Download Standard Form

Standard Form
Section 5-5
Goals
Goal
• To graph linear equations
using intercepts.
• To write linear equations in
standard form.
Rubric
Level 1 – Know the goals.
Level 2 – Fully understand the
goals.
Level 3 – Use the goals to
solve simple problems.
Level 4 – Use the goals to
solve more advanced problems.
Level 5 – Adapts and applies
the goals to different and more
complex problems.
Vocabulary
• x-intercept
• Standard form of a linear equation
Standard Form
Another way to determine whether a function is linear is to look
at its equation. A function is linear if it is described by a linear
equation. A linear equation is any equation that can be written in
the standard form shown below.
Standard Form
Notice that when a linear equation is written in standard
form.
Ax  By C
•x
and y are both on the same side of the equal sign.
•x
and y both have exponents of 1.
•x
and y are not multiplied together.
• x and y do not appear in denominators, exponents, or radical
signs.
Axial Intercepts
A y-intercept is the y-coordinate of
any point where a graph intersects
the y-axis. The x-coordinate of this
point is always 0.
An x-intercept is the x-coordinate of
any point where a graph intersects
the x-axis. The y-coordinate of this
point is always 0.
Example: Finding Intercepts
From a Graph
Find the x- and y-intercepts.
The graph intersects the y-axis
at (0, 1).
The y-intercept is 1.
The graph intersects the xaxis at (–2, 0).
The x-intercept is –2.
Example: Intercepts From Equation
Find the x- and y-intercepts.
5x – 2y = 10
To find the x-intercept, replace y with 0
and solve for x.
5x – 2y = 10
5x – 2(0) = 10
5x – 0 = 10
5x = 10
x=2
The x-intercept is 2.
To find the y-intercept, replace x with 0
and solve for y.
5x – 2y = 10
5(0) – 2y = 10
0 – 2y = 10
–2y = 10
y = –5
The y-intercept is –5.
Your Turn:
Find the x- and y-intercepts.
The graph intersects the yaxis at (0, 3).
The y-intercept is 3.
The graph intersects the xaxis at (–2, 0).
The x-intercept is –2.
Your Turn:
Find the x- and y-intercepts.
–3x + 5y = 30
To find the x-intercept, replace y with 0
and solve for x.
–3x + 5y = 30
–3x + 5(0) = 30
–3x – 0 = 30
–3x = 30
x = –10
The x-intercept is –10.
To find the y-intercept, replace x with 0
and solve for y.
–3x + 5y = 30
–3(0) + 5y = 30
0 + 5y = 30
5y = 30
y=6
The y-intercept is 6.
Your Turn:
Find the x- and y-intercepts.
4x + 2y = 16
To find the x-intercept, replace y with 0
and solve for x.
4x + 2y = 16
4x + 2(0) = 16
4x + 0 = 16
4x = 16
x=4
The x-intercept is 4.
To find the y-intercept, replace x
with 0 and solve for y.
4x + 2y = 16
4(0) + 2y = 16
0 + 2y = 16
2y = 16
y=8
The y-intercept is 8.
Graphing Using Standard
Form
For any two points, there is exactly one line that contains
them both. This means you need only two ordered pairs to
graph a line.
It is often simplest to find the ordered pairs that contain the x
and y intercepts.
The x and y intercepts can easily be found from standard
form. Therefore, to graph a linear equation in standard form,
use the x and y intercepts.
Example: Graphing Standard
Form
Use intercepts to graph the line given by the equation.
3x – 7y = 21
Step 1 Find the intercepts.
x-intercept:
3x – 7y = 21
3x – 7(0) = 21
3x = 21
x=7
y-intercept:
3x – 7y = 21
3(0) – 7y = 21
–7y = 21
y = –3
Example: Continued
x-intercept: x = 7 and y-intercept: y = -3
Step 2 Graph the line.
Plot (7, 0) and (0, –3).
x
Connect with a straight line.
Example: Graphing Standard
Form
Use intercepts to graph the line given by the equation.
y = –x + 4
Step 1 Write the equation in standard form.
y = –x + 4
+x +x
x+y=4
Add x to both sides.
Example: Continued
x+y=4
Step 2 Find the intercepts.
x-intercept:
x+y=4
x+0=4
x=4
y-intercept:
x+y=4
0+y=4
y=4
Example: Continued
x-intercept: x = 4 and y-intercept: y = 4
Step 3 Graph the line.
Plot (4, 0) and (0, 4).
Connect with a straight line.
Your Turn:
Use intercepts to graph the line given by the equation.
