Intro

Chapter 1

Section 1.1: What is a function?It is like a black box

InputThe input of a function is called the

“independent variable”We usually will use the letter x to refer to

the independent variable. Sometimes we will use the letter t but only if the independent variable is a measure of time.

The set of all values that the independent variable can take is called Domain.

OutputThe output of a function is called the

“dependent variable”We usually use the letter y to denote the

outputThe set of values that the output can take

is called Range

Section 1.2: Linear FunctionsFunctions that always

increase or decrease the same amount for each unit of the independent variable (x)

Functions that follow an equation of the form

Functions whose graph is straight line

bmxy

Linear function The quantities m and b determine the lineThe y-intercept is bThe slope is m

bmxy

The slopeIt is the most important characteristic of a

lineIt tells us how the function grows

Slope Positive

If , then the line is increasing

The bigger the slope, the steeper the line

0m

Slope negative

If , then the line is decreasing

The more negative the slope, the faster it decreases

0m

Slope Zero

If then the line is horizontal

0m

Line determined by 2 pointsGiven points find the slope and intercept of the line

passing through them.

),( 11 yx ),( 22 yx

12

12

xx

yym

22

11

xmyb

or

xmyb

Can the values of a table indicate a line?Yes, of course. It is very easy to know if the

values of a table indicate a line: If the y-value always grows by the same amount for a unit increase in x.

Section 1.3: Rates of ChangeHow can we measure how a function

grows ?... One way is with the Average Rate of Change

The change in the value of a quantity divided by the elapsed time. For a function, this is the change in the y-value divided by the change in the x-value for two distinct points on the graph.

ab

afbfxfArc

ba

)()(

)(,

Visualizing the ARCLooking at the

formula of the ARC we can see that it is the slope of the secant line that passes through the two points

ab

afbfxfArc

ba

)()(

)(,

ARC as a slopeNote that since the ARC is a slope, if the

quantity grows, then the ARC is positiveIf the quantity decreases, the ARC is

negative

Concave UpConcave Up: A graph or part of a graph

which looks like a right-side up bowl or part of an right-side up bowl.

Remember: Concave Up behaves like U.

Concave DownConcave Down: A graph or part of a graph

which looks like an upside-down bowl or part of an upside-down bowl.

Remember: Concave Down behaves like n

Types of intervals:

Section 1.5 Exponential Functions

taPtP 0)( 0P a

Initial valueY-intercept 0)0( PP

Tells us how fast the function grows

Increasing: Decreasing:

1a10 a

Exponential function

1a 10 a

%1 ra %1 ra

Population US

ttP 01.1303)( Where t is years since 2007 and P(t) is in millions of people