Intro. To Graphing Linear Equations
The Coordinate Plane
A. The coordinate plane has 4 quadrants.
B. Each point in the coordinate plain has an x coordinate (the abscissa) and a y coordinate (the
ordinate). The point is stated as an ordered pair (x,y).
C. Horizontal Axis is the X – Axis. (y = 0)
D. Vertical Axis is the Y- Axis (x = 0)
Plot the following points:
a) (3,7)
b) (-4,5)
c) (-6,-1)
d) (6,-7)
e) (5,0)
f) (0,5)
g) (-5,0)
f) (0, -5)
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 .
-2 – 3
-7 = 3(-2) + b
-7 = -6 + b
+6 +6
-1 = b
m = -15 .
-5
m= 3
III.
y = 3x – 1
Special Slopes
A. Zero Slope
* No change in Y
B. No Slope (undefined slope)
* 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,-5)
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
* Do not intersect
* Have same slopes
B. Perpendicular lines
* Intersect to form right angles (90˚)
* 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)
(5,30)
(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, 1)
(6,8)
8) (3,6)(3,-6)
(7,-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
y = 2x -7
23)
y-2= (3/5)(x-10)
y+11 =2(x+7)
20)
6x + 3y =21
12x + 16y = -48
24)
6x + 9y = 45
9x +15y = 75
21)
12x – 6y = -6
16x -8y = 40
25)
x=5
y-12 = -3(x+2)
22)
y= -4
x=7
26)
9x – 18y = 126
y = -4
Graphing Inequalities
When we solved and graphed inequalities with only one variable (ex: x > 3), we moved on to
compound inequalities (AND/OR). We would graph both inequalities on the same number line and
decide what to keep based on whether it was an AND or an OR problem. When we graphed linear
equations on the coordinate plane we moved on to solving systems of equations graphically.
When we graph inequalities in two variables on the coordinate plane, we do not graph compound
inequalities. We move on to solving systems of inequalities. It takes a little from both inequalities with
one variable and solving systems graphically.
Graph the Inequality:
y>¼x+3
Step 1: Graph the 1st inequality (graph
the line and test a point one up from the
from the y-int and one down from the y-int):
y>¼x+3
m = ¼ = ▲y = up 1
▲x
r4
y-int= (0,3)
(0, 2)
(0, 4)
2 > ¼ (0)+3
-3 > ¼ (0) + 3
2 > 3
4>3
FALSE
TRUE
Step 3: Shade towards the “true” point (0,4)

When you “test”, you must do it in the original equation!
Graphing the Parabola
a) Graph y = x2 – 4x + 3 for -1 < x < 5
b) Is the vertex a minimum or maximum?
c) Find the axis of symmetry, the vertex, and the roots.
1)
The first thing you should do is put the equation into the calculator. While you are in the y= screen, you
might as well put in y = 0 for the second line (you will need it to find the roots).
Min or Max?
LOOK AT THE PICTURE
Axis of Symmetry:
x = -b . x = -(-4). x = 4 . x = 2
[a is the coefficient of x2 (1)and b is the coef. of x (-4)]
2a
2(1)
2
a = 1 b = -4
Finding the vertex: If the axis of symm. is an integer, then you can find the ordered
pair for the vertex in the table. If not, we will use the calculator.
2nd CALC 1 (for value)
Type in the value for x (x= 2) and hit ENTER
The ordered pair for the vertex will appear on the bottom of the screen. (2,-1)
Finding the roots: The roots are where the equation or y = 0. (Where it crosses the x-axis).
This is why we put y = 0 into the calculator. The x-axis is y = 0.
There are usually 2 roots. To find the first:
2nd CALC 5 (for intersect) ENTER ENTER ENTER
The ordered pair will be at the bottom.
1st Root: _________
To find the 2nd root:
2nd CALC 5 (for intersect) DO NOT HIT ENTER!!!!!
You must move the cursor using the blue left and right arrows (not up and down) close to the 2nd root.
Once close: ENTER ENTER ENTER The ordered pair will be at the bottom. (1,0)
2nd root:
.
Fill out the table, plot the points, draw a smooth curve, and label. (use graph on back).
x
-1
0
1
2
3
4
5
y
2) a) Graph: y = 2x2 +3x – 5
b) Is the vertex a minimum or maximum?
c) Find the axis of symmetry.
d) Find the vertex.
e) Find the roots.
3) a) Graph: y = -x2 + 5x + 2
b) Is the vertex a min. or max.?
c) Find the axis of symmetry.
d) Find the vertex.
e) Find the roots.
Graphing the Parabola
when the numbers aren’t so nice
2
1)
a) Graph y = -3x + x + 2
b) Is the vertex a minimum or maximum?
c) Find the axis of symmetry, the vertex, and the roots.
The first thing you should do is put the equation into the calculator. While you are in the y= screen, you
might as well put in y = 0 for the second line (you will need it to find the roots).
Min or Max?
LOOK AT THE PICTURE
Axis of Symmetry:
x = -b .
x = -(1) .
2a
2(-3)
[a is the coefficient of x2 (-3)and b is the coef. of x (1)]
x = .16
Finding the vertex: If the axis of symm. is an integer, then you can find the ordered
pair for the vertex in the table. If not, we will use the calculator.
2nd CALC 1 (for value)
Type in the value for x (.16 ) and hit ENTER
The ordered pair for the vertex will appear on the bottom of the screen. (.16, 5.08)
Finding the roots:
The roots are where the equation or y = 0. (Where it crosses the x-axis).
This is why we put y = 0 into the calculator. The x-axis is y = 0.
There are usually 2 roots. To find the first:
2nd CALC 5 (for intersect) ENTER ENTER ENTER
The ordered pair will be at the bottom.
(1.47, 0)
To find the 2nd root:
2 CALC 5 (for intersect) DO NOT HIT ENTER!!!!!
You must move the cursor using the blue left and right arrows (not up and down) close to the 2nd root.
Once close: ENTER ENTER ENTER The ordered pair will be at the bottom. (-1.14,0)
nd
Graph the parabola:
x
y
-2
-1
0
1
2
-9
1
5
3
-5
-Plot the vertex. Estimate as best you
can.
-Plot the points from the
table.
-Draw as smooth a curve as possible.
-Label with the equation.
1) a) Graph: y = -3x2 + x + 2
b) Is the vertex a minimum or maximum?
c) Find the axis of symmetry.
d) Find the vertex.
e) Find the roots.
2) a) Graph: y = x2 – 3x – 4
b) Is the vertex a min. or max.?
c) Find the axis of symmetry.
d) Find the vertex.
e) Find the roots.
3) a) Graph: y = 2x2 + x - 3
b) Is the vertex a minimum or maximum?
c) Find the axis of symmetry.
d) Find the vertex.
e) Find the roots.
4) a) Graph: y = - x2 – 4x + 3
b) Is the vertex a min. or max.?
c) Find the axis of symmetry.
d) Find the vertex.
e) Find the roots.