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Chapter 2 – Functions18 Days
2.1 Definition of a Function 2.2 Graphs of Functions 2.3 Quadratic Functions 2.4 Operations on Functions 2.5 Inverse Functions 2.6 Variation
Table of Contents
2.1 Definition of a Function
Two Days
For the function
a) Find f(a)
b) Find f(a-1)
c) Find
d) Find
Functions*
x
xxf
5)(
)(
1
xf
x
f1
p 148 (# 5,8,14-28 even, 45,47,49,52,57)
Homework
Day 2 Def Function, Domain, Range, Increasing/Decreasing, Vert Line Test, Def Linear Function Evaluating (p148 #13)
A function F from a set D to a set E is a correspondence that assigns each element x of D to exactly one element y in E.
Definition of a Function
Domain – The set D is the domain of the function. Domain is the set of all possible inputs.
Range – The set E is the range of the function. Range is the set of all possible outputs.
The element y in E is the value of f at x also called the image of x under f.
Domain and Range
We say that f maps D into E.
Two functions f and g are equal if and only if f(x) = g(x) for all x in D.
Function Mapping
The graph of a function f is the graph of the equation y = f(x) for all x in the domain of f.
The vertical line test can be used to determine if a graph represents a function.◦ What does the vertical line test represent in terms
of a function mapping?
Graphs of Functions
-f is increasing whenf(a)<f(b) and a<b.
-f is decreasing whenf(b)>f(c) and b<c.
-f is constant when f(x)=f(y) for all x and y.
Increasing and Decreasing Functions
Given
Determine the domain of g. Evaluate g(-3)
Evaluate
Evaluating Functions
1
2)(
2 x
xxg
A
g1
What is the difference between sketching and graphing a function?
Why would we sketch a function as opposed to graph a function?
Sketching Functions
Sketch the following functions and determine the domain, range, and intervals of decreasing, increasing, and constant value:
Sketching Functions
4)( 2 xxf 225)( xxg
We can find linear functions in the same way that we find the equation of a line.
If f is a linear function such that f(-3)=6 and f(2)=-12, find f(x) where x is any real number.
Finding Linear Functions
Pg 150 #57, 59
Applications Problems
p 148 (# 15,32,34,35,46,48,50,53,54,60,63,65)
Homework
2.2 Graphs of Functions
Four Days
-Name of Family
-Parent Equation
-General Equation
-Locator Point
-Domain -Range
Parent Functions
Linear
xy
bmxy
),0(:int by
8
6
4
2
-2
-4
-6
-8
-15 -10 -5 5 10 15
f x = x
x y-2 -2-1 -10 01 12 2
-Name of Family
-Parent Equation
-General Equation
-Locator Point
-Domain -Range
Parent Functions
Value Absolute
xy
khxay
),(: khVertex
),0[
8
6
4
2
-2
-4
-6
-8
-15 -10 -5 5 10 15
f x = x
x y-2 2-1 10 01 12 2
-Name of Family
-Parent Equation
-General Equation
-Locator Point
-Domain -Range
Parent Functions
Quadratics
2xy
khxay 2)(
),(: khVertex
),0[
8
6
4
2
-2
-4
-6
-8
-15 -10 -5 5 10 15
f x = x2
x y-2 4-1 10 01 12 4
-Name of Family
-Parent Equation
-General Equation
-Locator Point
-Domain -Range
Parent Functions
Cubics
3xy
khxay 3)(
),(: khInflection
8
6
4
2
-2
-4
-6
-8
-15 -10 -5 5 10 15
f x = x3
x y-2 -8-1 -10 01 12 8
-Name of Family
-Parent Equation
-General Equation
-Locator Point
-Domain -Range
Parent Functions
Root Square
xy
khxay
),(:Endpoint kh
),0[ ),0[
8
6
4
2
-2
-4
-6
-8
-15 -10 -5 5 10 15
