Linear Functions
TLW identify linear
equations and intercepts.
A linear equation is the equation
of a line.
The standard form of a linear equation is
Ax + By = C
* A has to be positive and cannot be a
fraction.
Examples of linear equations
2x + 4y =8
6y = 3 – x
Equation is in Ax + By =C form
Rewrite with both variables
on left side … x + 6y =3
B =0 … x + 0 y =1
x=1
-2a + b = 5
4x  y
 7
3
Multiply both sides of the
equation by -1 … 2a – b = -5
Multiply both sides of the
equation by 3 … 4x –y =-21
Examples of Nonlinear Equations
The following equations are NOT in the
standard form of Ax + By =C:
4x2 + y = 5
x4
xy + x = 5
s/r + r = 3
The exponent is 2
There is a radical in the equation
Variables are multiplied
Variables are divided
Determine whether the equation is a
linear equation, if so write it in
standard form.
y = 5 – 2x
y = 5 – 2x
+ 2x
+ 2x
2x + y = 5
Rewrite the equation
Add 2x to each side
Simplify
A = 2, B= 1, C=5
This IS a linear equation.
Determine whether the equation is a
linear equation, if so write it in
standard form.
2xy -5y = 6
Since the term 2xy has two variables,
the equation cannot be written in the
form Ax + By =0. Therefore, this is NOT
a linear equation.
Determine whether the equation is a
linear equation, if so write it in
standard form.
y  x2  3
Since the term x is raised to the second
power, the equation cannot be written in
the form Ax + By =0. Therefore, this is
NOT a linear equation.
Determine whether the equation is a
linear equation, if so write it in
standard form.
y = 6 – 3x
Rewrite the equation
y = 6 – 3x
Add 3x to each side
+ 3x
+ 3x
3x +y = 6
Simplify
A = 3, B= 1, C=6
This IS a linear equation.
Determine whether the equation is a
linear equation, if so write it in
standard form.
1
x  5y  3
4
1
(4) ( x  5 y  3)
4
Multiply everything by the
denominator to get rid of the
fraction
x + 20y = 12
A = 1, B= 20, C=12
This IS a linear equation.
Determine whether the equation
is a linear equation, if so write it
in standard form.
-4x+7=2
X and Y intercepts
The x coordinate of the point at which the graph of an
equation crosses the x –axis is the x- intercept .
The y coordinate of the point at which the graph of an
equation crosses the y-axis is called the y- intercept.
y- intercept
(0, y)
X- intercept
(-x,0)
Graph the linear equation using the
x- intercept and the y intercept
3x + 2y = 9
To find the x- intercept, let y = 0
Original Equation
3x + 2y = 9
3x + 2(0) = 9
3x = 9
x=3
Replace y with 0
Divide each side by 3
To find the y- intercept, let x = 0
Original Equation
3x + 2y = 9
3(0) + 2y = 9
2y = 9
y = 4.5
Replace x with 0
Divide each side by 2
Plot the two points and connect
them to draw the line.
Graph the linear equation using the
x- intercept and the y intercept
2x + y = 4
To find the x- intercept, let y = 0
2x + y = 4
Original Equation
2x + (0) = 4
2x =4
x=2
Replace y with 0
Divide each side by 3
To find the y- intercept, let x = 0
2x + y = 4
Original Equation
2(0) + y = 4
y=4
Replace x with 0
Simplify
Plot the two points and connect them to draw the line.
Identify the x- and y- intercepts
given a table
X
Y
X
Y
-1
-6
-4
1
0
-4
-3
0
1
-2
-2
-1
2
0
-1
-2
3
2
0
-3
Find the x and y- intercepts
of x = 4y – 5
●
●
●
x-intercept:
Plug in y = 0
x = 4y - 5
x = 4(0) - 5
x=0-5
x = -5
(-5, 0) is the
x-intercept
●
●
y-intercept:
Plug in x = 0
x = 4y - 5
0 = 4y - 5
5 = 4y
5
=y
4
5
● (0, )
4
is the
y-intercept
Find the x and y-intercepts
of g(x) = -3x – 1*
●
●
●
x-intercept
Plug in y = 0
g(x) = -3x - 1
0 = -3x - 1
1 = -3x
1

=x
3
1
(  3 , 0) is the
x-intercept
*g(x) is the same as y
●
●
●
y-intercept
Plug in x = 0
g(x) = -3(0) - 1
g(x) = 0 - 1
g(x) = -1
(0, -1) is the
y-intercept
Find the x and y-intercepts of
6x - 3y =-18
●
●
●
x-intercept
Plug in y = 0
6x - 3y = -18
6x -3(0) = -18
6x - 0 = -18
6x = -18
x = -3
(-3, 0) is the
x-intercept
●
●
●
y-intercept
Plug in x = 0
6x -3y = -18
6(0) -3y = -18
0 - 3y = -18
-3y = -18
y=6
(0, 6) is the
y-intercept
Find the x and y-intercepts
of x = 3
●
●
x-intercept
Plug in y = 0.
There is no y. Why?
●
y-intercept
●A
vertical line never
crosses the y-axis.
●
There is no y-intercept.
x = 3 is a vertical line
so x always equals 3.
●
●
x
(3, 0) is the x-intercept.
y
Find the x and y-intercepts
of y = -2
●
x-intercept
Plug in y = 0.
y cannot = 0 because
y = -2.
● y = -2 is a horizontal
line so it never crosses
the x-axis.
●
●There
●
y-intercept
●
y = -2 is a horizontal line
so y always equals -2.
●
(0,-2) is the y-intercept.
x
is no x-intercept.
y
Graph by making a table
Graph
y
1
x 3
2
Select values from the domain and make a table. Then graph the order
pairs. Draw a line through the points
x
-2
0
2
1
x3
2
1
( 2)  3
2
1
(0)  3
2
1
( 2)  3
2
y
(x, y)
-4
-3
(-2, -4)
(0, -3)
-2
(2, -2)
Graph by making a table
Graph y   x  2
Select values from the domain and make a table. Then graph the order
pairs. Draw a line through the points
x
y  x  2
y
(x, y)
Questions??