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Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Section 1.3
Lines
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Find the slope of the line containing the points (-1, 4) and (2, -3).
3 4 7
2 1 3
m
− −
= = −
+
4 3 7
1 2 3
m
+
= = −
− −
The average rate of change of y with respect to x is
7
3
−
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Compute the slopes of the lines L1, L2, L3, and L4 containing the
following pairs of points. Graph all four lines on the same set of
coordinate axes.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Graph the equation: x = − 2
−4 −3 −2 −1 1 2
−1
1
2
3
4
x
y
−4 −3 −2 −1 1 2
−1
1
2
3
4
x
y
(-2,4)
(-2,2)
(-2,0)
(-2,1)
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Find the equation of a line with slope −3 and containing
the point (−1, 4).
( )( )4 3 1y x− = − − −
4 3 3y x− = − −
3 1y x= − +
−4 −3 −2 −1 1 2
−1
1
2
3
4
x
y
Run = 1
Rise = -3
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Find the equation of a horizontal line containing the
point (2, −4).
( ) ( )4 0 2y x− − = × −
4 0y + =
4y = −
−3 −2 −1 1 2 3
−4
−3
−2
−1
1
x
y
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Find an equation of the line L containing the points (−1, 4) and
(3, −1). Graph the line L.
−5 −4 −3 −2 −1 1 2 3 4 5
−5
−4
−3
−2
−1
1
2
3
4
5
(-1, 4)
(3, -1)
( )
1 4
3 1
m
− −
=
− −
5
4
= −
( )( )5
4 1
4
y x− = − − −
( )
5
4 1
4
y x− = − +
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
−5 −4 −3 −2 −1 1 2 3 4 5
−5
−4
−3
−2
−1
1
2
3
4
5
(0, -3)
(2, 0)
Find the slope m and y-intercept b of the equation
3x – 2y = 6. Graph the equation.
3
3
2
y x= −
3
2
3x – 2y = 6
– 2y = −3x+6
3
3
2
y x= −
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Graph the linear equation 3x + 2y = 6 by finding its
intercepts.
The x-intercept is at the point (2, 0)
The y-intercept is at the point (0, 3)
−5 −4 −3 −2 −1 1 2 3 4 5
−5
−4
−3
−2
−1
1
2
3
4
5
(0, 3)
(2, 0)
3x + 2(0) = 6
3x = 6
x = 2
3(0) + 2y = 6
2y = 6
y = 3
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
1 : 3 2 12L x y− + = 2 : 6 4 0L x y− =
2 3 12y x= +
3
6
2
y x= +
3
Slope ; -intercept 6
2
y= =
4 6y x− = −
3
2
y x=
3
Slope ; -intercept 0
2
y= =
−6 −5 −4 −3 −2 −1 1 2 3
−2
−1
1
2
3
4
5
6
x
y
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Find an equation for the line that contains the point ( 1,3) and is
parallel to the lin 3 4 1e .2x y =
−
−
4 3 12.y x− = − +
3
3
4
y x= −
3
So a line parallel to this one would have a slope of .
4
( )1 1y y m x x− = −
( )
3
3 ( 1)
4
y x− = − −
3 15
4 4
y x= +−8 −6 −4 −2 2 4 6
−6
−4
−2
2
4
6
8
x
y
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Find the slope of a line perpendicular to a line with slope .
3
4
perpendicular
4
3
m = −
−5 −4 −3 −2 −1 1 2 3 4 5
−5
−4
−3
−2
−1
1
2
3
4
5
1
3
4
m =
2
4
3
m = −
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Find an equation for the line that contains the point ( 1,3) and is
perpendicular to the li 2ne 3 1 .4x y− =
−
4 3 12.y x− = − +
3
3
4
y x= −
4
So a line perpendicular to this one would have a slope of .
3
−
( )1 1y y m x x− = −
( )
4
3 ( 1)
3
y x− = − − −
4 5
3 3
y x= − +−8 −6 −4 −2 2 4 6
−6
−4
−2
2
4
6
8
x
y

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Section 1.3 lines

  • 1. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Section 1.3 Lines
  • 2. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 3. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 4. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 5. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 6. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 7. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Find the slope of the line containing the points (-1, 4) and (2, -3). 3 4 7 2 1 3 m − − = = − + 4 3 7 1 2 3 m + = = − − − The average rate of change of y with respect to x is 7 3 −
  • 8. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Compute the slopes of the lines L1, L2, L3, and L4 containing the following pairs of points. Graph all four lines on the same set of coordinate axes.
