is the y-intercept

Equations of Lines

Example. Problem: Find an equation of

the horizontal line passing through the point ({1, 2)

Answer:

Point-Slope Form of a Line

Theorem. [Point-Slope Form of an Equation of a Line]An equation of a nonvertical line of slope m that contains the point (x1, y1) is

y { y1= m(x { x1)

Point-Slope Form of a Line

Example.

Problem: Find an equation of

the line with slope

passing through the point ({1,

2)

Answer:

Point-Slope Form of a Line

Example.

Problem: Find an equation of

the line containing the points

({1, 2) and (5,3).

Answer:

Slope-Intercept Form of a Line

Theorem. [Slope-Intercept Form of an Equation of a Line]An equation of a nonvertical line L with of slope m and y-intercept b

y = mx + b

Slope-Intercept Form of a Line

Example.

Problem: Find the slope-

intercept form of the line in

the graph

Answer:

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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

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.

Parallel Lines

Example.Problem: Find the line passing

through the point (1, {2) which is parallel to the line y = 3x + 2

Answer:

Perpendicular Lines

Perpendicular lines: Two lines that intersect at a right angle

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

Perpendicular Lines

Example.Problem: Find the line passing

through the point (1, {2) which is parallel to the line y = 3x + 2

Answer:

Key Points

Slope of a LineEquations of LinesPoint-Slope Form of a LineSlope-Intercept Form of a

LineGeneral Form of a LineParallel LinesPerpendicular Lines

Circles

Section 1.5

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

Standard Form of a Circle

Standard form of an equation of a circle with radius r and center (h, k) is

(x{h)2 + (y{k)2 = r2

Standard form of an equation centered at the origin with radius r is

x2 + y2 = r2

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Standard Form of a Circle

Example.Problem: Graph the equation (x{2)2 + (y+4)2 = 9Answer:

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 = 0if this equation has a circle for a graph

If given a general form, complete the square to put it in standard form

General Form of a Circle

Example.Problem: Find the center and

radius of the circle with equation

x2 + y2 + 6x { 2y + 6 = 0Answer:

Key Points

CirclesStandard Form of a CircleUnit CircleGeneral Form of a Circle