The Cartesian Plane and Functions

Calculus Chapter P 1

The Cartesian Plane and Functions

Calculus Chapter P


Real line

• Number line• X-axis

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Coordinate

• The real number corresponding to a point on the real line


Origin

• zero

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Positive direction

• To the right• Shown by arrowhead• Direction of increasing values of x

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Nonnegative

• Positive or zero


Nonpositive

• Negative or zero


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


Rational numbers

• Can be expressed as the ratio of two integers

• Can be represented by either a terminating decimal or a repeating decimal


Irrational numbers

• Not rational• Cannot be represented as terminating or

repeating decimals


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


Properties of inequalities

• Page 2


Set

• A collection of elements


Subset

• Part of a set


Set notation

• The set of all x such that a certain condition is true

• {x : condition on x}• Negative numbers : {x : x < 0}


Union of sets A and B

• The set of elements that are members of A or B or both

A B


Intersections of sets A and B

• The set of elements that are members of A and B

A B


Disjoint sets

• Have no elements in common


Open interval

• Endpoints are not included

, :a b x a x b


Closed Interval

• Endpoints are included

, :a b x a x b


Types of intervals

• See page 3


1. Example

• Exercise 16


2. Example

• Solve and sketch the solution on the real line. 2 7 3x


3. You try

• Solve and sketch the solution on the real line. 4 3 8x


4. Example

• Solve1 1

3x


5. You try

• Solve2 4 5 3x


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


6. Example

2 1 5x x


7. You try

22 1 9 3x x


Absolute value

• See page 6


Absolute value inequalities

• Rewrite as a double inequality


8. Example

9 2 1x


9. You try

3 1 4x


Distance between a and b

,d a b b a a b


Directed distances

• From a to b is b – a • From b to a is a – b


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


Midpoint of an interval

Midpoint of interval ,2

a ba b


To prove

• Show that the midpoint is equidistant from a and b


The Cartesian Plane

Calculus P.2


Cartesian Plane

• Rectangular coordinate system• Named after René Descartes• Ordered pair: (x, y)• Horizontal x-axis• Vertical y-axis• Origin: where axes intersect


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


Distance formula

2 222 1 2 1d x x y y

2 22 1 2 1d x x y y

2 22 1 2 1d x x y y

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(x1, y1)

(x1, y2) (x2, y2) x

y

• Pythagorean theorem

d


1. You try

• Find the distance between (-3, 2) and (3, -2)


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


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


Equation for a circleMath Composer 1. 1. 5ht tp: / / www. mathcomposer. com

(h, k)

(x, y) x

y

2 2r x h y k

2 2 2x h y k r

Standard form


Circles

• If the origin is at (0, 0), then

2 2 2x y r


Unit circle

• Center at origin and radius of 1


General Form

2 2 0Ax Ay 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.


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.


2. Example

2 23 3 6 1 0x y y

• Complete the square


3. You try

2 2 2 6 15 0x y x y • Complete the square


4. You try

2 216 16 16 40 7 0x y x y

• Complete the square


Graphs of Equations

Calculus P.3


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


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


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


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).


Tests for symmetry

• Page 20


2. You try

• Check the following equation for symmetry with respect to both axes and to the origin.

2 1xy

x


Points of Intersection

• Where two graphs cross• Points satisfy both equations• Find by solving equations simultaneously.


3. You try

• Find all points of intersection of the following graphs

2 2 51

x yx y


4. Example

• Exercise 72


Lines in the Plane

Calculus P.4


Slope of a line

• You can subtract in either order, as long as you are consistent

2 1

2 1

1 2

y yymx x x

x x


Point-slope form

1 1y y m x x


Slope-Intercept Form

• y-intercept at (0, b)

y mx b


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 14

y x

3 94 4

y x


Horizontal Line

y b


Vertical Line

x a


General Form

• Works for all equations – even vertical lines

0Ax By C


Parallel lines

• Have the same slope


Perpendicular lines

• Their slopes are negative reciprocals of each other

12

1mm


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 0x y

x y


Functions

Calculus P.5


functions

• For every x value there is exactly one y value.

• x is the independent variable• y is the dependent variable


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


Evaluating functions

• Replace each independent variable in the equation with the value for which you are evaluating the function


1. Example

2 2 2f x x x

evaluate

1f


2. You try

2 2 2f x x x

evaluate12

f

f c

f x x


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


3. Example

• Implied that t ≠ – 1

3 41

tf tt


Range of a function

• Possible y values• Determined from domain and function


4. Example

• Find the domain and range of the function

xg x

x


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


Vertical line test

• If a vertical line crosses the graph more than once, it is not a function


Horizontal line test

• If a horizontal line crosses a function more than once, it is not one-to-one


Six basic functions

• Page 37


Transformations of functions

• Page 38


Polynomial functions

• f(x) is a polynomial• Can use the leading coefficient test to

determine left and right behavior of graph• Page 39


Composites of functions

f g f g x


5. You try

• Find f ○ g and g ○ f

23 5

12

f x x

g x x


Zeros of a functions

• Values of x that make

0f x


Even functions

• Symmetric with respect to y-axis

f x f x


Odd functions

• Symmetric with respect to the origin

f x f x


Review of Trigonometric Functions

Calculus P.6


Angles

• Initial ray – beginning• Terminal ray – end• Vertex – where two rays meet• Standard position – initial ray at + x-axis

and vertex at origin


Coterminal angles

• Same terminal ray• 60° and –300°


Radian measure

• Length of arc of sector subtended by angle on unit circle

• 360° = 2pr• For other circles, s = rq


Evaluating trigonometric functions

• Unless it says to use a calculator or to approximate, you must find the exact answer using the unit circle.


Solving trigonometric equations

• Often there will be more than one possible answer. You must indicate this some how.


1. Example

tan 3q


2. Example

2Solve tan 3 for : 0 2q q q p


Graphs of Trigonometric Functions

• Pages 51 - 52


Examples

• Graph the following: • 3.

• 4.

2sin 2y x

3cosy xp

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The Cartesian Plane and Functions

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