Fst ch2 notes

2-1 The Language of Functions
univariate data:
data involving a single variable
data involving two variables
bivariate data:

mathematical description of a real situation
mathematical model:

relation:
any set of ordered pairs
function:
a set of ordered pairs (x,y) in which each value
of x is paired with exactly one value of y

independent variable (typically x):
the first variable in a relation (typically x)
dependent variable (typically y):
the second variable in a relation (typically y)

domain (think x values):
the first value in a set of ordered pairs
range (think y values):
the second value in a set of ordered pairs

vertical line test:
any vertical line drawn through a function will
intersect the graph in no more than one point

argument:
the variable x is sometimes called
the argument of a function

2-1 Examples
a. Which is true, “cost is a function of miles driven” or
“miles driven is a function of cost”?
b. Identify the independent and dependent variables.
c. State the domain and range of the function.
1. The cost of renting a car is $35 per day plus
$0.40/mile for mileage over 100 miles.

2-1 Examples
a. List three ordered pairs in the function.
b. Write an equation to express the relation between
miles and cost.
c. Graph the function.
2. Use the information in #1 above, assuming m 100≥
and the car is rented for one day only.

2-1 Examples
rule
whose
domain is the
set of real
numbers.
y = 2ξ2
+10

2-1 Examples
4. Supposef (t) = (2 − 3τ)(τ+ 6)
a. Evaluate f(1), f(2), and f(3).
b. Does f(1) + f(2) = f(3)?

2-1 Examples
5. If
, find
the following:
h(x) = ξ2
−
1
ξ
a. h(-2)
b. h(4t)
c. h(a+5)

2-2 Linear Models & Correlation
linear function:
a set of ordered pairs (x,y) that can be described by an
equation of the from y=mx+b where m and b are constants

linear regression:
the method of finding a line of best fit to a set of points

correlation coefficient:
a measure of the strength of the linear relation
between two variables, denoted by “r”.
perfect correlation:
r=1 or r=-1

positive relation:
a positive relation between variables
think “positive slope”
negative relation:
a negative relation between variables
think “negative slope”

strong relation:
a relation for which most of
the data fall close to a line
weak relation:
a relation for which most of
the data do not fall close to a line

2-2 Examples
a. Find a linear model between the cost and the rating.
b. Use the model to predict the rating of a $45 coffee maker.
c. How much faith do you have in the estimate in part b?
d. Is the prediction in part b interpolation or extrapolation?
1. The October, 1994, Consumer Reports listed the following
prices and overall ratings for drip coffee makers:
($27, 79), ($25, 77), ($60, 70), ($50, 66), ($22, 61), ($60, 61), ($35, 61), ($20, 60), ($35, 58),
($40, 54), ($22, 53), ($40, 51), ($30, 43), ($30, 35), ($20, 34), ($35, 32), ($19, 28).

2-2 Examples
a. Find a linear model for y in terms of w.
b. What is the correlation between y and w?
c. At this rate of weight loss, when will she reach her goal
of 132 pounds?
d. Is this prediction interpolation or extrapolation?
2. A 140-pound 5’2” tall woman diets, wishing to lose 0.5
pounds per week. Let y be her weight after w weeks,
where 0 ≤ w 20.≤

2-3 The Line of Best Fit
observed values:
data collected from experiments or surveys
predicted values:
points predicted by linear models
expected values:

errors in prediction:
deviations:
the differences between observed and predicted values
observed - predicted

line of best fit:
the line with the smallest value for
the sum of the squares of the errors

method of least squares:
the process of finding the line of best fit

center of gravity:
are the
mean of
the
observed
x-values
and the
mean of
the
observed
y-values
(x,y)

interpolation:
prediction between known values
extrapolation:
prediction beyond known values (more hazardous)

2-3 Examples
1. Find an equation for the line of best fit through the data
points.
2. Verify that the center of gravity of the four given points
is on the line of best fit.
3. Find the sum of the squares of the errors for the line of
best fit for these points.
In August 1990, Consumer Reports listed the following average
prices per pound for four grades of raw hamburger: regular
(72.5% lean), $1.57; chuck (80% lean), $1.85; round (85%
lean), $2.38; sirloin (92.5% lean), $2.93.

QuickTime™ and a
decompressor
are needed to see this picture.

