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8.1 More on Functions and Their Graphs Name: ________________________________

Obj.: I will be able to identify the parts of a function’s graph, including increasing/decreasing/constant areas, relative maxima and minima, symmetry, and whether the function is even or odd.

VocabularyIncreasing Function

Decreasing Function

Constant Function Relative Maximum

Relative Minimum

Even Function Odd Function

Notes Increasing, Decreasing, and Constant Functions

o A function is ________________ on an open interval, I, if ________________ whenever x1< x2 for any x1 and x2 in the interval.

o A function is ________________ on an open interval, I, if ________________ whenever x1< x2 for any x1 and x2 in the interval.

o A function is ________________ on an open interval, I, if ________________ for any x1 and x2 in the interval.

Relative Maxima and Relative Minimao A function value f (a ) is a ________________ ______________ of f if there exists an

open interval containing a such that _____________________ for all x≠a in the open interval.

o A function value f (b ) is a ________________ _______________ of f if there exists an open interval containing a such that _____________________ for all x≠b in the open interval.

o The word ________ may be substituted in place of ___________ when describing maxima and minima.

Tests for symmetryo A graph can be symmetrical with respect to the _________, ___________, or

______________.


Even and Odd Functionso ________ function

____________________ for all x in the domain of f Symmetric with respect to the _________.

o ________ function _____________________for all x in the domain. Symmetric with respect to the ____________.

o Neither Even nor Odd ______________and _________________ _________ __________________ with respect to the y-axis or the origin.


Practice

8.2 Piecewise Functions Name: ________________________________

Obj.: I will be able to recognize piecewise functions, graph them, and create equations for them from their graphs. I will be able to identify the domain and range of each segment of a piecewise function.

VocabularyPiecewise Function

Step Functions Greatest Integer Function

Notes A function that is defined by ______ (____ _________) equations over a _________________

____________ is called a piecewise function. o Ex:

o Graphs of piecewise functions have segments.o To graph a piecewise function, graph each of the

_________________ over their corresponding domains. Remember that if ____ or ____, use an ________

dot. If ____ or ____, use a ___________ dot.

Step Functionso Some piecewise functions are called __________ functions because their graphs form

__________________ steps. o One such function is called the greatest integer function, symbolized by _______

or________, where int(x) = the greatest integer that is less than or equal to x. In general, if_________________, where n is an integer, then ____________

Domain and Rangeo The __________ of a segment is its ______________ point to its _________________

point.o The __________ of a segment is the ___________________ point to the

________________ point.o Both are represented as _______________ or in _______________ _________________

________ if both ends ______ ______________. ________ if both ends ___________ _______ or ________ if only _________ ________ ______________.

o For either domain or range: If either end has an _______________, it continues to

______________________________. If either end has an___________ _______ that segment does _____ _________

that point, but everything up to it. If either end has a ___________ ______, that segment ___________ that point. If the graph is __________________, the ________-hand segment ______

____________ the connecting point. The ________-hand segment will ______. (Unless otherwise indicated).


Practice

8.3 Applications of Piecewise Functions Name: ________________________________

Obj.: I will be able to apply piecewise functions to real-world applications.

Notes There are many real-world applications of piecewise functions, including:

o Many ______________ ________ can be represented by piecewise functions. o ______________ ___________ that are based on weight classeso _______ ____ ______________ _______________ based on weights. o Effects of _________o ___________ and ___________________ ______________

Practice

8.3 Applications of Piecewise Functions Name: ________________________________

8.4 Power Functions Name: ________________________________

Obj.: I will be able to identify the pieces of a power function. I will be able to write statements of power functions and graph them. I will be able to use regression to find a power function for a set of data.

VocabularyPower Function Rational Functions Singularity Scaling factor

Notes Power Functions

o A variable is raised to a __________ exponent. o Similar to exponential functions, except this time, the _____________ is in the

________ instead of the exponent.o Parent function: __________________

___ serves as a simple ______________ ______________ (constant) ___ is called the _____________ or _____________, and

Determines the function’s rates of ___________ or ___________. Determines the _________ of the graph If ______________________ If ______, the function is a __________ function, and has ____

_______________. In this case, ________ is an example of a _______________.

________ Powers (Integers)o Any input for x will have a _____________ _____o Has the ___________ ____________ for _____ and _____, so is

symmetrical on the ___________o __________ function

_______Powers (Integers)o f(x) will have the _____________ ________ as x.o Outputs for ____ are the _____________ of ______o Symmetrical on the __________o ______ function

Rational Powers (Fractions)o If ¿¿, then take the _____ __________ of ¿¿.o _______________________o _______________ ___________ are determined based on

whether n is ________ or ______. Regression to find Power Functions

o Press ________o Select _____________o Type ____ values into _____ and ___ values into _____o Press _______o Go over to _______o Select ____________ and hit __________.

Variationo As one variable increases, the other decreases – inverse variation. xy = k where k is

constant.o As one variable increases, the other increases – direct variation. y = kx where k is

constant.


Practice

8.5 Best-Fit Models Name: ________________________________

Obj.: I will be able to use a calculator to find best-fit models, to draw conclusions, and make predictions.

Notes _______________ lines/curves may ______ __________ through ______________ _________

points.o Best ________________of variable relationships.

Find the regression for the best-fit line. o To make predictions of one variable, ________ _____ ___________ ____ to the

____________. o Unless told otherwise, may round to two decimal places.

Variationo If _______ in y=a xb, _____________ proportionalo If _______in y=a xb, ______________ proportional

Tip: If all y-values are negative, a is negative. May use positive values for regression & change the sign of a. (See #2 on 2nd page of practice)


Practice

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