panlilio 2008-2009. section 1.1 objectives find the slopes of lines write linear equations given...
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Panlilio 2008-2009
Panlilio 2008-2009
Section 1.1 ObjectivesFind the slopes of linesWrite linear equations given points on lines
and their slopesUse slope-intercept forms of linear equations
to sketch linesUse slope to identify parallel and
perpendicular lines
Panlilio 2008-2009
Slope =
Find slope for the following points:
m =riserun
=Δ in yΔ in x
=
Why is Slope represented by the letter “m”?
No one seems to know! One theory is that it stands for “modulus of slope”, another is that the French word for “climb” is “monster”, but nothing can be proven.
(-2,3) & (4,-2) (0,-7) & (5,-2)
Finding Slope
Panlilio 2008-2009
There are three main “forms” for linear equationsSlope-Intercept Form ___________________Point-Slope Form___________________Standard Form ___________________
Find a linear equation given the following:Passes thru (3,-7) with slope=-2
3Passes thru (-4,-2) and (1,3) Passes thru (5,1) and (5,-4)
Writing Equations for Lines
Panlilio 2008-2009
Slope is __________Equation: _________
Special LinesSlope is __________Equation: _________
Panlilio 2008-2009
Parallel LinesParallel Lines have __________ slopes
Write the equation for the line that passes thru (1,2) that is parallel to 4x-y=5
Write the equation for the line that passes thru (0,-4) that is parallel to -3x+4y=8
Panlilio 2008-2009
Perpendicular LinesPerpendicular Lines have __________ slopes“Flip it and Reverse It” m --> _______
Write the equation for the line that passes thru (-4,1) that is perpendicular to -x+3y=4
Write the equation for the line that passes thru (1,5) that is perpendicular to 5x-15y=10
Panlilio 2008-2009
Section 1.2 ObjectivesDecide whether relations between two
variables represent a functionUse function notation and evaluate functionsFind the domain of functionsUse the functions to model and solve real-life
problemsEvaluate difference quotients
Panlilio 2008-2009
What is a function?For every ________, there is exactly one
_________Domain: Set of all _____ valuesRange: Set of all _____ valuesDoes each relation represent a function?x -3 -2 -1 0 1
y 4 7 4 3 2
x -3 -2 -1 -2 -3
y 2 -1 3 5 2
(-2,3) (4,-2) (-2,3) (7,-2) (4,-1)
(0,1) (2,3) (5,-9) (4,4) (-5,-9)
Panlilio 2008-2009
Testing for FunctionsAlgebraically
Solve for y. It is a function if each x corresponds to _____ value of y.
x2 + y=1 −x + y2 = 1
GraphicallyUse the “Vertical Line Test”
Panlilio 2008-2009
Function NotationInput Output Equation
x y or f(x) f (x) =x3 −5
Evaluating Functions =
Plug AND Chug
h(x) =x2 −3x+ 2Let . Find h(1), h(-2), h(w), and h(x+1)
Panlilio 2008-2009
Finding DomainAgain, the Domain is the set of all ___ values
If given a list of points, the domain is all the ________
If given an equation, find the __________ valuesy =
x+ 3x2 −9
y = x
Interval Notation:
[ or ] means “includes” ( or ) means “does not include”
Always use ( or ) for ∞
Panlilio 2008-2009
Real-Life Functions
N(t) =10.75t−20.1, 5 ≤t≤720.11t−92.8, 8 ≤t≤11
⎧⎨⎩
The number N (in millions) of cellular phone subscribers in the United States increased in a linear pattern from 1995 to 1997. Then, in 1998, the number of subscribers took a jump, and until 2001, increased in a different linear pattern. These two patterns can be approximated by the function
Where t represents the year, with t=5 corresponding to 1995. Use this function to approximate the number of cellular phone subscribers for each year from 1995 to 2001.
Panlilio 2008-2009
Difference Quotients
To Solve, Plug AND Chug!
f (x +h)− f(x)h
,h≠0This ratio is called a difference quotient
For f (x) =x2 −2x+ 3, Find f(x+h) - f(x)
hFor f (x) =x2 + 4x, Find
f(x+h) - f(x)h
Panlilio 2008-2009
Section 1.3 ObjectivesFind the domains and ranges of functions and
use the Vertical Line Test for functionsDetermine intervals on which functions are
increasing, decreasing, or constantDetermine relative maximum and relative
minimum values of functionsIdentify and graph piecewise-defined
functionsIdentify even and odd functions
Panlilio 2008-2009
Domain and Rangef (x) = 2−x f (x) =x2 −5
Panlilio 2008-2009
Increasing and DecreasingRelative Max and Min Values
Increasing:
Decreasing
Rel Max:
Rel Min:
Increasing:
Decreasing
Rel Max:
Rel Min:
Increasing:
Decreasing
Rel Max:
Rel Min:
Panlilio 2008-2009
Piecewise-Defined FunctionsPiecewise Function - A function that is
defined by two or more equations over a specified domain
f (x) =−x2 , x≤13, 1 < x< 42x−5, x≥4
⎧
⎨⎪
⎩⎪
f (x) =x−4, x< −33x−1, x≥−3
