Chapter 4Chapter 4

Danica Reardon,Danica Reardon,

Caroline Song, andCaroline Song, and

Stephanie KnillStephanie Knill

4.1- Using an Equation 4.1- Using an Equation to Draw a Graphto Draw a Graph

• You can use the equation y=mx+b to draw graphs

• m is meaning the slope of the line• b is the y-intercept

y-intercept

slope

4.1 – Part One

• the slope is 2• y-intercept is 5• Start by locating the y-intercept and

marking it• Since slope = rise over run, and 2 =

2/1, count 2 spaces up and one to the right

• Draw the line it creates

4.1 – Part Two

• There are other ways of graphing the line

• One way is to take thecoordinates and makea table

• From this, you can alsoget the equation

x y

-3 -1

-2 1

-1 3

0 5

1 7

2 9

3 11

4.1 – Part Three• Another way is by

mapping• Draw 2 ovals• In one put the x

coordinates• In the other the y

coordinates• Draw arrows to

connect them

-3-2-1 0 1 2 3

-11357911

x y

Using A Graphing Calculator

Drawing Graphs

• Drawing graphs on these calculators are simple

• Click the button and type in your slope and y-intercept

Make sure the plot buttons are not shaded in like below. If yes, go to it and press

Drawing Graphs Cont’d

• Press the button• You should get something like

Stat Plots• Can also use your tables• Press then click • In put your x coordinates• In put your y-coordinates

1L

2L

Stat Plots Part 2

• Press and click on

• Now press graph, you should get something like that

4.2- The Slope of a Line4.2- The Slope of a Line

• Constant Slope Property:• The slopes of all line segments of a

line are equal

As you can tell, the slopes of all of these lines is ½. Therefore we can conclude that every segment on this line has a slope of ½, thus proving the statement we first made

ProofProofCalculate the slopes of

eachline segment:

• Slope of AB

• Slope of BC

• Slope of CDEquation: x-2y+4=0

1

1

1

2

2

2A(-4,0)

B(-2,1)

C(0,2)

D(2,3)

2

1

)4(2

01

2

1

)2(0

12

2

102

23

Relationship Between Slopes of Parallel Line

Segments• If the slopes of two lines are

equal, the lines are parallel• Therefore, if two non-

vertical lines are parallel, their slopes are equal

• Line AB:

• Line CD: C(0,-2)

B(0,2)

D(2,0)A(-

2,0)

Relationship Between Slopes of Perpendicular

Line Segments

• If the slopes of two lines are negative reciprocals, the lines are perpendicular

• Therefore, if two lines are perpendicular (and neither are vertical), their slopes are negative reciprocals

Relationship Between Slopes of Perpendicular

Line Segments Proof:• Line AB:

• Since parallel lines are equal, line CD also has a slope of 1

• Line AC:

• Since parallel lines are equal, line BD also has a slope of –1

• As you can see slope of AB/CD is 1 and the negative reciprocal of this is -1 which is the slope of AC/BD; This proves the statement we first made

C(0,-2)

B(0,2)

D(2,0)

A(-2,0)

4.3 The Equation of a Line: Part 1

Graphing an Equation: y=3x+2

Method 1:1. Draw table of

values2. Plot points on

grid3. Join points with a

straight line

x y

-2 -4

-1 -1

0 2

1 5

A(-2,-4)

D(1,5)

C(0,2)

B(-1,-1) By Increasing x by 1

It increases y by 3

Graphing an Equation: y=3x+2Method 2:

• There is also another method for graphing linear equations that does not need a table of values

• This method is based on two different numbers in the equation y=mx+b:

The Slope

•This is the coefficient of x• For y=3x+2, the slope is 3

• The equation y=mx+b is a straight line where m represents slope and b represents y-intercept

The y-intercept• This is the value of b, and the

value of y when x=o• For y=3x+2, the slope is 3

4.4- The Equation of a Line: 4.4- The Equation of a Line: Part 2Part 2

• In the last section, we got the equation of a line when it’s y-intercept and slope is known

• There are 4 cases you should be able to solve

4.4 – Case 1

• Let’s find the equation of a line with the coordinates(1,2) and (3,6)

4.4 – Case 1

• first find the slope

22

4

31

62

21

21

xx

yym

4.4 – Case 1

• Remember the formula• Plugging in your first coordinate, you

get the formula

bmxy

b )1(22y axis

x axis

slope

Unknown y-intercept

4.4 – Case 1

• Now, solve for b

• Go back to the formula

1b

bmxy

4.4 – Case 1

• This time, do not enter anything for y or x

• Instead, insert your slope and y-intercept

• You get the equation

12 xy

4.4 – Case 2

• Now, let’s find the equation of a line with the slope of –1 and a y-intercept of 4

4.4 – Case 2

• You have all the information necessary to make the equation

• So,

bmxy

4 xy

4.4 – Case 3

• Solving equations with 1 coordinate and the slope

• (1,5) and a slope of 2

1

4.4 – Case 3

• Put it into the equation like you did in Case 1 and solve for b

• So

b )1(2

15

b2

9

2

9

2

1 xy

4.4 – Case 4

• Slope of 5• x-intercept of 3

4.4 – Case 4

• At x-intercept y=0• Same as Case 2• Solve it the same

4.4 – Case 4

b

b

b

15

150

)3(50

155 xy

4.5- Interpreting the Equation 4.5- Interpreting the Equation

Ax+By+C=0Ax+By+C=0

• Determining the intercepts 1Determining the intercepts 1

-To find the y-intercept, substitute x for 0.

ex. 2x+4y-8=0 2(0)+4y-8=0

4y=8 y=4

4.5- Interpreting the Equation 4.5- Interpreting the Equation

Ax+By+C=0Ax+By+C=0

• The Standard Form:The Standard Form:

Ax+By+C=0Ax+By+C=0

• The Slope-Intercept Form: The Slope-Intercept Form: y=mx+b

Slope

y-intercept

4.5- Interpreting the Equation 4.5- Interpreting the Equation

Ax+By+C=0Ax+By+C=0

• Determining the intercepts 2Determining the intercepts 2

-To find the x-intercept, substitute y for 0.

ex. 2x+4y-8=0 2x+4(0)-8=0

2x=8 x=4

4.5- Interpreting the Equation 4.5- Interpreting the Equation

Ax+By+C=0Ax+By+C=0• Solving the equation for ySolving the equation for y

-In order to find the slope and y-intercept, you need to convert the

standard form to the slope-intercept form.

ex. 2x+4y-8=0 4y-8=-2x

4y=-2x+8 y=-0.5x+2

y-intercept

Slope

4.5- Interpreting the Equation 4.5- Interpreting the Equation

Ax+By+C=0Ax+By+C=0• Formulas for the slope, y-intercept, Formulas for the slope, y-intercept,

and x-interceptand x-interceptAx+By+C=0 Therefore: By+C=-Ax *Slope: -A/B By=-Ax-C *y-intercept:

-C/B y=-A/Bx-C/B *x-intercept: -C/A

This way, you just need to substitute the A, B and C values to find the slope, y-intercept and x-intercept.

BibliographyInternet Sources:

• http://en.wikipedia.org/wiki/Y%3Dmx%2Bb “Linear Equation” Wikipedia, 2007

• www.tea.state.tx.us/.../images/47graphicaa.gif “Grade 10 Math Online Test” Texas Education Agency, 2006

Book Sources:• Alexander, Robert & Kelly, Brendan.

Mathematics 10. Addison Welsey Longman Ltd. 1998

For all Math 10H For all Math 10H students…students…

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