Graphing Linear Equations



Graphing Linear Equations

To graph a line (linear equation), we first want to make sure the equation is in slope intercept form (y=mx+b). We will then use the slope and the y-intercept to graph the line.

Slope (m): Measures the steepness of a non-vertical line. It is sometimes refereed to as the rise/run or . It’s how fast and in what direction y changes compared to x.

y-intercept: The y-intercept is where a line passes through the y axis. It is always stated as an ordered pair (x,y). The x coordinate is always zero. The y coordinate can be taken from the “b” in y=mx+b.

Graphing The Linear Equation: y = 3x - 5

1) Find the slope: m = 3 ( m = 3 . = y .

1 x

2) Find the y-intercept: x = 0 , b = -5 ( (0, -5)

3) Plot the y-intercept

4) Use slope to find the next point: Start at (0,-5)

m = 3 . = y . ( up 3 on the y-axis

1 x ( right 1 on the x-axis

(1,-2) Repeat: (2,1) (3,4) (4,7)

5) To plot to the left side of the y-axis, go to y-int. and

do the opposite. (Down 3 on the y, left 1 on the x)

(-1,-8) Repeat: (-2,-11) (-3,-14)

6) Connect the dots.

Do Now on GP:

1) y = 2x + 1

2) y = -4x + 5

3) y = ½ x – 3

4) y = ⅔x + 2

Finding the equation of a line in slope intercept form (y=mx + b)

Example: Find the equation in slope intercept form of the line formed by (3,8) and (-2, -7).

A. Find the slope (m): B. Use m and one point to find b:

m = y2 – y1 y = mx + b

x2 – x1 m= 3 x= 3 y=8

 

m = -7 – 8 . -7 = 3(-2) + b

-2 – 3 -7 = -6 + b

+6 +6

m = -15 . -1 = b

-5

m= 3 y = 3x – 1

 

 

III. Special Slopes

A. Zero Slope B. No Slope (undefined slope)

* No change in Y * No change in X

* Equation will be Y = * Equation will be X =

* Horizontal Line * Vertical Line

 

Find equation in slope intercept form and graph:

1) (3,-2)(-6,-8) 6) m= 4 (-2,-5) 12) 16x -4y =36

2) (-6,10) (9,-10) 7) m= ⅔ (-6,-7) 13) 8x+24y = 96

3) (3,7) (3,-7) 8) m= -3/2 (8,-1) 14) y-7=2(x+1)

4) (7,-6)(-3,4) 9) m = 0 (4,3) 15) y+5=(2/5)(x-10)

5) (5,-9)(-5,-9) 10) m = undefined (-6, 5) 16) y-7= ¾ (x+12)

11) m=-3 (-4,19) 17) y-2=-3(x-2)

 

IV. Parallel and Perpendicular Lines:

A. Parallel Lines B. Perpendicular lines

* Do not intersect * Intersect to form right angles (90˚)

* Have same slopes * Slopes are negative reciprocals.

(Invert fraction and change sign)

(Products of slopes is –1)

 

Do in NB: For the given line, find a line that is parallel and passes through the given point. Then, find the equation of a line that is perpendicular and passes through the given point.

Given Line: Parallel: Perpendicular:

7) y = ⅓ x + 4 (6,1) ( -2,10)

8) y = 4x – 5 (2,13) (8.-5)

9) y = -⅔ x + 2 (-9,-11) (4,-1)

10) –5x + 6 (4,-27) (-10,6)

 

 

Practice Problems: a) Use the two points to find the equation of the line.

b)      For the line found in part a, find a line that is parallel and passes through the given point.

c)      Find the equation of a line that is perpendicular and passes through the given point.

 

Given Line: Parallel: Perpendicular:

1) (-5, 13) (3, -3) (4,-10) (2,7)

2) (-6,0) (3,6) (6,3) (6,-7)

3) (2,6)(-3,-19) (5,14) (5,5)

4) (-4,3) (-8,6) (-4, 10) (-6,-8)

5) (2,-5) (-2, -5) (8,-2) (4,-3)

6) (-9,-11)(6,9) (-3,-9) (-4,10)

7) (8,-3) (-4,9) (-2, 14) (6,-4)

8) (3,6)(3,-6) (11,-3) (5,2)

9) (4,-3)(-6,-8) (6,7) (-5,0)

10) (2,4)(-6,-12) (-3,-5) (-8,4)

11) Find the equation of the line parallel to y = 3x – 2, passing through (-2, 1).

12) Find the equation of the line perpendicular to y = -½x – 5, passing through (-2, -10)

13) Find the equation of the line parallel to y = -¼ x + 2, passing through (-8, 7)

14) Find the equation of the line perpendicular to y = (3/2)x + 6, passing through (-6, 1)

15) Find the equation of the line parallel to y = -5, passing through (2,7)

16) Find the equation of the line perpendicular to y = 5, passing through (6, -4).

17) Find the equation of the line parallel to x= 8, passing through (4, -9)

18) Find the equation of the line perpendicular to x = -3, passing through (6, -7).

Solve each system graphically:

19) y = -4x -5 23) y-2= (3/5)(x-10)

y = 2x -7 y+11 =2(x+7)

20) 6x + 3y =21 24) 6x + 9y = 45

12x + 16y = -48 9x +15y = 75

21) 12x – 6y = -6 25) x = 5

16x -8y = 40 y-12 = -3(x+2)

22) y= -4 26) 9x – 18y = 126

x = 7 y = -4

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