### Graphing Linear Equations

```An equation for
which the graph
is a line
Any ordered pair
of numbers that
makes a linear
equation true.
(9,0) IS ONE SOLUTION
FOR Y = X - 9
Example:
y=x+3
Step 1:
~ Three Point Method ~
Choose 3 values
for x
Step 2:
Find solutions using table
y=x+3
Y | X
0
1
2
Step 3:
Graph the points
from the table
(0,3) (1,4) (2,5)
Step 4:
Draw a line to
connect them
Graph
using a table (3
point method)
1) y = x + 3
2) y = x - 4
Where the line
crosses the xaxis
The x-intercept
has a y
coordinate of
ZERO
To find the xintercept, plug in
ZERO for y and
solve
Describes the
steepness of a
line
Equal to:
Rise
Run
The change
vertically, the
change in y
The change
horizontally or
the change in
x
Step 1:
Find 2 points on a
line
(2, 3) (5, 4)
(x , y ) (x , y )
1
1
2
2
Step 2:
Find the RISE
between these 2
points
Y-Y =
4-3=1
2
1
Step 3:
Find the RUN
between these 2
points
X-X =
5-2=3
2
1
Step 4:
Write the RISE over
RUN as a ratio
Y-Y
2
1
X-X
2
1
=
1
3
Where the line
crosses the yaxis
The y-intercept
has an xcoordinate of
ZERO
To find the yintercept, plug in
ZERO for x and
solve
y = mx + b
m = slope
b = y-intercept
Mark a point
on the yintercept
Define slope as
a fraction...
(RISE)
Denominator is
the horizontal
change
(RUN)
Graph at least
3 points and
connect the
dots
Definitions
 3 forms for a quad. function
 Steps for graphing each form
 Examples
 Changing between eqn. forms

A
function of the form y=ax2+bx+c
where a≠0 making a u-shaped
graph called a parabola.

The lowest or highest point
of a parabola.
Vertex
Axis of symmetry
Axis of
Symmetry
The vertical line through the vertex of the
parabola.
y=ax2 + bx + c
 If a is positive, u opens up
If a is negative, u opens down
b
 The x-coordinate of the vertex is at 2 a
 To find the y-coordinate of the vertex, plug the xcoordinate into the given eqn.
 The axis of symmetry is the vertical line x=
 Choose 2 x-values on either side of the vertex xcoordinate. Use the eqn to find the
corresponding y-values.
 Graph and label the 5 points and axis of symmetry
on a coordinate plane. Connect the points with a
smooth curve.
a=2 Since a is positive
the parabola will open
up.
b
 Vertex: use x 
2a
b=-8 and a=2
 (8) 8
x
 2
2(2)
4

y  2(2) 2  8(2)  6
y  8  16  6  2
Vertex is: (2,-2)
• Axis of symmetry is the
vertical line x=2
•Table of values for other
points:
x y
0 6
1 0
2 -2
3 0
4 6
* Graph!
x=2
(.5,12)
(-1,10)
(2,10)
(-2,6)
(3,6)
X = .5
y=a(x-h)2+k
 If a is positive, parabola opens up
If a is negative, parabola opens down.
 The vertex is the point (h,k).
 The axis of symmetry is the vertical line
x=h.
 Don’t forget about 2 points on either side
of the vertex! (5 points total!)
y=2(x-1)2+3
 Open
up or down?
 Vertex?
 Axis of symmetry?
 Table of values with 5 points?




a is negative (a = -.5), so parabola opens down.
Vertex is (h,k) or (-3,4)
Axis of symmetry is the vertical line x = -3
Table of values
x y
-1 2
Vertex (-3,4)
-2 3.5
(-4,3.5)
(-2,3.5)
-3 4
-4 3.5
(-5,2)
(-1,2)
-5 2
x=-3
(-1, 11)
(3,11)
X=1
(0,5)
(2,5)
(1,3)
y=a(x-p)(x-q)





The x-intercepts are the points (p,0) and (q,0).
The axis of symmetry is the vertical line x= p  q
2
pq
The x-coordinate of the vertex is 2
To find the y-coordinate of the vertex, plug the
x-coord. into the equation and solve for y.
If a is positive, parabola opens up
If a is negative, parabola opens down.



Since a is negative,
parabola opens
down.
The x-intercepts are
(-2,0) and (4,0)
To find the x-coord.
of the vertex, use p  q
24 2
x
 1
2
2

•The axis of symmetry
is the vertical line x=1
(from the x-coord. of
the vertex)
(1,9)
2
To find the y-coord.,
plug 1 in for x.
(-2,0)
(4,0)
y  (1  2)(1  4)  (3)(3)  9

Vertex (1,9)
x=1
y=2(x-3)(x+1)
 Open
up or down?
 X-intercepts?
 Vertex?
 Axis of symmetry?
x=1
(-1,0)
(3,0)
(1,-8)
The key is to FOIL! (first, outside, inside,
last)
 Ex: y=-(x+4)(x-9)
Ex: y=3(x-1)2+8
=-(x2-9x+4x-36)
=3(x-1)(x-1)+8
=-(x2-5x-36)
=3(x2-x-x+1)+8
y=-x2+5x+36
=3(x2-2x+1)+8
=3x2-6x+3+8
y=3x2-6x+11

```