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

2-1 The Language of Functions

mathematical model:

2-1 The Language of Functions

relation:

any set of ordered pairs

function:

a set of ordered pairs (x,y) in which each valueof x is paired with exactly one value of y

2-1 The Language of Functions

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):

2-1 The Language of Functions

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:

2-1 The Language of Functions

any vertical line drawn through a function willintersect the graph in no more than one point

argument:

2-1 The Language of Functions

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

3. Find the range of the function with rule whose domain is the set of real numbers.

y = 2ξ2 +10

2-1 Examples4. 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 Examples5. If , find the following:h(x) = ξ2 −

1ξ

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 anequation of the from y=mx+b where m and b are constants

2-2 Linear Models & Correlation

linear regression:

the method of finding a line of best fit to a set of points

2-2 Linear Models & Correlation

correlation coefficient:

a measure of the strength of the linear relationbetween two variables, denoted by “r”.

perfect correlation:

r=1 or r=-1

2-2 Linear Models & Correlation

positive relation:

a positive relation between variablesthink “positive slope”

negative relation:

a negative relation between variablesthink “negative slope”

2-2 Linear Models & Correlation

strong relation:

a relation for which most ofthe data fall close to a line

weak relation:

a relation for which most ofthe 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:

2-3 The Line of Best Fit

errors in prediction:

deviations:

the differences between observed and predicted values

observed - predicted

2-3 The Line of Best Fit

line of best fit:

the line with the smallest value forthe sum of the squares of the errors

2-3 The Line of Best Fit

method of least squares:

the process of finding the line of best fit

2-3 The Line of Best Fit

center of gravity:

the coordina

tes of this point

are the mean of

the observed x-values and the mean of

the observed y-values

(x , y )

2-3 The Line of Best Fit

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:

a function with a

formula of the form:

where a≠0, b>0, and b≠1.

a = beginning amount

b= growth factor

f (x) = αβξ

2-4 Exponential Functions

base:

the number “b” in f (x) = αβξ

Do you remember what “a” represents?

2-4 Exponential Functions

exponential growth function:

when b>1 inf (x) = αβξ

exponential growth curve: exponential decay curve:

2-4 Exponential Functions

growth rate:

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

2-4 Exponential Functions

strictly increasing/decreasing:

increasing (growth) decreasing (decay)

b>1 0<b<1

2-4 Exponential Functions

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

2. Compare and contrast the graphs of the three functions f, g,

and h, where , , and

for all values of x.

f (x) = 6ξ g(x) = (16

)ξh(x) = 6− ξ

2-5 Exponential Models

exponential model:

has the form

where

a≠0, b>0 and b≠1

y = αβξ

2-5 Exponential Models

initial value:

growth factor:

the “a” value in y = αβξthe “b” value in y = αβξ

2-5 Exponential Models

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

2-6 Quadratic Models

parabola:

a graph of a quadratic function

Show me y = ξ2

2-6 Quadratic Models

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

2-6 Quadratic Models

acceleration due to gravity:

g = 32 φτ / σεχ2 ορ 9.8µ / σεχ2

2-6 Quadratic Models

quadratic regression:

finding an equation for the best fitting parabola through a set of points

minimum of 3 sets of points needed

2-6 Quadratic Models

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) = −12

γτ2 + ϖ0τ+ η0

h(t) = −12

(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:

the function f such that for

every real

number x,

f(x) is the

greatest

integer

less than

or equal to x. Use this

symbol:

x or INT x (on calculator) HUH??

2-7 Step Functions

rounding-up function:

ceiling function:

the functi

on which pairs each

number

xwith the

smallest

integer

greater than

or equal to x. Use this

symbol:

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 =30044

b =σ

44

b = 6.82

b = 7 βυσεσ

7 * $50 per bus = $350