Post on 18-Dec-2015
Rectangular Coordinate System
Example. Problem: Plot the points (0,7),
({6,0), (6,4) and ({3,{5)Answer:
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-5
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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
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:
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
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
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
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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-10
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-5
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2.5
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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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2.5
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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
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:
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
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
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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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2.5
5
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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
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
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
-7.5 -5 -2.5 2.5 5 7.5
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2
4
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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
-2 -1 1 2
-2
-1.5
-1
-0.5
0.5
1
1.5
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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: