Download y - math-clix

Section 1.3
Linear Functions,
Slope, and
Applications
Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.
Objectives





Determine the slope of a line given two points on the
line.
Solve applied problems involving slope.
Find the slope and the y-intercept of a line given the
equation y = mx + b, or f (x) = mx + b.
Graph a linear equation using the slope and the
y-intercept.
Solve applied problems involving linear functions.
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.
Examples
Linear Function
y = mx + b
Identity Function
y = 1•x + 0 or y = x
Examples
Constant Function
y = 0•x + b or y = b
Not a Function
Vertical line: x = a
Horizontal and Vertical Lines
Horizontal lines are given by Vertical lines are given by
equations of the type y = b equations of the type x = a.
or f(x) = b. They are
They are not functions.
functions.
x=2
y=2
Slope
The slope m of a line containing the points (x1, y1) and
(x2, y2) is given by
rise
m
run
the change in y

the change in x
y2  y1 y1  y2


x2  x1 x1  x2
Example
Graph the function f (x)   2 x  1 and determine its slope.
3
Calculate two ordered pairs, plot the points, graph the
function, and determine its slope.
2
f (3)   (3)  1  2  1  1
3
2
f (9)    9  1  6  1  5
3
y2  y1
m
x2  x1
5  1 4
2



93
6
3
Types of Slopes
Positive—line slants up
from left to right
Negative—line slants down
from left to right
Horizontal Lines
If a line is horizontal, the change in y for any two points is
0 and the change in x is nonzero. Thus a horizontal line
has slope 0.
Vertical Lines
If a line is vertical, the change in y for any two points is
nonzero and the change in x is 0. Thus the slope is not
defined because we cannot divide by 0.
Example
Graph each linear equation and determine its slope.
a. x = –2
Choose any number for y ;
x must be –2.
x
y
2
0
2
2
2
2
Vertical line 2 units to the left of the y-axis. Slope is
not defined. Not the graph of a function.
Example (continued)
Graph each linear equation and determine its slope.
5
b. y 
2
Choose any number for x ; y
x
y
must be 5/2.
2
5/2
0
5/2
2
5/2
Horizontal line 5/2 units above the x-axis. Slope
0. The graph is that of a constant function.
Applications of Slope
The grade of a road is a number expressed as a percent
that tells how steep a road is on a hill or mountain. A 4%
grade means the road rises 4 ft for every horizontal
distance of 100 ft.
Example
Construction laws regarding access ramps for the
disabled state that every vertical rise of 1 ft requires a
horizontal run of 12 ft. What is the grade, or slope, of
such a ramp?
1
m
12
m  0.083  8.3%
The grade, or slope, of the ramp is 8.3%.
Average Rate of Change
Slope can also be considered as 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.
Example
The percent of travel bookings online has increased from
6% in 1999 to 55% in 2007. The graph below illustrates
this trend. Find the average rate of change in the percent
of travel bookings made online from 1999 to 2007.
Example
The coordinates of the two points on the graph are (1999,
6%) and (2007, 55%).
Change in y
Slope  Average rate of change 
Change in x
55  6
49
1


6
2007  1999
8
8
The average rate of change over the 8-yr period was
1
an increase of 6 % per year.
8
Slope-Intercept Equation
The linear function f given by f (x) = mx + b is written in
slope-intercept form. The graph of an equation in this
form is a straight line parallel to f (x) = mx.
The constant m is
called the slope,
and the y-intercept
is (0, b).
Example
Find the slope and y-intercept of the line with equation
y = – 0.25x – 3.8.
y = – 0.25x – 3.8
Slope = –0.25;
y-intercept = (0, –3.8)
Example
Find the slope and y-intercept of the line with equation
3x – 6y  7 = 0.
We solve for y:
3x  6y  7  0
6y  3x  7
1
1
 (6y)   (3x  7)
6
6
1
7
y x
2
6
1
7

Thus, the slope is
and the y-intercept is  0,   .

2
6
Example
2
Graph y   x  4
3
The equation is in slope-intercept
form, y = mx + b.
The y-intercept is (0, 4). Plot this
point, then use the slope to locate
a second point.
rise change in y 2  move 2 units down
m


run change in x
3  move 3 units right
Example
An anthropologist can use linear functions to estimate the
height of a male or a female, given the length of the
humerus, the bone from the elbow to the shoulder. The
height, in centimeters, of an adult male with a humerus of
length x, in centimeters, is given by the function
M x   2.89x  70.64
The height, in centimeters, of an adult female with a
humerus of length x is given by the function .
F x   2.75x  71.48
A 26-cm humerus was uncovered in a ruins.
Assuming it was from a female, how tall was she?
Example(cont)
We substitute into the function:
f 26   2.75 26   71.48
 142.98
Thus, the female was 142.98 cm tall.