# Linear Function - Biloxi Public Schools

```Bell Ringer
1) If (x) = 3x + 2, then what is the
solution of f(2). Hint: substitute
2 in for x.
2) If f(x) = 2x2 – 3x + 4, then what
is f(3), or what’s the solution
when you substitute 3 in for x?
Linear Functions
Function Notationa linear function written in the form
y = mx + b where y is written as a function f.
x-coordinate
as ‘f of x’
f(x) = mx + b
slope
y-intercept
f(x) is another name for y.
It means “the value of f at x.”
g(x) or h(x) can also be used to name functions
Domain and Range
= values of ‘x’ for which the
function is defined.
 Domain
= the values of f(x) where ‘x’
is in the domain of the function f.
 Range
 The
graph of a function f is the set
of all points (x, f(x)).
Graphing a Function
 To
graph a function:
(1)
make a table by substituting into the
function.
(2) plot the points from your table and
connect the points with a line.
(3) identify the domain and range, (if
restricted)
Graph a Function
Graph the Function
SOLUTION
STEP 1
STEP 2
Make a table by
choosing a few values
for x and then finding
values for y.
-2 -1
STEP 3
Plot the points. Notice
the points appear on a
line. Connect the points
drawing a line through
them.
f ( x)  2 x  3
x
f(x) = 2x – 3
0
1
2
f(x) -7 -5 -3
-1
1
f ( x)  2 x  3
The domain and
range are not
restricted
therefore, you do
not have to
identify.
Graph a Function
1
–
Graph the function f(x) = 2 x + 4 with domain x ≥ 0.
Then identify the range of the function.
STEP 1
x
0
2
4
6
8
Make a table.
y
4
3
2
1
0
STEP 2
Plot the points.
Connect the points with a ray
because the domain is restricted.
f ( x)  
1
x4
2
STEP 3
Identify the range. From the graph, you can see that all points have
a y-coordinate of 4 or less, so the range of the function is y ≤ 4.
Family of Functions
is a group of functions with
similar characteristics. For
example, functions that have
the form f(x) = mx + b
constitutes the family of linear
functions.
Parent Linear Function

The most basic linear function in the family of all
linear functions is called the PARENT LINEAR
FUNCTION which is:
f(x) = x
f(x) = x
x
-5
-2
0
1
3
f(x)
-5
-2
0
1
3
Real-Life Functions
A cable company charges new
customers \$40 for installation and \$60
per month for its service. The cost to
the customer is given by the function
f(x) = 60x +40 where x is the number of
months of service. To attract new
customers, the cable company reduces
the installation fee to \$5. A function for
the cost with the reduced installation
fee is g(x) = 60x + 5. Graph both
functions. How is the graph of g related
Real-Life Functions
The graphs of both functions are shown. Both
functions have a slope of 60, so they are parallel.
The y-intercept of the graph of g is 35 less than
the graph of f. So, the graph of g is a vertical
translation of the graph of f.
Write an Equation Given a
Slope and a Point
Write the Equation using Point-Slope Form
1
m   and (1,6)
3
Step 1: Plug it in
y  y1  m( x  x1 )
Point- Slope Form
A Challenge
Write the equation of a line in point-slope form that
passes through (-3,6) and (1,-2)
Hint: Find the change in y and the change in x.
Change is determined by subtraction.
Reminder that slope is rise or run.
A Challenge
Can you write the equation of a line in point-slope
form that passes through (-3,6) and (1,-2)
6+2 =- 8 =- 2
m=
-3 – 1
4
1
y – y1 = m(x – x1)
y + 2 = - 2 (x – 1)
Calculate the slope.
Use m = - 2 and the point (1, -2).
Point-Slope Form
y = -2(x – 1) - 2
y = -2x + 2 - 2
y = -2x + 0
Slope-Intercept Form
y = -2x
Equations of Parallel Lines
Write an equation for the line that contains
(5, 1) and is parallel to
3
m=
5
3
y  1   x  5
5
3
y  x4
5
Equations of Perpendicular Lines
Find the equation of the line that contains
(0, -2) and is perpendicular to y = 5x + 3
1
m=
5
1
y2 x
5
```