  • 9. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 10. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 11. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 12. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 13. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 14. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 15. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Graph the equation: x = − 2 −4 −3 −2 −1 1 2 −1 1 2 3 4 x y −4 −3 −2 −1 1 2 −1 1 2 3 4 x y (-2,4) (-2,2) (-2,0) (-2,1)
  • 16. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 17. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 18. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 19. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Find the equation of a line with slope −3 and containing the point (−1, 4). ( )( )4 3 1y x− = − − − 4 3 3y x− = − − 3 1y x= − + −4 −3 −2 −1 1 2 −1 1 2 3 4 x y Run = 1 Rise = -3
  • 20. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Find the equation of a horizontal line containing the point (2, −4). ( ) ( )4 0 2y x− − = × − 4 0y + = 4y = − −3 −2 −1 1 2 3 −4 −3 −2 −1 1 x y
  • 21. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 22. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 23. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Find an equation of the line L containing the points (−1, 4) and (3, −1). Graph the line L. −5 −4 −3 −2 −1 1 2 3 4 5 −5 −4 −3 −2 −1 1 2 3 4 5 (-1, 4) (3, -1) ( ) 1 4 3 1 m − − = − − 5 4 = − ( )( )5 4 1 4 y x− = − − − ( ) 5 4 1 4 y x− = − +
  • 24. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 25. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 26. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 27. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 28. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 29. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. −5 −4 −3 −2 −1 1 2 3 4 5 −5 −4 −3 −2 −1 1 2 3 4 5 (0, -3) (2, 0) Find the slope m and y-intercept b of the equation 3x – 2y = 6. Graph the equation. 3 3 2 y x= − 3 2 3x – 2y = 6 – 2y = −3x+6 3 3 2 y x= −
  • 30. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 31. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 32. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Graph the linear equation 3x + 2y = 6 by finding its intercepts. The x-intercept is at the point (2, 0) The y-intercept is at the point (0, 3) −5 −4 −3 −2 −1 1 2 3 4 5 −5 −4 −3 −2 −1 1 2 3 4 5 (0, 3) (2, 0) 3x + 2(0) = 6 3x = 6 x = 2 3(0) + 2y = 6 2y = 6 y = 3
  • 33. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 34. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 35. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 1 : 3 2 12L x y− + = 2 : 6 4 0L x y− = 2 3 12y x= + 3 6 2 y x= + 3 Slope ; -intercept 6 2 y= = 4 6y x− = − 3 2 y x= 3 Slope ; -intercept 0 2 y= = −6 −5 −4 −3 −2 −1 1 2 3 −2 −1 1 2 3 4 5 6 x y
  • 36. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Find an equation for the line that contains the point ( 1,3) and is parallel to the lin 3 4 1e .2x y = − − 4 3 12.y x− = − + 3 3 4 y x= − 3 So a line parallel to this one would have a slope of . 4 ( )1 1y y m x x− = − ( ) 3 3 ( 1) 4 y x− = − − 3 15 4 4 y x= +−8 −6 −4 −2 2 4 6 −6 −4 −2 2 4 6 8 x y
  • 37. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 38. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 39. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
  • 40. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Find the slope of a line perpendicular to a line with slope . 3 4 perpendicular 4 3 m = − −5 −4 −3 −2 −1 1 2 3 4 5 −5 −4 −3 −2 −1 1 2 3 4 5 1 3 4 m = 2 4 3 m = −
  • 41. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Find an equation for the line that contains the point ( 1,3) and is perpendicular to the li 2ne 3 1 .4x y− = − 4 3 12.y x− = − + 3 3 4 y x= − 4 So a line perpendicular to this one would have a slope of . 3 − ( )1 1y y m x x− = − ( ) 4 3 ( 1) 3 y x− = − − − 4 5 3 3 y x= − +−8 −6 −4 −2 2 4 6 −6 −4 −2 2 4 6 8 x y