coordinate geometry and functions. the principal goal of education is to create individuals who are...

57
Coordinate Geometry and Functions

Upload: george-fields

Post on 31-Dec-2015

216 views

Category:

Documents


1 download

TRANSCRIPT

Page 1: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

Coordinate Geometry and Functions

Page 2: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating what other generations have done.

Jean Piaget

Swiss psychologist

Page 3: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

Introduction to the Cartesian Coordinate (x,y) Plane

Before the end of the European Renaissance, math was cleanly divided into the two separate subjects of geometry and algebra. You didn't use algebraic equations in geometry, and you didn't draw any pictures in algebra.

Then, around 1637, a French guy named René Descartes came up with a way to put these two subjects together

Page 4: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

Seating ChartThe following seating chart has 5 rows of desks with 4 desks in each row

Mary sits in the third row at the second desk

Row 1 Row 2 Row 3 Row 4 Row 5

Page 5: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

Who is sitting in desk (4,2)?

A B C D E

F G H I J

K

P

L M N O

Q R S T

4

3

2

11 2 3 4 5

N

Row 4

Seat 2

Page 6: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

Mary sits in the third row at the second desk (3,2) and George sits in the second row at the third desk (2,3). Are these seats the same?

No!! The seats (3,2) and (2,3) are called ordered pairs because the order in which the pair of numbers is written is important!!

Page 7: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

This was a handy way to reference individuals in the room, telling you how far over (row 2) and how far up (seat 4) you needed to go. Descartes did something similar for mathematics

Coordinate Plane

Page 8: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

Descartes' breakthrough was in taking a second number line, standing it up on its end, and crossing the first number line at zero:

Coordinate Plane

Page 9: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

Definitionx axis – a horizontal number line on a

coordinate grid.

1 2 3 4 50 6x

Page 10: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

Definitiony axis – a vertical number line on a

coordinate grid.

12345

0

6

y

Page 11: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

Coordinates – an ordered pair of numbers that give the location of a point on a grid. (3, 4)

12345

0

6

1 2 3 4 50 6

(3,4)

Page 12: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

Hint The first number is always the x or first letter in

the alphabet. The second number is always the y the second letter in the alphabet.

1

32

45

0

6

1 2 3 4 50 6

(3,4)

Page 13: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

How to Plot Ordered PairsStep 1 – Always find the x value first, moving horizontally either right (positive) or left (negative).

1

32

45

0

6

1 2 3 4 50 6

(2, 3)

y

x

Page 14: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

How to Plot Ordered PairsStep 2 – Starting from your new position find the y value by moving vertically either up (positive) or down (negative).

1

32

45

0

6

1 2 3 4 50 6

(2, 3)

(2,3)y

x

Page 15: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

What is the ordered pair?

1

32

45

0

6

1 2 3 4 50 6

(3,5)

y

x

Page 16: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

The x-axis and y-axis separate the coordinate plane into four regions, called quadrants.

II(-, +)

I(+, +)

IV(+, -)

III(-, -)

Page 17: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

How to Plot in 4 QuadrantsStep 1 - Plot the x number first moving to the left when the number is negative.

-2

0-1

12

-3

3

-2 -1 0 1 2-3 3

(-3, -2)(-3, -2)y

x

Page 18: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

How to Plot in 4 QuadrantsStep 2 - Plot the y number moving from your new position down 2 when the number is negative.

-2

0-1

12

-3

3

-2 -1 0 1 2-3 3

(-3, -2)(-3, -2)y

x

Page 19: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

Write the ordered pairs that name points A, B, C, and D.

A = (1, 3)B = (3, -2)C = (0, -4)D = (-6, -1)

5

5

-5

-5

• A

• B• C

• D

Page 20: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating
Page 21: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

Linear Equation: an equation whose graph forms a line in an x,y coordinate plane

x + y = 1

3x – 4y = 2

In linear equations, all variables are taken to the first power.

Page 22: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

Suppose you were asked to graph the solution set of the linear equation,

You could start by creating a table of values.

x + y = 3

Page 23: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

Choose any number to use as x or y

x + y = 3

x

0

3

-3

y

3

0

6

Substitute into equation and solve

0 + y = 3

y = 3

3 + y = 3

y = 0

-3 + y = 3

y = 6

Page 24: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

Plot the points and graph

x

0

3

-3

y

3

0

6

x + y = 3

(0,3)

(3,0)

(-3,6)

connect the points with a straight line

Page 25: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

Let’s graph another line:

3x – y = 2

x

-1

0

2

y

-5

-2

4

Substitute into equation and solve

3(-1) – y = 2

y = -5

3(0) – y = 2

y = -2

3(2) – y = 2

y = 4

Page 26: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

Plot the points and graph

x

-1

0

2

y

-5

-2

4

3x – y = 2

(0,-2)

(2,4)

(-1,-5)

connect the points with a straight line

Page 27: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

Graphing Using InterceptsThe x-intercept of a line is the x-coordinate of the point where the line intercepts the x-axis.

-8 -6 -4 -2

2

42 6 8

4

6

-4

-6

-8

-2

8The line shown intercepts the x-axis at (2,0).

We say that the x-intercept is 2.

Page 28: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

Graphing Using InterceptsThe y-intercept of a line is the y-coordinate of the point where the line intercepts the y-axis.

-8 -6 -4 -2

2

42 6 8

4

6

-4

-6

-8

-2

8The line shown intercepts the y-axis at (0,-6).

We say that the y-intercept is -6.

Page 29: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

Graph using intercepts.

x y-40

4x – 3y = 12

*To find the y-intercept, let x = 0.

4(0) – 3y = 12

0 – 3y = 12

-3y = 12-3 -3

y = -4

4x – 3y = 12

Page 30: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

x y-4

0

0

3

4x – 3y = 12

*To find the x-intercept, let y = 0.

