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Graphs Linear Equations

Y

ymxb

X

Example of a Linear Function

Y

77

(11,77)

A Dogs Humans Equivalent Age

yf(x)7x

35

(5,35)

21

(3,21)

X

(0,0)

3 5 11

A Dogs Actual Age

Major Elements of Graphing Lines

Graphing Ordered Pairs Graphing

Equations Linear Equations Slope

Equations Finding Equations of Lines Fitting

Equations to Lines Parallel Perpendicular

Lines

A Point

Y

(X,Y) (4,3)

X

(X,Y) is called an Ordered Pair The X value or X

Coordinate is the location of a point in the X

direction The Y value or Y Coordinate is the

location of a point in the Y direction

How to Graph a Point

Y

(X,Y) (4,2)

321

X

1 2 3 4

-4-3-2-1 0

X is the distance along the xaxis Y is the

distance along the yaxis

Y

3210

HINT Think of the x-axis as the Number Line

X

-1

1 2 3 4

-4-3-2-1 0

-2

HINT Think of the y-axis a vertical Number Line

Important Vocabulary for Graphs

The Graph itself is called the x-y plane (ie.

Plane surface) or The Coordinate Plane or

Cartesian Coordinate Plane after Renee Descartes

Y

y-axis

1st Quadrant

2nd Quadrant

(X,Y)

(-X,Y)

X

x-axis

(X,-Y)

(-X,-Y)

3rd Quadrant

4th Quadrant

Quadrants start with Positive X Y and go

Counter Clockwise

Graphing Linear Equations(Find 3 Domain Range

Points)

First Degree Equations are Lines (ymxb) and you

calculate 3 (X,Y) values Make sure the points

line up on a x-y graph and connect the dots.

RECALL X-Domain Y-Range

Graphing Lines is just like finding the Range of

3 Domain Points(Substitute each Domain value

into the equation)

- y 2x-7 when the Domain is -2, 0, 2
- f(-2) 2(-2) -7 -4 -7 -11 (-2,-11)
- f(0) 2(0) -7 0 -7 -7 (0,-7)
- f(2) 2(2) -7 4 -7 -3 (2,-3)

Answer RANGE -11, -7, -3

Practice Finding 3 PointsGiven a Linear Equation

Find any 3 (X,Y) points for the following

equations y5x y4x-5 y3x1 (Hint Try x0)

Sample Solutions

- x y 5x
- 0 0
- 5
- 10

x y 4x-5

x y 3x1

Now Graph the 3 Points

- x y 5x
- 0 0
- 5
- 10

(2,10)

(1,5)

(0,0)

What is Intercept in Math?

Y

X

An Intercept is the coordinate where a line

crosses the x or y axis

Using XY Intercepts to Graph a Line

Y

y-axis

The Y intercept is the y coordinate (where a line

crosses the y axis).

(0,2)

(3,0)

X

x-axis

The X intercept is the x coordinate (where a line

crosses the x axis).

Name the XY Intercepts

Y

y-axis

(0,2)

(3,0)

X

x-axis

Name the XY Intercepts

Y

y-axis

(0,2)

X

x-axis

(-2,0)

Name the XY Intercepts

Y

y-axis

X

x-axis

Name the XY Intercepts

Y

y-axis

X

x-axis

What is the value of x at the y intercept? What

is the value of Y at the x-intercept?

Y

y-axis

X

x-axis

Graph y 2x - 6 using xy intercepts

Graph Linear Eq.

- X Y 2x - 6

1st Make x-y table

Y

y-axis

X

x-axis

Graph y 2x - 6 using xy intercepts

Graph Linear Eq.

- X Y 2x - 6
- 0 -6

1st Make x-y table 2nd Set x 0 and solve for

y

Y

y-axis

X

x-axis

(0,-6)

Graph y 2x - 6 using xy intercepts

Graph Linear Eq.