f x = x
x y0 01 14 29 316 4
-Name of Family
-Parent Equation
-General Equation
-Locator Point
-Domain -Range
Parent Functions
Root Cube
3 xy
khxay 3
),(:Inflection kh
8
6
4
2
-2
-4
-6
-8
-15 -10 -5 5 10 15
f x = x1
3
x y-8 -2-1 -10 01 18 2
Parent: Shift up k units: Shift down k units:
Shift right h units: Shift left h units
Combined Shift:◦ (right h units, up k units)
Graph Shifting and Reflections
xy
kxy
kxy
hxy
hxy
khxy
)(xfy
khxfy )(
)( hxfy
)( hxfy
kxfy )(
kxfy )(
Parent: Reflection in x-axis:
Vertical Stretch a>1 Vertical Shrink 0<a<1
Horizontal Stretch 0<c<1 : Horizontal Compression c>1:
Combined Transformation:
Graph Shifting and Reflections
xy
xay
xcy
xay
)(xfy
)(xfay
)( xcfy
)(xfay
khxay khxfay )(
Graph the following using translations:
Graph Shifting and Reflections
12 xy
33 xy
31 2 xy
542 xxy
33 xy
83 xy
242 xy
22xy
142 2 xy
32 xy
Shifts and Reflections WS
Homework
Day 2 – Even and Odd functions. Vertical and Horizontal stretching and compressing of graphs.
f is an even function if f(-x)=f(x) for all x in the domain.◦ Even functions have symmetry with respect to the
y-axis.◦ Ex:
f is an odd function if f(-x)=-f(x) for all x in the domain. ◦ Odd functions have symmetry with respect to the
origin. ◦ Ex:
Even and Odd Functions
2)( xxf
3)( xxf
A parent function is the simplest function in a family of certain characteristics.
A translation shifts the graph horizontally, vertically, or both. Resulting in a graph of the same shape in a different location.
A reflection over the x-axis changes y-values to their opposites.
Family Functions and Shifts
A vertical stretch multiplies all y-values by the same factor greater than 1.
A vertical shrink reduces all y-values by the same factor between 0 and 1.
Each member of a family of functions is a transformation, or change, of the parent function.
A horizontal compression divides all x-values by the same factor greater than 1.
A horizontal stretch divides all x-values by the same factor between 0 and 1.
Family Functions and Shifts
Parent: Shift up k units: Shift down k units:
Shift right h units: Shift left h units
Combined Shift:◦ (right h units, up k units)
Graph Shifting and Reflections
xy
kxy
kxy
hxy
hxy
khxy
)(xfy
khxfy )(
)( hxfy
)( hxfy
kxfy )(
kxfy )(
Parent: Reflection in x-axis:
Vertical Stretch a>1 Vertical Shrink 0<a<1
Horizontal Stretch 0<c<1 : Horizontal Compression c>1:
Combined Transformation:
Graph Shifting and Reflections
xy
xay
xcy
xay
)(xfy
)(xfay
)( xcfy
)(xfay
khxay khxfay )(
pg 164 (# 2,3,5,7,8,13,15,17,20,31-36,39 a-f, 41,42,45)
Homework
Day 3 – Piecewise functions and questions from the previous 2 days. Application of Piecewise functions (pg 168 #66)
Piecewise functions are defined by more than one expression over different intervals.
Absolute Value is actually a piecewise defined function.
Piecewise Functions
Lets graph the following piecewise defined function.
Piecewise Functions
2x;
2x2;
2x;
32
4
)( 2
x
x
x
xf
Lets graph the following piecewise defined function.
Piecewise Functions
2x;
2x2;
2x;
2
42
)(2
x
x
x
xf
An electric company charges its customers $0.0577 per kWh for the first 1000kWh, $0.0532 for the next 4000kWh, and $0.0511 for any over 5000kWh. Write a piecewise defined function C for a customer’s bill of x kWhs.