–3x + 4y = –12
Step 1 Find the intercepts.
x-intercept:
y-intercept:
–3x + 4y = –12
–3x + 4y = –12
–3x + 4(0) = –12
–3x = –12
–3(0) + 4y = –12
4y = –12
x=4
y = –3
Your Turn: Continued
x-intercept: x = 4 and y-intercept: y = -3
Step 2 Graph the line.
Plot (4, 0) and (0, –3).
Connect with a straight line.
Your Turn:
Use intercepts to graph the line given by the equation.
Step 1 Write the equation in standard form.
Multiply both sides by 3, the LCD of the
fractions, to clear the fraction.
3y = x – 6
–x + 3y = –6
Write the equation in standard form.
Your Turn: Continued
–x + 3y = –6
Step 2 Find the intercepts.
x-intercept:
–x + 3y = –6
–x + 3(0) = –6
–x = –6
y-intercept:
–x + 3y = –6
–(0) + 3y = –6
3y = –6
x=6
y = –2
Your Turn: Continued
x-intercept: x = 6 and y-intercept: y = -2
Step 3 Graph the line.
Plot (6, 0) and (0, –2).
Connect with a straight line.
Equations of Horizontal
and Vertical Lines
Equations of Horizontal
and Vertical Lines
Equation of a Horizontal Line
A horizontal line is given by an equation of the form
y=b
where b is the y-intercept. Note: m = 0.
Equation of a Vertical Line
A vertical line is given by an equation of the form
x=a
where a is the x-intercept. Note: m is undefined.
Y
Equations of
Horizontal Lines
• Let’s 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.
(0,b)
Y-axis
(x,b)
X-axis
X
Y
Horizontal Line
• The equation for the horizontal line is still
y = mx + b ( Slope Intercept Form ).
Where m is:
m=
DY
DX
=
Y-axis
(b – b)
(x – 0)
DY = 0
=0
(0,b)
(x,b)
DX
X-axis
X
Y
Horizontal Line
• Because the value of m is 0,
y = mx + b becomes
Y-axis
y=b
(A Constant Function)
(0,b)
(x,b)
X-axis
X
Y
Example 1: Horizontal Line
• Let’s find the equation for the line passing through the
points (0,2) and (5,2)
y = mx + b ( Slope Intercept Form ).
Where m is:
Y-axis
m=
DY
DX
=
(2 – 2)
(5 – 0)
=0
(0,2)
(5,2)
DX
DY = 0
X-axis
X
Y
Example 1: Horizontal Line
• Because the value of m is 0,
y = 0x + 2 becomes
Y-axis
y=2
(A Constant Function)
(0,2)
(5,2)
X-axis
X
Y
Your Turn:
• Find the equation for the lines passing through the
following points.
1.) (3,2) & ( 8,2)
y=2
2.) (-5,4) & ( 10,4)
y=4
3.) (1,-2) & ( 7,-2)
y = -2
4.) (4,3) & ( -2,3)
y=3
X
Y
Equations of Vertical Lines
• Let’s 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.
Y-axis
(a,y)
(a,0)
X-axis
X
Y
Vertical Line
• The equation for the vertical line is
x = a ( a is the X-Intercept of the line).
Because m is:
Y-axis
(a,y)
m=
DY
DX
=
(y – 0)
= Undefined
(a – a)
(a,0)
X-axis
X
Y
Vertical Line
• Because the value of m is undefined, caused by the division
by zero, there is no slope m.
x = a becomes the equation
Y-axis
x=a
(The equation of a vertical line)
(a,y)
(a,0)
X-axis
X
Y
Example 2: Vertical Line
• Let’s look at a line
with no y-intercept
b, an x-intercept a,
passing through
(3,0) and (3,7).
Y-axis
(3,7)
(3,0)
X-axis
X
Y
Example 2: Vertical Line
• The equation for the vertical line is:
x = 3 ( 3 is the X-Intercept of the line).
Because m is:
Y-axis
(3,7)
m=
DY
DX
=
(7 – 0)
(3 – 3)
=
7
0
= Undefined
(3,0)
X-axis
X
Y
Your Turn:
• Find the equation for the lines passing through the
following points.
1.) (3,5) & ( 3,-2)
x=3
2.) (-5,1) & ( -5,-1)
x = -5
3.) (1,-6) & ( 1,8)
x=1
4.) (4,3) & ( 4,-4)
x=4
X
Linear Equations
Joke Time
• Why don’t blind people go skydiving?
• Because it scares the bejesus out of the dogs!
• What kind of horses go out after dark?
• Nightmares!
• Why did the skeleton go to the party alone?
• He had no body to go with him.
Assignment
• 5-5 Exercises Pg. 354 - 356: #8 – 72 even