Transcript Chapter 1 lecture slides

Chapter 1
Linear Functions
1- 1
Section 1.1
Slopes and
Equations of
Lines
1- 2
Figure 1
Figure 2
Your Turn 1
Find the slope of the line through (1,5) and (4,6).
Solution: Let ( x1 , y1 )  (1,5) and ( x2 , y2 )  (4,6).
Use the definition of the slope m 
y
.
x
65

4 1
1

3
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).
0  6 6 3


4  0 4 2
Substituting these values into y = mx + b, we have
m
3
y  x  6.
2
Your Turn 3
Find the slope of the line whose equation is 8 x  3 y  5.
Solution: To find the slope, solve the equation for y.
8x  3 y  5
3 y  8 x  5
Subtract 8 x from both sides.
8
5
y   x  Divide both sides by 3.
3
3
The coefficient of x is  8 / 3, which is the slope of the line.
Your Turn 4
Find the equation of the line through (2,9) and (5,3).
Put your answer in slope-intercept form.
Solution: Begin by using the definition of slope of the line that
passes through the given points.
3  9 6
m

 2
52 3
Either (2,9) or (5,3) can be used in the point-slope form with m  2.
If ( x1 , y1 )  (2,9), then
y  y1  m( x  x1 )
y  9  2( x  2)
y1  9, m  2, x1  2
y  9  2 x  4
Distributive Property
y  2 x  13
Add 9 both sides
Your Turn 5
Find the equation of the line that passes through the point (4,5)
and is parallel to the line 3 x  6 y  7.
Solution: The slope of 3 x  6 y  7 can be found by writing the
equation in slope-intercept form.
3x  6 y  7
6 y  3 x  7
3
7
y
x
6
6
1
7
y  x
2
6
subtract 3 x from both sides and
and divide both sides by  6.
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.
continued
Your Turn 5 continued
This line passes through (4,5). Substituting m  1/ 2,
x1  4 and y1  5 into the point-slope form gives
y  y1  m( x  x1 )
This line passes through (4,5). Substituting m  1/ 2,
x1  4 and y1  5 into the point-slope form gives
y  y1  m( x  x1 ).
1
y  x25
2
1
y  x3
2
Add 5 to both sides.
Your Turn 6
Find the equation of the line that passes through the point (3, 2)
and is perpendicular to the line 2x  3 y  4.
Solution: To find the slope, write 2x  3 y  4 in
slope-intercept form:
2x  3y  4
3 y  2 x  4
subtract 2x from both sides and
2
4
y
x
and divide both sides by 3.
3
3
The slope is  2 / 3. Since the lines are perpendicular, if line L
2
has slope m, then
m  1
3
m  3 / 2.
continued
Your Turn 6 continued
Now subtitute m  3 / 2, x1  3 and y1  2 into the point-slope form.
y  y1  m( x  x1 )
3
y  2  ( x  3)
2
3
9
y2  x
2
2
3
9
y  x  +2
2
2
3
5
y  x
2
2
Distributive property
Add 2 to both sides and get a
common denominator.
Section 1.2
Linear Functions
and Applications
1 - 19
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
p  D(q)  9  0.75q.
Demand function
(a) Find the quantity of watermelons demanded at a price of
$3.30 per watermelon.
Solution: 3.30  9  0.75q
 5.70  0.75q
Subtract 9 from both sides.
7.6  q
Divide both sides by  0.75.
Thus, at a price $3.30, the 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
p  S (q)  0.75q.
Supply function
(b) Find the quantity of watermelons supplied at a price of
$3.30 per watermelon.
Solution: 3.30  0.75q
4.4  q
Divide both sides by 0.75.
Thus, at a price $3.30, the quantity supplied is 4400 watermelons.
Your Turn 3
Find the equilibrium quantity and price for the watermelons
using the demand equation D(q)  10  0.85q
and the supply equation
S (q)  0.4q.
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.
10  0.85q  0.4q
10  1.25q
Add 0.85q to both sides.
8q
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
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,
p  D(8)  10  0.85(8)  3.2.
The equilibrium price is $3.20.