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Calculus Chapter P 1
The Cartesian Plane and Functions
Calculus Chapter P
Calculus Chapter P 2
Real line
• Number line• X-axis
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Calculus Chapter P 3
Coordinate
• The real number corresponding to a point on the real line
Calculus Chapter P 4
Origin
• zero
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Calculus Chapter P 5
Positive direction
• To the right• Shown by arrowhead• Direction of increasing values of x
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Calculus Chapter P 6
Nonnegative
• Positive or zero
Calculus Chapter P 7
Nonpositive
• Negative or zero
Calculus Chapter P 8
One-to-one correspondence
• Type of relationship• Example: each point on the real line
corresponds to one and only one real number, and each real number corresponds to one and only one point on the real line
Calculus Chapter P 9
Rational numbers
• Can be expressed as the ratio of two integers
• Can be represented by either a terminating decimal or a repeating decimal
Calculus Chapter P 10
Irrational numbers
• Not rational• Cannot be represented as terminating or
repeating decimals
Calculus Chapter P 11
Order and inequalities
• Real numbers can be ordered• If a and b are real numbers, then a is less
than b if b – a is positive• Shown with inequality a < b
Calculus Chapter P 12
Properties of inequalities
• Page 2
Calculus Chapter P 13
Set
• A collection of elements
Calculus Chapter P 14
Subset
• Part of a set
Calculus Chapter P 15
Set notation
• The set of all x such that a certain condition is true
• {x : condition on x}• Negative numbers : {x : x < 0}
Calculus Chapter P 16
Union of sets A and B
• The set of elements that are members of A or B or both
A B
Calculus Chapter P 17
Intersections of sets A and B
• The set of elements that are members of A and B
A B
Calculus Chapter P 18
Disjoint sets
• Have no elements in common
Calculus Chapter P 19
Open interval
• Endpoints are not included
, :a b x a x b
Calculus Chapter P 20
Closed Interval
• Endpoints are included
, :a b x a x b
Calculus Chapter P 21
Types of intervals
• See page 3
Calculus Chapter P 22
1. Example
• Exercise 16
Calculus Chapter P 23
2. Example
• Solve and sketch the solution on the real line. 2 7 3x
Calculus Chapter P 24
3. You try
• Solve and sketch the solution on the real line. 4 3 8x
Calculus Chapter P 25
4. Example
• Solve1 1
3
x
Calculus Chapter P 26
5. You try
• Solve2 4 5 3x
Calculus Chapter P 27
Polynomial inequalities
• Remember that a polynomial can change signs only at its real zeros
• Find zeros, then use them to divide real line into test intervals
• Test one value in each interval to determine if it makes the inequality true or not
Calculus Chapter P 28
6. Example
2 1 5x x
Calculus Chapter P 29
7. You try
22 1 9 3x x
Calculus Chapter P 30
Absolute value
• See page 6
Calculus Chapter P 31
Absolute value inequalities
• Rewrite as a double inequality
Calculus Chapter P 32
8. Example
9 2 1x
Calculus Chapter P 33
9. You try
3 1 4x
Calculus Chapter P 34
Distance between a and b
,d a b b a a b
Calculus Chapter P 35
Directed distances
• From a to b is b – a • From b to a is a – b
Calculus Chapter P 36
10. You try
• Find the distance between –5 and 2
• Find the directed distance from –5 to 2
• Find the directed distance from 2 to –5
Calculus Chapter P 37
Midpoint of an interval
Midpoint of interval ,2
a ba b
Calculus Chapter P 38
To prove
• Show that the midpoint is equidistant from a and b
Calculus Chapter P 39
The Cartesian Plane
Calculus P.2
Calculus Chapter P 40
Cartesian Plane
• Rectangular coordinate system• Named after René Descartes• Ordered pair: (x, y)• Horizontal x-axis• Vertical y-axis• Origin: where axes intersect
Calculus Chapter P 41
QuadrantsMath Composer 1. 1. 5http: / /www. mathcomposer. com
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-5
-4
-3
-2
-1
1
2
3
4
5
x
y
I
IVIII
II
Calculus Chapter P 42
Distance formula
2 222 1 2 1d x x y y
2 2
2 1 2 1d x x y y
2 2
2 1 2 1d x x y y
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(x1, y1)
(x1, y2) (x2, y2) x
y
• Pythagorean theorem
d
Calculus Chapter P 43
1. You try
• Find the distance between (-3, 2) and (3, -2)
Calculus Chapter P 44
Midpoint formula
• To find the midpoint of the line segment joining two points, average the x-coordinates and average the y-coordinates.
