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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