y x equations of lines y x. at the end of this lesson you will be able to: write equations for...

Y

X

Equations of LinesEquations of Lines

Y

X

At the end of this lessonAt the end of this lessonyou will be able to:you will be able to:

Write equations for non-vertical lines.Write equations for non-vertical lines.

Write equations for horizontal lines.Write equations for horizontal lines.

Write equations for vertical lines.Write equations for vertical lines.

Use various forms of linear equations.Use various forms of linear equations.

Calculate the slope of a line passing Calculate the slope of a line passing through two points.through two points.

Y

X

Before we begin.Before we begin.

Let’s review some vocabulary.Let’s review some vocabulary.

Y

X

Slope (Slope (mm) =) = Vertical change (YY))

Y-interceptY-intercept ( (bb): The ): The yy-coordinate of the point where the -coordinate of the point where the graph of a line crosses the graph of a line crosses the yy-axis.-axis.

Slope Slope ((mm): The measure of the steepness of a ): The measure of the steepness of a lineline; it is the ratio ; it is the ratio of vertical change (of vertical change (YY) to horizontal change () to horizontal change (XX).).

Horizontal change (XX))

X-interceptX-intercept ( (aa): The ): The xx-coordinate of the point where the graph -coordinate of the point where the graph of a line crosses the of a line crosses the xx-axis.-axis.

Equations ofEquations ofNon-vertical Lines.Non-vertical Lines.

Let’s look at a line with a Let’s look at a line with a y-intercepty-intercept of of b, b, a a slopeslope m m and letand let (x,y) be any point (x,y) be any point on the line.on the line.

Y

X

Y-axis

X-axis

(0,b)

(x,y)

Slope Intercept FormSlope Intercept FormThe equation for the non-vertical line is:The equation for the non-vertical line is:

Y

X

Y-axis

X-axis

(0,b)

(x,y)

YY

XX

y = y = mmx + x + b b ( Slope Intercept Form )

Where Where mm is: is:

m =Y

X=

(y – b)

(x – 0)

More Equations ofMore Equations ofNon-vertical Lines.Non-vertical Lines.

Let’s look at a line passing through Let’s look at a line passing through Point 1 Point 1 (x(x11,y,y11)) and Point 2 and Point 2 (x(x22,y,y22).).

Y

X

Y-axis

X-axis

(x1,y1)

(x2,y2)

Point Slope FormPoint Slope FormThe equation for the non-vertical line is:The equation for the non-vertical line is:

Y

X

Y-axis

X-axis

YY

XX

y – yy – y11 = = mm(x – x(x – x11) ) ( Point Slope Form )

Where Where mm is: is:

m =Y

X=

(y2 – y1)

(x2 – x1)(x1,y1)

(x2,y2)

Equations ofEquations ofHorizontal Lines.Horizontal Lines.

Let’s look at a line Let’s look at a line with a with a y-intercepty-intercept of of b, b, a a slopeslope m = 0m = 0,, andand letlet (x,(x,bb) be any point on ) be any point on the Horizontal line.the Horizontal line.

Y

X

Y-axis

X-axis

(0,b) (x,b)

Horizontal LineHorizontal LineThe equation for the horizontal line is stillThe equation for the horizontal line is still

Y

X

Y-axis

X-axis

y = y = mmx + x + b b ( Slope Intercept Form ).

Where Where mm is: is:

m =Y

X=

(b – b)

(x – 0)

Y = 0Y = 0XX

(0,b)(x,b)

= 0

Horizontal LineHorizontal Line

Because the value of Because the value of mm is 0, is 0,

Y

X

y = y = mmx + x + b b becomes becomes

y = b y = b (A Constant Function)(A Constant Function)

Y-axis

X-axis

(0,b)(x,b)

Equations ofEquations ofVertical Lines.Vertical Lines.

Let’s look at a line with Let’s look at a line with no no y-intercepty-intercept bb, an , an x-x-interceptintercept aa, , an undefined an undefined slopeslope mm,, andand letlet ((aa,y) be ,y) be any point on the vertical any point on the vertical line.line.

Y

X

Y-axis

X-axis(a,0)

(a,y)

Vertical LineVertical LineThe equation for the vertical line isThe equation for the vertical line is

Y

X

Y-axis

X-axis

x = x = a a ( a is the X-Intercept of the line).

Because Because mm is: is:

m =Y

X=

(y – 0)

(a – a)= Undefined

(a,0)

(a,y)

Vertical LineVertical LineBecause the value of Because the value of mm is undefined, caused by the is undefined, caused by the division by zero, there is no slope division by zero, there is no slope mm..

Y

X

x = x = a a becomes the equation becomes the equation

x = ax = a (The equation of a vertical line)(The equation of a vertical line)

Y-axis

X-axis(a,0)

(a,y)

Example 1: Slope Intercept FormExample 1: Slope Intercept FormFind the equation for the line Find the equation for the line with with m = 2/3m = 2/3 and and b = 3b = 3

Y

X

Y-axis

X-axis

Because Because b = 3b = 3

Y = 2

X = 3

(0,3)X = 3X = 3

The line will pass through (0,3)

Because Because m = 2/3m = 2/3

The Equation for the line is:The Equation for the line is:

y = y = 2/32/3 x + x + 33

Y = 2Y = 2

Slope Intercept Form PracticeSlope Intercept Form PracticeWrite the equation for the lines using Write the equation for the lines using Slope Intercept Slope Intercept form.form.