• Midpoint has coordinates
1 2 1 2,2 2
x x y y
Calculus Chapter P 45
Circle
• The set of all points in a plane that are equidistant from a fixed point.
• Center: the fixed point• Radius: distance from fixed point to point
on circle
Calculus Chapter P 46
Equation for a circleMath Composer 1. 1. 5http: / / www. mathcomposer. com
(h, k)
(x, y) x
y
2 2r x h y k
2 2 2x h y k r
Standard form
Calculus Chapter P 47
Circles
• If the center is at (0, 0), then
2 2 2x y r
Calculus Chapter P 48
Unit circle
• Center at origin and radius of 1
Calculus Chapter P 49
General Form
2 2 0Ax By Cx Dy F • Obtained from standard form by squaring and
simplifying.• To convert from general form to standard form,
you must complete the square.• If you get a radius of 0, then it is a single point.• If you get a negative radius, then the graph does
not exist.
Calculus Chapter P 50
Completing the square
1. Get coefficients of x2 and y2 to be 1.
2. Get variable terms on one side of the equation and constant terms on the other.
3. Add the square of half the coefficient of x and the square of half the coefficient of y to both sides.
4. Factor and simplify.
Calculus Chapter P 51
2. Example
2 23 3 6 1 0x y y
• Complete the square
Calculus Chapter P 52
3. You try
2 2 2 6 15 0x y x y • Complete the square
Calculus Chapter P 53
4. You try
2 216 16 16 40 7 0x y x y
• Complete the square
Calculus Chapter P 54
Graphs of Equations
Calculus P.3
Calculus Chapter P 55
Sketching a graph
• Solve the equation for y• Construct a table with different x values• Plot the points in the table• Connect with a smooth curve
Calculus Chapter P 56
Using a calculator to graph
• Excellent tool• Make sure your viewing window is
appropriate so you see the whole graph• You may have to solve for y and plot two
equations• 1. Example: 2 29 9x y
Calculus Chapter P 57
Intercepts of a Graph
• Have 0 as one of the coordinates• x-intercepts: y is 0• y-intercepts: x is 0• To find the x-intercepts, let y be zero and
solve for x• To find the y-intercepts, let x be zero and
solve for y
Calculus Chapter P 58
Symmetry of a Graph
• Symmetric with respect to the y-axis if whenever (x, y) is a point on the graph, (-x, y) is also a point on the graph.
• Symmetric with respect to the x-axis if whenever (x, y) is on the graph, so is (x, -y).
• Symmetric with respect to the origin if whenever (x, y) is on the graph, so is (-x, -y).
Calculus Chapter P 59
Tests for symmetry
• Page 20
Calculus Chapter P 60
2. You try
• Check the following equation for symmetry with respect to both axes and to the origin.
2 1
xy
x
Calculus Chapter P 61
Points of Intersection
• Where two graphs cross• Points satisfy both equations• Find by solving equations simultaneously.
Calculus Chapter P 62
3. You try
• Find all points of intersection of the following graphs
2 2 5
1
x y
x y
Calculus Chapter P 63
4. Example
• Exercise 72
Calculus Chapter P 64
Lines in the Plane
Calculus P.4
Calculus Chapter P 65
Slope of a line
• You can subtract in either order, as long as you are consistent
2 1
2 1
1 2
y yym
x x x
x x
Calculus Chapter P 66
Point-slope form
1 1y y m x x
Calculus Chapter P 67
Slope-Intercept Form
• y-intercept at (0, b)
y mx b
Calculus Chapter P 68
1. You try
• A line passes through the point (1, 3) and has a slope of ¾. Write its equation in point-slope form and slope-intercept form.
