1 - 1 Chapter 1 Linear Functions. 1 - 2 Section 1.1 Slopes and Equations of Lines.
-
Upload
rosamond-flowers -
Category
Documents
-
view
218 -
download
2
Transcript of 1 - 1 Chapter 1 Linear Functions. 1 - 2 Section 1.1 Slopes and Equations of Lines.
Your Turn 1
Find the slope of the line through (1,5) and (4,6).
1 1 2 2 Let ( , ) (1,5) andSol ( ,ut )ion: (4,6). x y x y
Use the definition of the slope .y
mx
6 54 1
13
Your Turn 2
Find the equation of the line with x-intercept − 4 and
y-intercept 6.
Solution: Notice that b = 6.
To find m, use the definition of the slope after writing the
x-intercept as (− 4, 0) and y-intercept as (0,6).
Substituting these values into y = mx + b, we have
0 6 6 34 0 4 2
m
36.
2y x
Your Turn 3
Find the slope of the line whose equation is 8 3 5.x y
To find the slope, solve the equatiSolution on f: or y.
8 3 5x y
Subtract 8 from3 8 5 both sides. y x x
Divide both si8 5
3
des b 33
y . y x
The coefficient of is 8 / 3, which is the slope of the line. x
Your Turn 4
Find the equation of the line through (2,9) and (5,3).
Put your answer in slope-intercept form. Begin by using the definition of slope of the line that
passes through the give
So
n
lution
poi
:
nts.3 9 6
25 2 3
m
Either (2,9) or (5,3) can be used in the point-slope form with 2.m
1 1
1 1
1 1
If ( , ) (2,9), then
( )
9 2( 2)
9, 2, 2 y x
x y
y y m x x
y m x
9 2 4 Distributiv e Propertyy x
2 13 Add 9 b ot h sidesy x
Your Turn 5
Find the equation of the line that passes through the point (4,5)
and is parallel to the line 3 6 7.x y The slope of 3 6 7 can be found by writing the
equation in slope-int
Solutio
ercept form.
n
: x y
subtract 3 from both sides a
3 6 7
6 3 7
3 7
nd
and divid 6 6
e both sides by 6.
1 72 6
x
x y
y x
y x
y x
The result shows that the slope is 1/ 2. Since the lines
are parallel, 1/ 2 is also the slope of the line whose equation
we want. cont inued
Your Turn 5 continued
1 1
1 1
This line passes through (4,5). Substituting 1/ 2,
4 and 5 into the point-slope form give
(
s
)y
m
x y
y m x x
1 1
1 1
This line passes through (4,5). Substituting 1/ 2,
4 and 5 into the point-slope form giv s
( ).
e
y x
x
y
m
m x
y
12 5 Add 5 to both sid e .
2sy x
13
2y x
Your Turn 6
Find the equation of the line that passes through the point (3,2)
and is perpendicular to the line 2 3 4.x y To find the slope, write 2 3 4 in
slope-int
Solut
erce
ion
pt r
:
m:
fo
x y
subtract 2 from both sides
2 3 4
3 2 4
2 4
3 3
and
and divide both sides by 3.
x y
y x
y
x
x
The slope is 2 / 3. Since the lines are perpendicular, if line
2has slope , then 1
33 / 2. continued
L
m m
m
Your Turn 6 continued
1 1
1 1
Now subtitute 3 / 2, 3 and 2 into the point-slope
( )
form.
y y m
m x y
x x
32 ( 3)
2y x
3 92 Distri
2butive propert
2yy x
Add 2 to both sides and ge3 9
+2 2 23 5
t a
common de 2 2
nominator.
y x
y x
Your Turn 1
Let ( ) 4 5. Find ( 5).g x x g
To find ( 5), substituteSolution: 5 for .g x
( 5) 4( 5) 5 g
( 5) 20 5 25g
Your Turn 2(a) Suppose that Greg Tobin, manager of a giant supermarket
chain, has studied the supply and demand for watermelons. He
has noticed that the demand increases as the price decreases.
He has determined that the quantity (in thousands) demanded
weekly, q, and the price (in dollars) per watermelon, p, are
related by the linear function
(a) Find the quantity of watermelons demanded at a price of $3.30 per watermelon.
( ) 9 0.75 . Demand functionp D q q
Subtract 9 from both sides.
3.30 9 0.75
5.70 0.75
7.6 Divide both sides by 0.7
Thus, at
Sol
a
u
price $3.30,
t
ion:
the
5.
q
q
q
quantity demanded is 7600 watermelons.
Your Turn 2(b)
Greg also noticed that the quantity of watermelons supplied
decreased as the price decreased. Price p and supply q are
related by the linear function
(b) Find the quantity of watermelons supplied at a price of $3.30 per watermelon.
( ) 0.75 . Supply functionp S q q
Divide both sides by 0.75.
3.30 0.75
4.4
Thus, at a price $3.30, the quantity supplied is 4400 watermelons
S
.
olution: q
q
Your Turn 3 Find the equilibrium quantity and price for the watermelons
using the demand equation
and the supply equation
Solution: The equilibrium quantity is found when the prices
from both supply and demand are equal. Set the two
expressions for p equal to each other and solve.
The equilibrium quantity is 8000 watermelons.
The equilibrium price can be found by plugging the value of q = 8
into either the demand or the supply function.
Continued
( ) 10 0.85D q q ( ) 0.4 .S q q
10 0.85 0.4
10 1.25 Add 0.85 to both sides.
8
q q
q
q
q
Your Turn 3 continued
The equilibrium price can be found by plugging the value of q = 8
into either the demand or the supply function.
Using the demand function,
The equilibrium price is $3.20.
(8) 10 0.85(8) 3.2.p D
Your Turn 4
The marginal cost to make x batches of a prescription medication
is $15 per batch, while the cost to produce 80 batches is $1930.
Find the cost function C(x), given that it is linear.
Solution: Since the cost function is linear, it can be expressed in the
form C(x) = mx+b. The marginal cost is $15 per batch, which gives
the value for m. Using x = 80 and C(x) = 1930 in the point-slope
form of the line gives
( ) 1930 15( 80)
( ) 1930 15 1200
( ) 15 730 Ad d 1930 to both sides.
C x x
C x x
C x x
Your Turn 5A firm producing poultry feed finds that the total cost C(x) in
dollars of producing x units is given by
Management plans to charge $58 per unit for the feed.
How many units must be sold to produce a profit of $8030?
Solution: Since R(x) = p x and p = 58, R(x) = 58x.
Use the formula for profit P(x) = R(x) – C(x).
( ) 35 250.C x x
Solve for8030 58 (35 250)
8030 58 35 250
8030 23
250
.xx x
x x
x
8030 250 23
8280 23
Add 250 t
360
Sales of 360 units will produce $8030 profi
o both sides.
Divide both sides by 23.
t.
x
x
x