Equations of Lines

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Equations of Lines

• Honors Geometry
• Mr. Brown

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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.

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Before we begin.

• Lets review some vocabulary.

Slope (m) The measure of the steepness of a
line it is the ratio of vertical change (DY) to
horizontal change (DX).
Vertical change (DY)
Slope (m)
Y-intercept (b) The y-coordinate of the point
where the graph of a line crosses the y-axis.
X-intercept (a) The x-coordinate of the point
where the graph of a line crosses the x-axis.
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Equations ofNon-vertical Lines.

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

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Slope Intercept Form

• The equation for the non-vertical line is

• y mx b ( Slope Intercept Form )

• Where m is

DY
DY
m
(x 0)
DX
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More Equations ofNon-vertical Lines.

• Lets look at a line passing through Point 1
  (x1,y1) and Point 2 (x2,y2).

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Point Slope Form

• The equation for the non-vertical line is

• y y1 m(x x1) ( Point Slope Form )

• Where m is

DY
m
DY
(x2 x1)
DX
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Equations ofHorizontal Lines.

• Lets 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.

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Horizontal Line

• The equation for the horizontal line is still

• y mx b ( Slope Intercept Form ).

• Where m is

DY
m
0
(x 0)
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Horizontal Line

• Because the value of m is 0,

y mx b becomes
y b (A Constant Function)
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Equations ofVertical Lines.

• Lets 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.

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Vertical Line

• The equation for the vertical line is

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

• Because m is

DY
m
Undefined
(a a)
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Vertical Line

• Because 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)
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Example 1 Slope Intercept Form

• Find the equation for the line with m 2/3 and b
  3

Because b 3
The line will pass through (0,3)
Because m 2/3
DY 2
The Equation for the line is
y 2/3 x 3
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Slope Intercept Form Practice

• Write the equation for the lines using Slope
  Intercept form.

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
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Example 2 Point Slope Form

• Lets find the equation for the line passing
  through the points (3,-2) and (6,10)

• First, Calculate m

(6,10)
DY
m
DY
(3,-2)
DX
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Example 2 Point Slope Form

• To find the equation for the line passing through
  the points (3,-2) and (6,10)

Next plug it into Point Slope From
y y1 m(x x1)
Select one point as P1
(6,10)
Lets use (3,-2)
The Equation becomes
y -2 4(x 3)
DY
(3,-2)
DX
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Example 2 Point Slope Form

• Simplify the equation / put it into Slope
  Intercept Form

Distribute on the right side and the equation
becomes
y 2 4x 12
Subtract 2 from both sides gives.
(6,10)
y 4x 14
DY
(3,-2)
DX
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Point Slope Form Practice

• Find the equation for the lines passing through
  the following points using Point Slope form.

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)
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Example 3 Horizontal Line

• Lets find the equation for the line passing
  through the points (0,2) and (5,2)

y mx b ( Slope Intercept Form ).

• Where m is

DY
m
0
(5 0)
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Example 3 Horizontal Line

• Because the value of m is 0,

y 0x 2 becomes
y 2 (A Constant Function)
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Horizontal Line Practice

• Find the equation for the lines passing through
  the following points.

1.) (3,2) ( 8,2) 2.) (-5,4) ( 10,4) 3.)
(1,-2) ( 7,-2) 4.) (4,3) ( -2,3)
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Example 4 Vertical Line

• Lets look at a line with no y-intercept b, an
  x-intercept a, passing through (3,0) and (3,7).

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Example 4 Vertical Line

• The equation for the vertical line is

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

• Because m is

DY
m
Undefined
0
(3 3)
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Vertical Line Practice

• Find the equation for the lines passing through
  the following points.

1.) (3,5) ( 3,-2) 2.) (-5,1) ( -5,-1) 3.)
(1,-6) ( 1,8) 4.) (4,3) ( 4,-4)
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Equation Internet Activity

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

Slope Intercept Form Information and Practice
SparkNotes Slope Intercept Form
Point Slope Form Information and Practice
SparkNotes Point Slope Form
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Graphing Calculator Activity

• Using a TI-83 calculator, graph the following
  equations.

y1 4x 5 y2 (1/2)x 3 Y3 -2x 2 y4
-(1/4)x 1 y5 4x 0
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Graphing Calculator Activity

• Describe the graphs of each of the lines.
• Include any similarities or differences you see
  in the graphs.
• Be sure to Zoom Standard and Zoom Square
  before you answer these questions.

y1 4x 5 y2 (1/2)x 3 Y3 -2x 2 y4
-(1/4)x 1 y5 4x 0
Press the space bar to compare your graphs with
mine. The equation and its graph are color
coded.
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Graphing Calculator Activity

• Using a TI-83 calculator, graph the following
  equations.

y1 2x 3 y2 ? Y3 -3x -1 y4 ? y5
7 y6 ?

• Now, graph each line given and a line that is
  Parallel to it on the calculator.
• Record the equations you use on your sheet.

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Graphing Calculator Activity

• Compare the graphs of each set of lines.
• Be sure to Zoom Standard and Zoom Square
  before you compare graphs.

y1 2x 3 y2 ? Y3 -3x -1 y4 ? y5
7 y6 ?
Press the space bar to compare your graphs with
mine. The equations and their graphs are color
coded.
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Graphing Calculator Activity

• Using a TI-83 calculator, graph the following
  equations.

y1 2x 3 y2 ? Y3 -3x -1 y4 ? y5
7 y6 ?

• Now, graph each line given and a line that is
  Perpendicular to it on the calculator.
• Record the equations you use on your sheet.

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Graphing Calculator Activity

• Compare the graphs of each set of lines.
• Be sure to Zoom Standard and Zoom Square
  before you compare graphs.

y1 2x 3 y2 ? Y3 -3x -1 y4 ? y5
7 y6 ?
Press the space bar to compare your graphs with
mine. The equations and their graphs are color
coded.
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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.

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Equation Summary

• Slope

Vertical change (DY)
Slope (m)

• Slope-Intercept Form

y mx b

• Point-Slope Form

y y1 m(x x1)