Transcript of David Raju 1.1 Lines. At the end of this lesson you will be able to: Write equations for...
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- David Raju 1.1 Lines
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- At the end of this lesson you 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. David Raju Y X
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- Lets review some vocabulary. David Raju Y X Slope (m) = Y)
Vertical change ( Y) 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 ( Y) to horizontal change ( X). X) Horizontal change ( X)
X-intercept (a): The x-coordinate of the point where the graph of a
line crosses the x-axis.
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- Equations of Non-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.
David Raju Y X Y-axis X-axis (0,b) (x,y)
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- Slope Intercept Form The equation for the non-vertical line is:
David Raju Y X Y-axis X-axis (0,b) (x,y) YYYY XXXX y = mx + b y =
mx + b ( Slope Intercept Form ) Where m is: m = YY XX = (y b) (x
0)
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- More Equations of Non-vertical Lines. Lets look at a line
passing through Point 1 (x 1,y 1 ) and Point 2 (x 2,y 2 ). David
Raju Y X Y-axis X-axis (x 1,y 1 ) (x 2,y 2 )
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- Point Slope Form The equation for the non-vertical line is:
David Raju Y X Y-axis X-axis YYYY XXXX y y 1 = m(x x 1 ) y y 1 =
m(x x 1 ) ( Point Slope Form ) Where m is: m =m = YY XX = (y 2 y 1
) (x 2 x 1 ) (x 1,y 1 ) (x 2,y 2 )
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- Equations of Horizontal 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. David Raju Y X Y-axis X-axis (0,b) (x,b)
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- Horizontal Line The equation for the horizontal line is still
David Raju Y X Y-axis X-axis y = mx + b y = mx + b ( Slope
Intercept Form ). Where m is: m =m = YY XX = (b b) (x 0) Y = 0 XXXX
(0,b) (x,b) = 0
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- Horizontal Line Because the value of m is 0, David Raju Y X y =
mx + b becomes y = b (A Constant Function) Y-axis X-axis (0,b)
(x,b)
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- Equations of Vertical 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. David Raju Y X Y-axis
X-axis (a,0) (a,y)
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- Vertical Line The equation for the vertical line is David Raju
Y X Y-axis X-axis x = a x = a ( a is the X-Intercept of the line).
Because m is: m =m = YY XX = (y 0) (a a) = Undefined (a,0)
(a,y)
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- Vertical Line Because the value of m is undefined, caused by
the division by zero, there is no slope m. David Raju Y X x = a
becomes the equation x = a (The equation of a vertical line) Y-axis
X-axis (a,0) (a,y)
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- Example 1: Slope Intercept Form Find the equation for the line
with m = 2/3 and b = 3 David Raju Y X Y-axis X-axis Because b = 3 Y
= 2 X = 3 (0,3) X = 3 The line will pass through (0,3) Because m =
2/3 The Equation for the line is: y = 2/3 x + 3 Y = 2
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- Slope Intercept Form Practice Write the equation for the lines
using Slope Intercept form. David Raju 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
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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) David Raju Y X Y-axis
X-axis YYYY XXXX First, Calculate m : m =m = YY XX = (10 -2) (6 3)
(3,-2) (6,10) 3 12= =4
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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) David Raju Y X Y-axis
X-axis YYYY XXXX y y 1 = m(x x 1 ) Next plug it into Point Slope
From : (3,-2) (6,10) y -2 = 4(x 3) Select one point as P 1 : Lets
use (3,-2) The Equation becomes:
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- Example 2: Point Slope Form Simplify the equation / put it into
Slope Intercept Form David Raju Y X Y-axis X-axis YYYY XXXX 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
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- Point Slope Form Practice Find the equation for the lines
passing through the following points using Point Slope form. David
Raju 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)
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- Example 3: Horizontal Line Lets find the equation for the line
passing through the points (0,2) and (5,2) David Raju Y X Y-axis
X-axis y = mx + b y = mx + b ( Slope Intercept Form ). Where m is:
m =m = YY XX = (2 2) (5 0) Y = 0 XXXX (0,2) (5,2) = 0
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- Example 3: Horizontal Line Because the value of m is 0, David
Raju Y X y = 0x + 2 becomes y = 2 (A Constant Function) Y-axis
X-axis (0,2) (5,2)
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- Horizontal Line Practice Find the equation for the lines
passing through the following points. David Raju Y X 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).
David Raju Y X Y-axis X-axis (3,0) (3,7)
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- Example 4: Vertical Line The equation for the vertical line is:
David Raju Y X Y-axis X-axis x = 3 x = 3 ( 3 is the X-Intercept of
the line). Because m is: m =m = YY XX = (7 0) (3 3) = Undefined
(3,0) (3,7) = 7 0
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- Vertical Line Practice Find the equation for the lines passing
through the following points. David Raju Y X 1.) (3,5) & (
3,-2) 2.) (-5,1) & ( -5,-1) 3.) (1,-6) & ( 1,8) 4.) (4,3)
& ( 4,-4)
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- Graphing Calculator Activity Using a TI-84 calculator, graph
the following equations. David Raju y 1 = 4x + 5 y 2 = ( 1/2 )x + 3
Y 3 = -2x + 2 y 4 = -(1/4)x + 1 y 5 = 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. David Raju y 1 = 4x + 5 y 2 = ( 1/2 )x + 3 Y 3 =
-2x + 2 y 4 = -(1/4)x + 1 y 5 = 4x + 0 Y-axis X-axis 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-84 calculator, graph
the following equations. David Raju y 1 = 2x + 3 y 2 = ? Y 3 = -3x
+ -1 y 4 = ? y 5 = 7 y 6 = ? 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. David Raju Y-axis X-axis Press the space bar to compare
your graphs with mine. The equations and their graphs are color
coded. y 1 = 2x + 3 y 2 = ? Y 3 = -3x + -1 y 4 = ? y 5 = 7 y 6 =
?
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- Graphing Calculator Activity Using a TI-84 calculator, graph
the following equations. David Raju y 1 = 2x + 3 y 2 = ? Y 3 = -3x
+ -1 y 4 = ? y 5 = 7 y 6 = ? 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. David Raju Y-axis X-axis Press the space bar to compare
your graphs with mine. The equations and their graphs are color
coded. y 1 = 2x + 3 y 2 = ? Y 3 = -3x + -1 y 4 = ? y 5 = 7 y 6 =
?
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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. David Raju
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- Equation Summary David Raju Slope (m) = Y) Vertical change ( Y)
X) Horizontal change ( X) Slope-Intercept Form: y = mx + b
Point-Slope Form: y y 1 = m(x x 1 )