- X Y 2x - 6
- 0 -6
- 0

1st Make x-y table 2nd Set x 0 and solve for

y 3rd Set y 0 and solve for x

Y

y-axis

X

(3,0)

x-axis

(0,-6)

Graph y 2x - 6 using xy intercepts

Graph Linear Eq.

- X Y 2x - 6
- 0 -6
- 0

1st Make x-y table 2nd Set x 0 and solve for

y 3rd Set y 0 and solve for x 4th Plot these 2

points and draw line

Y

y-axis

X

(3,0)

x-axis

(0,-6)

Graph y 2x - 6 using xy intercepts

Graph Linear Eq.

- X Y 2x - 6
- 0 -6
- 0
- 4 2

1st Make x-y table 2nd Set x 0 and solve for

y 3rd Set y 0 and solve for x 4th Plot these 2

points and draw line 5th Use 3rd point to check

Y

y-axis

(4,2)

X

(3,0)

x-axis

(0,-6)

Graphing Horizontal Vertical Lines

This line has a y value of 4 for any x-value.

Its equation is y 4 (meaning y always equals

4)

Y

y-axis

X

x-axis

Graphing Horizontal Vertical Lines

This line has a x value of 1 for any y-value.

Its equation is x 1 (meaning x always equals

1)

Y

y-axis

X

x-axis

The Equation of a Vertical Line is XConstant

Y

y-axis

x 1

X

x-axis

The Equation of a Horizontal Line is YConstant

Y

y-axis

y 3

X

x-axis

Graph the following lines

Y -4 Y 2 X 5 X -5 X 0 Y 0

Answers

x -5

x 5

Y

y-axis

X

x-axis

Answers

Y

y-axis

y 2

X

x-axis

y -4

Answers

Y

y-axis

y 0

X

x-axis

x 0

SLOPE

NEGATIVE-DOWN

POSITIVE-UP

Slope is a measure of STEEPNESS

The Symbol forSLOPE m

NEG. Slope is -m

POS. Slope is m

Think of m for Mountain

SLOPE

(6,4)

4

3

(3,2)

2

1

(0,0)

2

1

3

4

5

6

How much does this line rise? How much does

it run?

SLOPE

2

3

(6,4)

4

3

(3,2)

2

1

(0,0)

2

1

3

4

5

6

How much does this line rise? How much does

it run?

2

3

mSLOPE

x2y2

(6,4)

4

x1y1

3

(3,2)

2

1

(0,0)

2

1

3

4

5

6

Switch points and calculate slopeMake (3,2)

(x2,y2) (6,4) (x1,y1)

(x2,y2)(64)

(x1,y1)(6,4)

(x1,y1)(3,2)

(x2,y2)(3,2)

Recalculation with points switched

(x1,y1)(6,4)

(x2,y2)(3,2)

Same slope as before

It doesnt matter what 2 points you choose on a

linethe slope must come out the same

Keeping Track of Signs When Finding The Slope

Between 2 Points

Be Neat Careful Use (PARENTHASES) Double

Check Your Work as you Go Follow 3 Steps

3 Steps for finding the Slope of a line between 2

Points(3,4)(-2,6)

1st Step Write x1,y1,x2,y2 over numbers 2nd

Step Write Formula and Substitute x1,x2,y1,y2

values. 3rd Step Calculate Simplify

x1 y1

x2 y2

(3,4) (-2,6)

Find the Slopes of Lines containing these 2 Points

1. (1,7) (5,2)

2. (3,5) (-2,-8)

3. (-3,-1) (-5,-9)

4. (4,-2) (-5,4)

5. (3,6) (5,-5)

6. (1,-4) (5,9)

ANSWERS

1. (1,7) (5,2)

2. (3,5) (-2,-8)

3. (-3,-1) (-5,-9)

4. (4,-2) (-5,4)

5. (3,6) (5,-5)

6. (1,-4) (5,9)

Solve for y if (9,y) (-6,3) m2/3

Review Finding the Slopes of Lines Given 2 Points

1st Step Write x1,x2,y1,y2 over numbers 2nd

Step Write Formula and Substitute x1,x2,y1,y2

values. 3rd Step Calculate Simplify

NOTE Be Neat, Careful, and Precise and Check

your work as you go..