Y

X

1.) m = 3 & b = 3

2.) m = 1 & b = -4

3.) m = -4 & b = 7

4.) m = 2 & b = 0

5.) m = 1/4 & b = -2

Example 2: Point Slope FormExample 2: Point Slope FormLet’s find the equation for the line passing through Let’s find the equation for the line passing through the points the points (3,-2)(3,-2) and and (6,10)(6,10)

Y

X

Y-axis

X-axis

YY

XX

First, Calculate First, Calculate mm : :

m =Y

X=

(10 – -2)

(6 – 3)

(3,-2)

(6,10)

33

1212== == 44

Example 2: Point Slope FormExample 2: Point Slope FormTo find the equation for the line passing through To find the equation for the line passing through the points the points (3,-2)(3,-2) and and (6,10)(6,10)

Y

X

Y-axis

X-axis

YY

XX

y – yy – y11 = = mm(x – x(x – x11))

Next plug it into Point Slope From :Next plug it into Point Slope From :

(3,-2)

(6,10)

y – y – -2-2 = = 44(x – (x – 33))

Select one point as PSelect one point as P11 : :Let’s use Let’s use (3,-2)(3,-2)

The Equation becomes:The Equation becomes:

Example 2: Point Slope FormExample 2: Point Slope FormSimplify the equation / put it into Slope Intercept FormSimplify the equation / put it into Slope Intercept Form

Y

X

Y-axis

X-axis

YY

XX

y + 2 = y + 2 = 44x – 12x – 12

Distribute on the right side and the equation becomes:Distribute on the right side and the equation becomes:

(3,-2)

(6,10)

Subtract 2 from both sides gives.Subtract 2 from both sides gives.

y + 2 = y + 2 = 44x – 12x – 12-2 = - 2-2 = - 2

y = 4x – 14y = 4x – 14

Point Slope Form PracticePoint Slope Form PracticeFind the equation for the lines passing through the Find the equation for the lines passing through the following points using following points using Point SlopePoint Slope form. form.

Y

X

1.) (3,2) & ( 8,-2)

2.) (-5,4) & ( 10,-12)

3.) (1,-5) & ( 7,7)

4.) (4,2) & ( -8,-4)

5.) (5,3) & ( 7,9)

Example 3: Horizontal LineExample 3: Horizontal LineLet’s find the equation for the line passing through Let’s find the equation for the line passing through the points the points (0,2)(0,2) and and (5,2)(5,2)

Y

X

Y-axis

X-axis

y = y = mmx + x + b b ( Slope Intercept Form ).

Where Where mm is: is:

m =Y

X=

(2 – 2)

(5 – 0)

Y = 0Y = 0XX

(0,2)(5,2)

= 0

Example 3: Horizontal LineExample 3: Horizontal Line

Because the value of Because the value of mm is 0, is 0,

Y

X

y = y = 00x + x + 2 2 becomes becomes

y = 2y = 2 (A Constant Function)(A Constant Function)

Y-axis

X-axis

(0,2)(5,2)

Horizontal Line PracticeHorizontal Line PracticeFind the equation for the lines passing through the Find the equation for the lines passing through the following points.following points.

Y

X

1.) (3,2) & ( 8,2)

2.) (-5,4) & ( 10,4)

3.) (1,-2) & ( 7,-2)

4.) (4,3) & ( -2,3)

Example 4: Vertical LineExample 4: Vertical Line

Let’s look at a line with no Let’s look at a line with no y-y-interceptintercept bb, an , an x-interceptx-intercept aa, , passing through (3,0) and passing through (3,0) and (3,7).(3,7).

Y

X

Y-axis

X-axis(3,0)

(3,7)

Example 4: Vertical LineExample 4: Vertical LineThe equation for the vertical line is:The equation for the vertical line is:

Y

X

Y-axis

X-axis

x = x = 3 3 ( 3 is the X-Intercept of the line).

Because Because mm is: is:

m =Y

X=

(7 – 0)

(3 – 3)= Undefined

(3,0)

(3,7)

=7

0

Vertical Line PracticeVertical Line PracticeFind the equation for the lines passing through the Find the equation for the lines passing through the following points.following points.

Y

X

1.) (3,5) & ( 3,-2)

2.) (-5,1) & ( -5,-1)

3.) (1,-6) & ( 1,8)

4.) (4,3) & ( 4,-4)

Equation Internet ActivityEquation Internet Activity

Click on each of the links below and follow Click on each of the links below and follow the directions to complete problems.the directions to complete problems.

SparkNotes: Slope Intercept Form

SparkNotes: Point Slope Form

Slope Intercept Form Information and Practice

Point Slope Form Information and Practice

Graphing Equations ConclusionsGraphing Equations Conclusions

What are the similarities you see in the What are the similarities you see in the

equations for Parallel lines? equations for Parallel lines?

What are the similarities you see in the What are the similarities you see in the

equations for Perpendicular lines?equations for Perpendicular lines?

Record your observations on your sheet.Record your observations on your sheet.

Equation SummaryEquation SummarySlope:Slope:

Slope (Slope (mm) =) = Vertical change (YY))

Horizontal change (XX))

Slope-Intercept Form:Slope-Intercept Form:y = y = mmx + x + bb

Point-Slope Form:Point-Slope Form:

y – yy – y11 = = mm(x – x(x – x11))