panlilio 2008-2009. section 1.1 objectives find the slopes of lines write linear equations given...

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”

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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

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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