2-4 Exponential Functions
exponential function:
where a≠0,
b>0, and b≠1.
a = beginning
amount
b= growth
factor
f (x) = αβξ

base:
the number
“b” in
f (x) = αβξ
Do you remember what “a” represents?

exponential growth function:
when b>1 in
f (x) = αβξ
exponential growth curve: exponential decay curve:

growth rate:
the factor by which a quantity changes during a
given time period. It is the “b” value.

strictly increasing/decreasing:
increasing (growth) decreasing (decay)
b>1 0<b<1

asymptote (a-simp-tote):
a line that the curve approaches but never touches
Example 1Example 2

2-4 Examples
1. With these assumptions, give a formula for the U.S.
population x years after 1995.
2. From the formula, estimate the population in 2010.
1. In 1995, the U.S. population was estimated at 264,000,000
people and was predicted to grow about 0.9% a year for the
near future.

2-4 Examples
functions f,
g,
and h,
where
,
, and
for all
values of x.
f (x) = 6ξ
g(x) = (
1
6
)ξ
h(x) = 6− ξ

2-5 Exponential Models
exponential model:
where
a≠0, b>0
and b≠1
y = αβξ

initial value:
growth factor:
the “a”
value in
y = αβξthe “b”
value in
y = αβξ

doubling time:
how long it takes for a quantity to double;
growth factor b = 2
half-life:
how long it takes for a quantity to decay to
half its original amount
growth factor b = .5

2-5 Examples
1. The population of a certain cell type was observed to be 100
on the second day, and 2700 on the fifth day. Assuming the
growth is exponential, find the number of cells present initially,
and the number of cells expected on the seventh day.

2-5 Examples
a. Give a formula for A(t), the amount the note is worth
after t years.
b. How much is in the account after 15 years?
3. Use the compound interest formula to
answer the following questions: $1500 is put in a treasury note
paying 5.5%.
A(t) = Π(1+ ψ)
τ

2-6 Quadratic Models
quadratic model:
a model based on quadratic functions

parabola:
a graph of a quadratic function
Show
me
y = ξ2

maximum point:
if the parabola opens down, it has a maximum
minimum point:
if the parabola opens up, it has a minimum
the point is called the vertex

acceleration due to gravity:
g = 32 φτ/ σεχ2
ορ 9.8µ / σεχ2

quadratic regression:
finding an equation for the best fitting
parabola through a set of points
minimum of 3 sets of points needed

impressionistic model:
non-theory-based model:
no theory exists that explains why the
model fits the data

2-6 Examples
a. Find its x- and y-intercepts.
b. Tell whether the parabola has a maximum or minimum
point and find its coordinates.
1. Consider the function f with rule .f (x) = 2ξ2
− 9ξ + 3

2-6 Examples
2. A projectile is shot from a tower 10 feet high with an
upward velocity of 100 feet per second.
a. Approximate the relationship between height h (in feet) and
time t (in seconds) after the projectile is shot.
b. How long is the projectile in the air?
h(t) = −
1
2
γτ2
+ ϖ0τ+ η0
h(t) = −
1
2
(32)τ2
+100τ+10
h(t) = −16τ2
+100τ+10
Hint: find the 2nd x-intercept since x represents time

2-6 Examples
3. A student was doodling and drew the following patterns of
dots. In counting the number of dots, the student found there
were 1, 5, 13, and 25 dots. Let y be the number of dots in the
xth pattern. Find a quadratic model linking x and y. (If needed,
give the following hint: The graph must contain the points (1,
1), (2, 5), (3, 13), and (4, 25).
Hint: quadratic model … hmmm, maybe a calculator!

2-7 Step Functions
step function:
a function whose graph looks like steps

2-7 Step Functions
greatest integer function:
rounding down function:
floor function:
numb
er x,
f(x) is
the
great
est
integ
er
less
than
or
equal
to x.
Use
this
symb
x or INT x (on calculator)
HUH??

2-7 Step Functions
rounding-up function:
ceiling function:
r
xwith
the
smalle
st
intege
r
greate
r than
or
equal
to x.
Use
this
symbo
l:
x  But, how does
it work?

2-7 Step Functions
discontinuous:
the graph cannot be drawn without lifting your
pencil off the paper
point of discontinuity:
the value of x at which you lift your pencil
continuous:
a graph that has no point of discontinuity

2-7 Examples
1. Evaluate each expression.
a. 33.4 
β. ΙΝΤ(2π)
χ. − 7



δ. 100.4 + 98.3+ 6.4 
34
6
-3
205

2-7 Examples
3. How many buses b are needed to transport s students if each
bus can hold 44 students and no other means of transportation
is used?
Think: how many buses are needed to transport 50 students?
50
44
= 1.14
However, we can’t have 1.14 buses, so we must round up! So …
b =
σ
44





2-7 Examples
4. Suppose it costs $50 to rent a bus in the above situation.
What will it cost to transport 300 students?
b =
300
44




b =
σ
44




b = 6.82 
b = 7 βυσεσ
7 * $50 per bus = $350

Fst ch2 notes

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Fst ch2 notes