33 1
4y x
3 9
4 4y x
Calculus Chapter P 69
Horizontal Line
y b
Calculus Chapter P 70
Vertical Line
x a
Calculus Chapter P 71
General Form
• Works for all equations – even vertical lines
0Ax By C
Calculus Chapter P 72
Parallel lines
• Have the same slope
Calculus Chapter P 73
Perpendicular lines
• Their slopes are negative reciprocals of each other
12
1m
m
Calculus Chapter P 74
2. You try
• Write the general form of equations of the lines through the given point and • Parallel to the given line• Perpendicular to the given line
2,1
4 2 3x y 2 3 0
2 4 0
x y
x y
Calculus Chapter P 75
Functions
Calculus P.5
Calculus Chapter P 76
functions
• For every x value there is exactly one y value.
• x is the independent variable• y is the dependent variable
Calculus Chapter P 77
Function notation
• Independent variable is in parentheses• Say “f of x”
2
2
2 4 1
instead of
2 4 1
f x x x
y x x
Calculus Chapter P 78
Evaluating functions
• Replace each independent variable in the equation with the value for which you are evaluating the function
Calculus Chapter P 79
1. Example
2 2 2f x x x
evaluate
1f
Calculus Chapter P 80
2. You try
2 2 2f x x x
evaluate
1
2f
f c
f x x
Calculus Chapter P 81
Domain of a function
• Explicitly defined: they tell you possible values of x using an inequality
• Implicitly defined: implied to be the set of all real numbers for which the equation is defined
Calculus Chapter P 82
3. Example
• Implied that t ≠ – 1
3 4
1
tf t
t
Calculus Chapter P 83
Range of a function
• Possible y values• Determined from domain and function
Calculus Chapter P 84
4. Example
• Find the domain and range of the function
xg x
x
Calculus Chapter P 85
One-to one function
• To each y-value in the range there corresponds exactly one x-value in the domain.
2
3 2 is one-to-one
f is not
f x x
x x
Calculus Chapter P 86
Vertical line test
• If a vertical line crosses the graph more than once, it is not a function
Calculus Chapter P 87
Horizontal line test
• If a horizontal line crosses a function more than once, it is not one-to-one
Calculus Chapter P 88
Six basic functions
• Page 37
Calculus Chapter P 89
Transformations of functions
• Page 38
Calculus Chapter P 90
Polynomial functions
• f(x) is a polynomial• Can use the leading coefficient test to
determine left and right behavior of graph• Page 39
Calculus Chapter P 91
Composites of functions
f g f g x
Calculus Chapter P 92
5. You try
• Find f ○ g and g ○ f
23 5
12
f x x
g x x
Calculus Chapter P 93
Zeros of a functions
• Values of x that make
0f x
Calculus Chapter P 94
Even functions
• Symmetric with respect to y-axis
f x f x
Calculus Chapter P 95
Odd functions
• Symmetric with respect to the origin
f x f x
Calculus Chapter P 96
Review of Trigonometric Functions
Calculus P.6
Calculus Chapter P 97
Angles
• Initial ray – beginning• Terminal ray – end• Vertex – where two rays meet• Standard position – initial ray at + x-axis
and vertex at origin
Calculus Chapter P 98
Coterminal angles
• Same terminal ray• 60° and –300°
Calculus Chapter P 99
Radian measure
• Length of arc of sector subtended by angle on unit circle
• 360° = 2pr• For other circles, s = rq
Calculus Chapter P 100
Evaluating trigonometric functions
• Unless it says to use a calculator or to approximate, you must find the exact answer using the unit circle.
Calculus Chapter P 101
Solving trigonometric equations
• Often there will be more than one possible answer. You must indicate this some how.
Calculus Chapter P 102
1. Example
tan 3
Calculus Chapter P 103
2. Example
2Solve tan 3 for : 0 2
Calculus Chapter P 104
Graphs of Trigonometric Functions
• Pages 51 - 52
Calculus Chapter P 105
Examples
• Graph the following: • 3.
• 4.
2sin 2y x
3cosy x