SECTION 4.1 – 4.2
LINEAR FUNCTIONS AND
APPLICATIONS
LINEAR FUNCTIONS
A function f is a linear function if it can be written as 𝑓 𝑥 = 𝑚𝑥 + 𝑏,
where m and b are constants.
If m = 0, the function is a constant function 𝑓 𝑥 = 𝑏
If m = 1 and b = 0, the function is the identity function 𝑓 𝑥 = 𝑥
*The graph of a linear function is a line.
Linear Function
Constant Function
Identity Function
𝑓 𝑥 = 4𝑥 − 5
𝑓 𝑥 =6
𝑓 𝑥 =𝑥
𝑓 𝑥 = −3𝑥 + 1
𝑓 𝑥 = −7
𝑓 𝑥 = 2𝑥
𝑓 𝑥 =0
Horizontal Line
Diagonal in I and III quadrants
ZEROS, SOLUTIONS, 𝑥-INTERCEPTS
The following are equivalent:
• The 𝑥-intercepts of a graph
• The real solutions to 𝑓 𝑥 = 0
• The real zeros of 𝑓(𝑥).
ZEROS, SOLUTIONS, 𝑥-INTERCEPTS
1. Find the zero of 𝑓 𝑥 = 3𝑥 − 7
𝑓 𝑥 =0
3𝑥 − 7 = 0
3𝑥 = 7
7
𝑥=
3
𝑥-intercept:
7
,0
3
Soultion to 𝑓 𝑥 = 0: 𝑥 =
Zero:
7
3
7
3
LINEAR FUNCTIONS
1
𝑑
33
2. Pressure at Sea Depth. The function 𝑃, given by 𝑃 𝑑 =
+ 1, gives
the pressure, in atmospheres (atm) at a depth 𝑑, in feet under the sea.
(a) Find the value an explain the meaning of each of the following.
1
𝑃 0 = 33 (0)  1  1
𝑃 10 =
At the surface there is 1 atm of pressure
1
(33)  1  2
𝑃 33 =
33
1
(10)  1  1  1.3
33
At 10 ft below the surface there is 1.3 atm of pressure
1
(200)  1  7.1
𝑃 200 =
33
At 33 ft below the surface there is 2 atm of pressure
At 200 ft below the surface there is 7.1 atm of pressure
(b) What is the domain for this function? 𝑑 ≥ 0
AVERAGE RATE OF CHANGE OF LINEAR FUNCTION
A slope is an average rate of change. To find the average
rate of change between any two data points on a graph, we
determine the slope of the line that passes through the two
points.
change in y
m
change in x
y
m
x
In words, you will be asked to
“find the (average) rate of change in OUTPUT (OVER INPUT)”
AVERAGE RATE OF CHANGE OF LINEAR FUNCTION
**The average rate of change is
CONSTANT for linear functions, i.e.
you can choose any two points on
the line and the slope does not
change!
y
m
x
AVERAGE RATE OF CHANGE OF LINEAR FUNCTION
3. Decline in Teen Smoking. The percent of 10th grade students who have
smoked daily in the last 30 days has greatly decreased, from 16.3% in 1995 to
8.3% in 2004. Find the average rate of change over the 9-yr period in the
percent of 10th grade students who have smoked daily in the last 30 days.
The average rate of change in y over x.
The average rate of change in % over yrs.
( yrs, %)
1995,16.3
(2204, 8.3)
%
16.3  8.3

yr 1995  2004
8

 0.89% / yr
9
m
Over the 9-yr period, there was a 0.89%/yr. decrease.
LINEAR FUNCTIONS
4. Phone Bills For interstate calls, AT&T charges 7 cents per minute
plus a base charge of $4.95 each month. Write an expression for the
monthly charge y as a linear function of the number of minutes of use.
𝑥 = number of minutes used in a month
𝑦 = 𝑓(𝑥) = monthly charge, $
y  f ( x)  mx  b
y
$
m

x  min
m  0.07
b  4.95
𝑓 𝑥 = 0.07𝑥 + 4.95
LINEAR FUNCTIONS
5. Cost A company buys and retails baseball caps. The total cost
function is linear. The total cost for 200 caps is $2680 and the total cost
of 500 caps is $3530. Model this cost function.
𝐶 𝑥 = 𝑚𝑥 + 𝑏, (#𝑐𝑎𝑝𝑠, 𝑡𝑜𝑡𝑎𝑙 𝑐𝑜𝑠𝑡$)
m  2.83
(200, 2680)
(500, 3530)
2680  3530
200  500
 850

 2.83
 300
m
y  2680  2.83( x  200)
y  2680  2.83 x  566
y  2.83 x  2114
𝑓 𝑥 ≈ 2.83𝑥 + 2114