P.o.D. – Find the inverse of each
function.
1.) f(x)=3x+1
2.) 𝑓(𝑥 ) = 𝑥 2
3.) 𝑓(𝑥 ) = √4 − 𝑥 2
1.) 𝑦 = 3𝑥 + 1 → 𝑥 = 3𝑦 + 1 →
𝑥−1
𝑥 − 1 = 3𝑦 →
=𝑦→
3
𝑥−1
−1 ( )
𝑓 𝑥 =
3
2.) 𝑦 = 𝑥 2 → 𝑥 = 𝑦 2 → √𝑥 = 𝑦 →
𝑓 −1 (𝑥 ) = √2
3.) 𝑦 = √4 − 𝑥 2 → 𝑥 = √4 − 𝑦 2 →
𝑥2 = 4 − 𝑦2 → 𝑦2 = 4 − 𝑥2 →
𝑦 = √4 − 𝑥 2 → 𝑓 −1 (𝑥 ) = √4 − 𝑥 2
1.10: Mathematical Modeling and
Variation
Learning Target: be able to write
mathematical models for direct variation
EX: The median purchase price (in
thousands of dollars) for homes in the
United States from 1990 to 1998 are
given by the following order pairs:
(1990,$131.2),
(1991,$134.3),(1992,$141.0),
(1993,$141.9), (1994, $145.4),
(1995,$147.7), (1996,$153.2),
(1997,$159.7), (1998,$167.9)
A linear model that approximates this
data is y=130.0+4.22t where y represents
the median purchase price and t=0
represents 1990. Plot the actual data and
the model on the same graph. How
closely do they relate?
The model is a good fit for
the data.
Let’s confirm the equation for the model
by using a Linear Regression.
Direct Variation:
- y varies directly as x
- can also be stated as “directly
proportional”
- y=kx for some nonzero constant k
- k is known as the constant of variation
EX: The simple interest on an investment
is directly proportional to the amount of
the investment. By investing $2500 in a
certain bond issue, you obtained an
interest payment of $187.50 at the end of
1 year. Find a mathematical model that
gives the interest I for this bond issue at
the end of 1 year in terms of the amount
invested P.
𝑦 = 𝑘𝑥 → 𝐼 = 𝑘𝑃 → 187.50 = 𝑘(2500)
187.5
→
= 𝑘 → .075 = 𝑘 →
2500
𝐼 = .075𝑃
Direct Variation as an nth Power:
- y varies directly as the nth power of x
- y is directly proportional to the nth
power of x
- 𝑦 = 𝑘𝑥 𝑛 for some constant k.
EX: Neglecting air resistance, the
distance s an object falls varies directly
as the square of the duration t of the fall.
An object falls a distance of 144 feet in 3
seconds.
a. Write an equation relating distance s
and duration t.
b. How far will an object fall in 6
seconds?
a. 𝑦 = 𝑘𝑥 𝑛 → 𝑠 = 𝑘𝑡 𝑛 →
144 = 𝑘(3)2 → 144 = 9𝑘 →
144
= 𝑘 → 16 = 𝑘 → 𝑠 = 16𝑡 2
9
b. 𝑠 = 16(6)2 = 16(36) = 576 𝑓𝑒𝑒𝑡
Inverse Variation:
- y varies inversely as x.
- y is inversely proportional to x
𝑘
- 𝑦 = for some constant k.
𝑥
EX: A company has found that the
demand for its product varies inversely
as the price of the product. When the
price is $2.75, the demand is 600 units.
a. Write an equation relating the
demand d and the price p
b. Approximate the demand when the
price is $3.25.
a. 𝑦 =
𝑘
𝑥
→𝑑=
𝑘
𝑝
→ 600 =
𝑘
2.75
→
600(2.75) = 𝑘 → 1650 = 𝑘 →
1650
𝑑=
𝑝
b. 𝑑 =
1650
3.25
= 507.6923 ≈ 508 𝑢𝑛𝑖𝑡𝑠
Joint Variation:
- z varies jointly as x and y
- z is jointly proportional to x and y
- z=kxy for some constant k
EX: The maximum load that can be
safely supported by a horizontal beam is
jointly proportional to the width of the
beam and the square of its depth, and
inversely proportional to the length of the
beam. Determine the change in the
maximum safe load under the following
conditions:
a. The width of the beam is doubled.
b. The depth of the beam is doubled.
a. Let m=maximum load, w=width,
d=depth, and b=length of the beam.
𝑚=
𝑘𝑤𝑑 2
𝑏
. If the width of the beam
is doubled, then 𝑚 =
2
𝑘𝑤𝑑 2
𝑏
𝑘(2𝑤)𝑑 2
𝑏
=
. The safe load is doubled.
b. 𝑚 =
𝑘𝑤(2𝑑)2
𝑏
=
𝑘𝑤(4𝑑 2 )
𝑏
=4
𝑘𝑤𝑑 2
𝑏
.
The safe load is quadrupled, or 4
times as great.
HW Pg. 109 2,8, 28, 30, 40, 46, 60, 66,
82-86E