Y

X

Write and GraphEquations of Lines

GeometryLesson 3.5

At the end of this lessonyou will be able to:

Write equations for non-vertical lines.Write equations for horizontal lines.Write equations for vertical lines.Use various forms of linear equations.Calculate the slope of a line passing through two points.

Before we begin.Let’s review some vocabulary.

X

Slope (m) = Vertical change (DY)

Y-intercept (b): The y-coordinate of the point where the graph of a line crosses the y-axis.

Slope (m): The measure of the steepness of a line; it is the ratio of vertical change (DY) to horizontal change (DX).

Horizontal change (DX)

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

Standard Form

Ax + By = CGraphing find x and y interceptsEx 3x + 2y = 6

Equations ofNon-vertical Lines.

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

X

Y-axis

X-axis

(0,b)

(x,y)

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

X

Y-axis

X-axis

(0,b)

(x,y)

DY

DX

y = mx + b ( Slope Intercept Form )

Where m is:

m = DYDX

=(y – b)

(x – 0)

More Equations ofNon-vertical Lines.

Let’s look at a line passing through Point 1 (x1,y1) and Point 2 (x2,y2).

X

Y-axis

X-axis

(x1,y1)

(x2,y2)

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

Y-axis

X-axis

DY

DX

y – y1 = m(x – x1) ( Point Slope Form )

Where m is:

m = DYDX

=(y2 – y1)

(x2 – x1)(x1,y1)

(x2,y2)

Equations of Horizontal Lines

Let’s look at a line with a y-intercept of b, a slope m = 0, and let (x,b) be any point on the Horizontal line.

Y

X

Y-axis

X-axis

(0,b) (x,b)

Horizontal LineThe equation for the horizontal line is still

Y-axis

X-axis

y = mx + b ( Slope Intercept Form ).

Where m is:

m = DYDX

=(b – b)

(x – 0)

DY = 0DX(0,b) (x,b)

= 0

Horizontal Line

Because the value of m is 0,y = mx + b becomes

y = b (A Constant Function)

Y-axis

X-axis

(0,b) (x,b)

Equations ofVertical Lines.

Let’s look at a line with no y-intercept b, an x-intercept a, an undefined slope m, and let (a,y) be any point on the vertical line.

Y-axis

X-axis(a,0)

(a,y)

Vertical LineThe equation for the vertical line is

Y-axis

X-axis

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

Because m is:

m = DYDX

=(y – 0)

(a – a)= Undefined

(a,0)

(a,y)

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

x = a becomes the equation

x = a (The equation of a vertical line)

Y-axis

X-axis(a,0)

(a,y)

Mr Brown Honors Geometry

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

Y-axis

X-axis

Because b = 3

DY = 2

DX = 3(0,3)

DX = 3

The line will pass through (0,3)

Because m = 2/3

The Equation for the line is:y = 2/3 x + 3

DY = 2

Mr Brown Honors Geometry

Slope Intercept Form PracticeWrite the equation for the lines using Slope Intercept form and then graph the equation.

1.) m = 3 & b = 3

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

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

Y-axis

X-axis

DY

DX

First, Calculate m :

m = DYDX

=(10 – -2)

(6 – 3)

(3,-2)

(6,10)

312= = 4

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

Y-axis

X-axis

DY

DX

y – y1 = m(x – x1)Next plug it into Point Slope From :

(3,-2)

(6,10)

y – -2 = 4(x – 3)

Select one point as P1 :Let’s use (3,-2)

The Equation becomes:

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

Y-axis

X-axis

DY

DX

y + 2 = 4x – 12

Distribute on the right side and the equation becomes:

(3,-2)

(6,10)

Subtract 2 from both sides gives.y + 2 = 4x – 12

-2 = - 2

y = 4x – 14

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

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

2.) (5,3) & ( 7,9)

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

Y-axis

X-axis

y = mx + b ( Slope Intercept Form ).Where m is:

m = DYDX

=(2 – 2)

(5 – 0)

DY = 0DX(0,2) (5,2)

= 0

Example 3: Horizontal Line

Because the value of m is 0,y = 0x + 2 becomes

y = 2 (A Constant Function)

Y-axis

X-axis

(0,2) (5,2)

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

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

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

Example 4: Vertical LineLet’s look at a line with no y-intercept b, an x-intercept a, passing through (3,0) and (3,7).

Y-axis

X-axis(3,0)

(3,7)

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

Y-axis

X-axis

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

Because m is:

m = DYDX

=(7 – 0)

(3 – 3)= Undefined

(3,0)

(3,7)

=70

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

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

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

Graphing Equations Conclusions

What are the similarities you see in the

equations for Parallel lines?

What are the similarities you see in the

equations for Perpendicular lines?

Record your observations on your sheet.

Equation SummarySlope:

Slope (m) = Vertical change (DY)

Horizontal change (DX)

Slope-Intercept Form:y = mx + b

Point-Slope Form:y – y1 = m(x – x1)

Mr Brown Honors Geometry

HOMEWORK

Pages 184-185 #3-8, 16-21, 23-44, 46-51