⎧⎨⎩
Panlilio 2008-2009
Even and Odd FunctionsEven Odd
Symmetric to _________
f(-x)=f(x) for all x’s
Symmetric to _________
f(-x)=-f(x) for all x’s
Panlilio 2008-2009
Even and Odd FunctionsDetermine whether a function is even, odd,
or neither, by evaluating f(-x). If f(-x)=-f(x), it’s ______. If f(-x)=f(x), it’s ______. If not, it’s neither. g(x) =x3 −x g(x) =x2 + 4 g(x) =x3 −1
Panlilio 2008-2009
Section 1.4 ObjectivesRecognize graphs of common functionsUse vertical and horizontal shifts and
reflections to graph functionsUse nonrigid transformations to graph
functions
Panlilio 2008-2009
Common FunctionsConstant Function f(x)=c Identity Function f(x)=x Abs Value Function f(x)=|x|
Cubic Function f(x)=x3Square Root Function f(x)=Quadratic Function f(x)=x2 x
Panlilio 2008-2009
Vertical and Horizontal ShiftsStart with f(x)Vertical Shift --> Add to or Subtract
from __Horizontal Shift --> Add to or Subtract
from __
f (x)±c
f (x ±c)
y=x2
y=
y=x2 y=
Panlilio 2008-2009
Reflecting GraphsReflection in the x-axis: h(x) = -f(x)Reflection in the y-axis: h(x) = f(-x)
y=x2
y=
y=x+1
y=
Panlilio 2008-2009
Nonrigid TransformationsNonrigid - Cause a distortion
y=cf(x) Multiply Y by
Vertical ________
c > 1
Vertical ________
0 < c < 1
y=f(cx) Multiply X by
Horizontal ________
0 < c < 1
Horizontal ________
c > 1
Panlilio 2008-2009
Nonrigid TransformationsCompare y=x2
to
y=x2
y=|x|
y =(3x)2 and y=12
x⎛⎝⎜
⎞⎠⎟
2Compare y=|
x| toy =2 x and y=13
x
Panlilio 2008-2009
Section 1.5 ObjectivesAdd, subtract, multiply, and divide functionsFind compositions of one function with
another functionUse combinations of functions to model and
solve real-life problems
Panlilio 2008-2009
Combining Functions( f + g)(x) = f(x) + g(x)( f −g)(x) = f(x)−g(x)( fg)(x) = f(x) • g(x)
fg
⎛⎝⎜
⎞⎠⎟(x) =
f(x)g(x)
SumDifferenceProductQuotient
Panlilio 2008-2009
Combining FunctionsFor each set of equations, find (f+g)(x), (f-g)
(x), (fg)(x), and (f/g)(x)
f (x) =3x−1 & g(x) =x+ 2 f (x) =2x2 & g(x) =x−1 f (x) = x & g(x) = x−1
Panlilio 2008-2009
Composition of FunctionsThe composition of function f with function g
is:
For each set of equations, find when x=0,1, and 2
f og( )(x) and (gof )(x)
f (x) =2x−3 & g(x) =x+ 5 f (x) =x2 -1 & g(x) = x−3
f og( )(x) =_____________
Panlilio 2008-2009
Real-Life CompositionsN(t) =20T 2 −80T + 500, 2 ≤T ≤14
The number N of bacteria in a refrigerated food is given by
where T is the temperature of the food in degrees Celsius. When the food is removed from refrigeration, the temperature of the food is given by
T (t) =4t+ 2, 0 ≤T ≤3Where t is the time (in hours). Find the composition N(T(t)) and interpret its meaning. Find the number of bacteria in the food when t = 2 hours. Find the time when the bacterial count reaches 2000.
Panlilio 2008-2009
Section 1.6 ObjectivesFind inverse functions informally and verify
that two functions are inverse functions of each other
Use graphs of functions to decide whether functions have inverse functions
Determine if functions are one-to-oneFind inverse functions algebraically
Panlilio 2008-2009
Finding Inverse FunctionsInverse Functions: When the domain of f is
equal to the ________ of f -1 , and vice versa.Inverse Functions “undo” each other.Examples:
f (x) =x+ 3 : (1,4),(2,5),(3,6),(4,7)
f−1(x) =x−3 :(4,__),(5, __),(6, __),(7, __)
f (x) =2x: (1,__),(2,__),(3,__),(4,__)
f−1(x) =x__
: (__,1),(__,2),(__,3),(__,4)
Panlilio 2008-2009
Graphs of Inverse FunctionsIf the point (a,b) lies on f, then the point (b,a)
must lie on f -1. That means that inverse functions are symmetrical about ______
Panlilio 2008-2009
Verifying Inverse FunctionsInverse Functions “undo” each other, so
verify that ( f og)(x) =x and (gof )(x) =x
f (x) =x3 +1 and h(x) = x−13 g(x) =5x−2 and m(x) =x+ 25
Panlilio 2008-2009
One-to-One FunctionsOne-to-one functions: Every X has only one Y,
and Every Y has only one XOne-to-one functions pass the Horizontal Line
Test For one-to-one functions, f(a)=f(b) implies
that a=bf (x) = x−3 f (x) =x2
Panlilio 2008-2009
Finding Inverse FunctionsUse the Horizontal Line Test to test whether f
is a one-to-one function and has an inverse function
Switch the x’s and y’sSolve for y. Replace y with f -1 f (x) =2x−5 f (x) =
x2
2f (x) =
x−34
Panlilio 2008-2009
Homework1.1: P.11 #1,19,25,33,37,43,51,53,55,65,69,831.2: P.24
#1,2,7,8,13,19,29,35,37,38,49,53,55,69,73, 83,861.3: P.38 #1,3,13-19 odd,41,45,47,49,531.4: P.48 #1-11 odd,15-25 odd,67,681.5: P.58 #5-25 EOO,35,45,47,49,51-54,57,67,69,
77,78,821.6: P.69 #9-13 odd,21-24,25,43,45,49,51,83Chapter Review P.82 #1-45 EOO,47,65,69-72,85-
93 odd,97,107