4x – 3(0) = 12

4x - 0 = 12

4x = 124 4

x = 3

Graph using intercepts. 4x – 3y = 12

Page 31: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

-8 -6 -4 -2

2

42 6 8

4

6

-4

-6

-8

-2

8

x y-4

0

0

3

Graph using intercepts. 4x – 3y = 12

Page 32: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

Slope

Graphs of any line can easily be classified by its y-intercept and by its steepness.

The steepness is often referred to as slope

Page 33: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

Types of Slope

PositiveNegative

Zero

Undefinedor

No Slope

Page 34: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

Slope

To walk up the steps, the man must go up one step and over one step

1

1

1

1

To walk up the steps, the man must go up two steps and over one step

To walk up the steps, the man must go up one step and over two steps

Page 35: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

-4 -2

2

42 6 8 10

4

6

-4

-6

-8

-2

-10

10

8

rise

runSlope =riserun

Slope

Page 36: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

Graph the line containing points (2,1) and (7,6) and find the slope.

-4 -2

2

42 6 8 10

4

6

-4

-6

-8

-2

-10

10

8

rise

runSlope =riserun

55

1

Find the slope

Page 37: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

Slope =riserun

change in y coordinateschange in x coordinates

2 1

2 1

y ym

x x

1 2

1 2

y ym

x x

or

Slope

Page 38: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

Find the slope of the line containing points (1,6) and (5,4).

2 1

2 1

y ym

x x

4 6m

5 1

2m

4

1m

2

Page 39: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

Slope of a Horizontal Line

-4 -2

2

42 6 8 10

4

6

-4

-6

-8

-2

-10

10

8What is the rise?

What is the run?

0

6

Slope =riserun

06

0

The slope of ANY horizontal line is 0.

Page 40: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

Slope of a Vertical Line

-4 -2

2

42 6 8 10

4

6

-4

-6

-8

-2

-10

10

8What is the rise?

What is the run?

5

0

Slope =riserun

50

The slope of ANY vertical line is undefined.

Page 41: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

If given an equation of a line, there is a nice way to find the slope and y-intercept. Simply write the equation in slope-intercept form, which is:

y = mx + b slope

y-intercept

Page 42: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

Find the slope andy-intercept of the

following equations.

y = 3x + ½

slope= 3

y-intercept = ½

Page 43: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

3x + 5y = 10

First, solve the equation for y.

5y = -3x + 10

y = -3/5 x + 2

m= -3/5 b = 2

Find the slope and y-intercept:

Page 44: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

To find the equation, consider slope intercept form: y = mx + b

• Slope• If given, just substitute it into the

equation• Otherwise calculate the slope

• y-intercept• If given, just substitute it into the

equation• Otherwise use a point and

substitute into equation for x and for y, then solve for b

Find the equation of a line:

Page 45: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

Always find slope first!

Find the slope using the formula.m = 2 – 6 4 – 3 m = -4

Find the equation of a line containing the following points:

(4,2) and (3,6)

y = -4x + b

m = ?b = ?

Page 46: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

(4, 2) and (3, 6) m = -4

Find the y-intercept.y = mx + b2 = -4 H 4 + b2 = -16 + b18 = b

Pick a point to plug into the equation

(4,2) → x = 4y = 2

Write equation:y = -4x + 18

Page 47: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

Relations A relation is a mapping, or pairing, of

input values with output values.

The set of input values is called the domain (x values)

The set of output values is called the range (y-values)

Page 48: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

Functions

A relation is a function provided there is exactly one output for each input.

It is NOT a function if at least one input has more than one output

Page 49: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

INPUT

(DOMAIN)

OUTPUT (RANGE)

FUNCTIONMACHINE

In order for a relationship to be a function…

EVERY INPUT MUST HAVE AN OUTPUT

TWO DIFFERENT INPUTS CAN HAVE THE SAME OUTPUT

ONE INPUT MUST HAVE ONLY ONE OUTPUT

Page 50: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

No two ordered pairs can have the same first coordinate

(and different second coordinates).

Which of the following relations are functions?

R= {(9,10, (-5, -2), (2, -1), (3, -9)}

S= {(6, a), (8, f), (6, b), (-2, p)}

T= {(z, 7), (y, -5), (r, 7) (z, 0), (k, 0)}

Page 51: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

Identify the Domain and Range. Then tell if the relation is a function.

Input Output

-3 3

1 1

3 -2

4

Domain = {-3, 1,3,4}Range = {3,1,-2}

Function?Yes: each input is mapped onto exactly one output

Page 52: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

Input Output

-3 3

1 -2

4 1

4

Identify the Domain and Range. Then tell if the relation is a function.

Domain = {-3, 1,4}Range = {3,-2,1,4}

Function?No: input 1 is mapped onto both -2 & 1

Page 53: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

The Vertical Line Test

If it is possible for a vertical line to intersect a graph at more than one point, then the graph is NOT the graph of a function.

Page 117

Page 54: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

Which are functions?

Function FunctionNot a Function

Page 55: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

“f of x”

Input = x

Output = f(x) = y

Function Notationf (x)

Page 56: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

y = 6 – 3x

-2

-1

0

1

2

12

9

6

0

3

x y

f(x) = 6 – 3x

-2

-1

0

1

2

12

9

6

0

3

x f(x)

Before… Now…

(x, y) (x, f(x))

Page 57: Coordinate Geometry and Functions. The principal goal of education is to create individuals who are capable of doing new things, not simply of repeating

f(x) = 2x2 – 3

Find f(0), f(3)

Evaluate the function:

f(0) = 2(0)2 – 3

= 0 – 3

= - 3

f(3) = 2(3)2 – 3

= 18 – 3

= 15