Negative Slope Is Down the Hill

Positive Slope Is Up the Hill

NO Slope Vertical Drop

ZERO Slope Horizontal

NO Slope Vertical Drop

ZERO Slope Horizontal

Equations of a Line

There are 3 Forms of Line Equations Standard

Form axbyc Slope Intercept

Form ymxb Point-Slope Form y-y1m(x-x1) All

3 describe the line completely but are used for

different purposes. You can convert from one form

to another.

Converting fromStandard Form axbycto Slope

Intercept Form

JUST SOLVE FOR Y

Slope Intercept Form ymxb

Slope Intercept Form ymxb

The great thing about this form is b is the

y-intercept. This makes graphing a line

incredibly easy. Check it out. If

The y intercept is 1

(0,1)

Almost a free point on graph

Slope Intercept Form ymxb

All you have to do now is use the slope to rise

and run from the intercept connect the points.

(0,1)

Rise 2 and Run 3 from the y-intercept connect

points.

ymxb when m is negative

All you have to do now is use the slope to rise

and run from the intercept connect the points.

(0,1)

Rise -2 and Run 3 from the y-intercept connect

points.

Slope Intercept Form ymxbGRAPH THESE LINEAR

EQUATIONS

Label y-intercept Use one big graph

If linear equation is not in ymxb form solve

for y

Solution Steps to Solve for y Divide by 2

Now it is This line has an y intercept of -2 and

rises 5 and runs 2.

Graphing a line with slope intercept equation

Graph

- 1. Solve for y
- Y-Intercept is 1st Point.
- From the y-intercept
- Rise 5 and run 2 for
- Second Point.
- 4. Connect Points with line.

2

5

(0,-2)

Now it is easy to graph

(0,-2)

Put into slope-intercept form and graph

Review Steps of Graphing from the Slope Intercept

Equation

- Make sure equation is in ymxb form
- Plot b(y-intercept) on graph (0,b)
- From b, Rise and Run according to the slope to

plot 2nd point. - Check sign of slope visually

NEG. Slope is -m

POS. Slope is m

Find the Equation of a Line(Given Pt. Slope)

Given a point (2,5) m5 Write the Equation

- Write Slope-Intercept Equation
- 2. Plug-in (x,y) m values
- Solve for b
- Plug m b into Slope-Int. Eq.

Find the Equation of a Line(Given Pt. Slope)

Method 2Using the Pt.-Slope Eq.

Given a point (2,5) m5 Write the Equation

- Write Pt.-Slope Equation
- 2. Plug-in (x,y) m values
- Solve for y

Find the Equation of a Line(Given 2 Points)

Given a point (x1,y1) (x2,y2) (2,5) (3,10)

- Find Slope using
- Write Slope-Intercept Equation
- Plug-in (x,y) m values
- Solve for b
- Plug m b into Slope-Int. Eq.

Parallel LinesHave the Same Slope

5

4

3

2

1

(0,0)

2

1

3

4

5

6

Perpendicular LinesHave Neg. Reciprocal Slopes

3

2

1

(0,0)

2

1

3

4

5

6

Systems of Equations

Given 2 linear equations The single point where

they intersect is a solution to either

equation It is also the solution to both

equations or what we call the solution to the

SYSTEM OF EQUATIONS

(0,3)

3

2

Solution

(2,1)

1

(0,0)

2

1

3

4

5

6

-1

(0,-1)

(0,3)

3

2

Solution

(2,1)

1

(0,0)

2

1

3

4

5

6

-1

(0,-1)

Systems of Equations

The Solution is where the two lines meet (or

intersect)

(0,3)

3

2

Solution

(2,1)

1

(0,0)

2

1

3

4

5

6

-1

(0,-1)

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