# Intercepts

```Intercepts
An Introduction
The y-intercept of a function is the ordered
pair when x is 0.
An x-intercept of a function is any ordered pair
when y is 0.
Although the x-intercept and the y-intercept
are ordered pairs, in practice
 0, 8  people often
refer to just the x-coordinate of the x intercept
or just the y-coordinate of the y-intercept.
  6, 0 
Time traveling
in car
(hours)
Distance from
home
(miles)
0
0
288
288
2.25
172
4.5
81
6
6
0
What is the general meaning of the y-intercept?
What is the specific meaning of the y-intercept?
What is the general meaning of any x-intercept?
What is the specific meaning of any x-intercepts?
Intercepts
An Introduction
Applying Intercepts
Graphs and Tables
C a rs
W a sh e d
0
4
6
8
12
P ro fit
(d o lla rs)
 60
 20
0
20
60
What is the general meaning of the y-intercept?
What is the specific meaning of the y-intercept?
What is the general meaning of any x-intercept?
What is the specific meaning of any x-intercepts?
What is the general meaning of the y-intercept?
What is the specific meaning of the y-intercept?
What is the general meaning of any x-intercept?
Why won’t there be any x-intercepts?
What is the general meaning of the y-intercept?
What is the specific meaning of the y-intercept?
What is the general meaning of any x-intercept?
What is the specific meaning of each x-intercept?
What is the general meaning of the y-intercept?
What is the specific meaning of the y-intercept?
What is the general meaning of any x-intercept?
Why won’t there be any x-intercepts?
What is the general meaning of the y-intercept?
What is the specific meaning of the y-intercept?
What is the general meaning of any x-intercept?
What is the specific meaning of each x-intercept?
Applying Intercepts
Graphs and Tables
Graph the data and extend the line to estimate the
intercepts. Use 0 to 20 for x values and 0 to 1000 for y
values. Discuss the meaning of any intercept or explain
why the intercept doesn’t exist.
M o n th s sin ce co u ch
w as bought
R e sa le v a lu e
(D o lla rs)
4
600
5
550
10
300
Graph the data and extend the line to estimate the
intercepts. Use 0 to 20 for x values and 0 to 100 for y
values. Discuss the meaning of any intercept or explain
why the intercept doesn’t exist.
M o n th s sin ce co u ch
w as bought
R e sa le v a lu e
(D o lla rs)
4
600
5
550
10
300
Graph the data and extend the line to estimate the
intercepts. Discuss the meaning of any intercept
or explain why the intercept doesn’t exist.
A grade school is putting on a carnival. When people
enter they buy tokens which they use to play the
carnival games. Scale the x-axis from 0 to 2000 and
the y-axis from -400 to 400.
T o k e n s so ld
P ro fit m a d e (\$ )
400
100
700
50
1000
200
Graph the data and extend the line to estimate the
intercepts. Discuss the meaning of any intercept or
explain why the intercept doesn’t exist. Scale the xaxis from 0 to 80 and the y-axis from 0 to 80.
M in u te s
p la yin g th e
slo t m a ch in e
M o n e y le ft
(D o lla rs)
10
64
25
4 7 .5 0
40
31
Applying Intercepts
Equations
Part 1
To find the y-intercept of an equation substitute 0 for x
and solve for y.
To find the x-intercept(s) of an equation substitute 0 for
y and solve for x.
The function
y  6 .5 x  2 0
returns y, the profit (in dollars) made from
a car wash, given x, the number of cars that have been washed.
W h a t's th e sp e cific m e a n in g
o f th e y-in te rce p t?
W h a t's th e sp e cific m e a n in g
o f th e x-in te rce p t?
y  6 . 5 x  20
y  6 . 5 x  20
y  6 . 5  0   20
0  6 . 5  x   20
y   20
3 . 077  x
Before any cars are washed
the loss is \$20.
To make up the \$20 loss
(your profit is \$0) you need to
wash 4 cars.
The function y  2 x  13 estimates the percent of one-person
households in the U.S., y, using x, the year since 1960.
What is the general meaning of the y-intercept?
What is the specific meaning of the y-intercept?
What is the general meaning of any x-intercept?
Why won’t there be any x-intercepts?
y  2 x  13
y  2 x  13
O per
y  2 0  13
In v
^2
y  0  13
2
2
y  13
13
13
In 1960 around 13%
of households were
one-person
households.
0  2 x  13
0  2 x  13
 13
 13
0  2 4 2 .2 5  1 3
13  2 x
2
2
 6 .5 
  6 .5 
2

0  18 . 09 ...
x

42 . 25  x
0  2 6 .5  1 3
x

2
You can find y, the income you make on a product
(in \$10,000) by supplying x, the price you plan to
charge for the product (in dollars) using the
function y   5( x  4 ) 2  80 .
What is the general meaning of the y-intercept?
What is the specific meaning of the y-intercept?
What is the general meaning of any x-intercepts?
What is the specific meaning of the x-intercepts?
y   5( x  4 )
y   5( 0  4 )
2
2
 80
 80
y   5 1 6   8 0
y  80  80
y  0
Setting the price to \$0
leads to an income of \$0.
O p e r In v
4
4
^2

5 5
 80  80
Setting the income to
\$0 leads to a price of
\$0 or \$8.
y   5( x  4 )
0   5( x  4 )
2
2
 80
 80
 8 0   5( x  4 )
16   x  4 
2
2
 16  x  4
44  x
x  8 or x  0
Applying Intercepts
Equations
Part 1
Applying Intercepts
Equations
Part 2
y
The function y  3 1,5 0 0 e  0 . 15 x predicts the current
value for a hybrid car of age x.
What is the general meaning of the y-intercept?
What is the specific meaning of the y-intercept?
What is the general meaning of any x-intercepts?
What is the specific meaning of the x-intercepts?
y  3 1, 5 0 0 e
y  3 1, 5 0 0 e
 0 . 15 x
 0 . 15 ( 0 )
y  3 1, 5 0 0 e 0
y  3 1, 5 0 0 (1 )
y  31, 500 e
O per
In v
  0 .1 5   0 .1 5
e^
ln
 3 1, 5 0 0  3 1, 5 0 0
0  31, 500 e
0

The value doesn’t go to \$0.
The car initially cost
\$31,500.
e
0  e
 0
 0 . 15 x
3 1, 5 0 0 e
3 1, 5 0 0
y  3 1, 5 0 0
 0 . 15 x
 0 . 15 x
3 1, 5 0 0
 0 . 15 x
ln 0  ln e
 0 . 15 x
e rro r   0 . 1 5 x
x
y
The number of years since 1985 can be used to predict the
pounds of beef consumed per person per year in the United
States using the formula y   5 . 5 ln ( x  1)  79 .
What is the general meaning of the y-intercept?
What is the specific meaning of the y-intercept?
What is the general meaning of any x-intercepts?
What is the specific meaning of the x-intercepts?
y   5 . 5 ln ( x  1)  7 9
y   5 . 5 ln ( 0  1)  7 9
y   5 . 5 ln (1)  7 9
y   5 .5  0   7 9
y  79
In 1985 approximately
79 pounds of beef was
consumed per person.
y   5 . 5 ln ( x  1)  79
O per
In v
1
1
ln
e^
  5 .5   5 .5
79
79
0   5 . 5 ln ( x  1)  79
 79
 79
 79   5 . 5 ln ( x  1)
 5 .5
 5 .5
 79  5 . 5  ln ( x  1)
The pounds of beef
consumed will not
drop to 0.