How much will a customer’s bill be if they used 4300kWh of electricity?
Applications of Piecewise Functions
pg 167 (# 47-50,53,54,55,56,63-65)
Homework
Day 4 – Graphing Piecewise functions WS. Working day for students.
Graphing Piecewise Functions WS
Homework
2.3 Quadratic Functions
Two Days
Day 1 – Standard form of a quadratic. Vertex form of a quadratic. Completing the square. Finding x and y intercepts.
Standard form of a Quadratic:
Vertex form of a Quadratic:
Quadratic Functions
0 ; 2 acbxaxy
),( :Vertex ; )( 2 khkhxay
Competing the Square
463 2 xxy
4) 2(3 2 xxy222 )1(34))1( 2(3 xxy
3a (-1,-7),:Vertex 7)1(3 2 xy
● To find the x-intercept, set y=0. Solve for x.
● To find the y-intercept, set x=0. Solve for y.
Find the x and y intercepts of the following:
Finding x and y intercepts
14 xy 652 xxy
Vertex and Intercepts WS
Homework
Day 2 – Vertex formula and Theorem on Max/Min Values.
Standard Form of a Quadratic
Factored Form of a Quadratic
Vertex Formula
a
bf
a
bcbxaxy
2,
2 :Vertex ; 2
2
,2
:Vertex ; )()( 212121
xxf
xxxxxxay
Theorem on the Maximum or Minimum of a Quadratic Function
is 2
then 0,a where)( If 2
a
bfcbxaxxf
0a if of valuemaximum the)1( f
0a if of valueminimum the)2( f
Find the vertex of the following and determine if it is a max or a min:
Finding the Vertex
54 xy
1242 xxy
1052 xxy
The length of a frog’s leap is 9ft and has a maximum height of 3ft off the ground. Assuming the frog’s path through the air is parabolic, find an equation that describes the path of the frog through the air.
Applications of Quadratics
pg 179 (# 1,6,8,10,13,17,20,23,26,27,30,39)
pg 179 (# 7,15,31,38,40,43,44,49,54)
Homework
2.4 Operations on Functions
One Day
We can perform several operation on functions just as we perform the same operation on real numbers. Consider f(x) and g(x):
Operation on Functions
)()())(( xgxfxgf
)()())(( xgxfxgf
)()())(( xgxfxgf
)(
)()(
xg
xfx
g
f
The composite function of two functions f and g is defined by:
The domain of is the set of all x in the domain of g such that g(x) is in the domain of f.
Essentially, the range of g(x) is the domain of f(x) minus any possible restrictions.
Composition of Functionsgf
))(())(( xgfxgf
gf
Let and
Find:
Examples23)( xxf xxxg 2)( 2
))(( xgf
))(( xgf
))(( xgf
)(x
g
f
))(( xgf
))(( xfg
Is ????
Consider
Question..fggf
xxf )( 4)( xxg
)4)(( gf
)5(
f
g
)3)(( gf
)2)(( fg
pg 192 (# 2,3,6,7,9,12,15,21,25,29,30,35,36,39,45,46)
Homework
Graph each function belowa) Exact x and y interceptsb) Give the domain/rangec) Intervals of increase, decrease, constant
1)
2)
3)
4)
Graph Shifting Review
1)1(2 2 xy
653 xy
1)4( 3 xy
462 xy
2.5 Inverse FunctionsTwo Days
f(x) and g(x) are inverse functions that “undo” one another if and only if .
Notation:Original Function:
Inverse Function:
Inverses are NOT reciprocals!!
Definition of an Inverse Function
xxfgxgf ))(())((
)(xf
)(1 xf
Inverses switch the x and y values of a function. (x,y) -> (y,x)
The domain of f is the range of its inverse. The range of f is the domain of its inverse.
Graphically, the inverse of a function is a reflection over y=x.
Properties of Inverses
Can you come up with a function that when reflected over the line y=x will no longer be a function?
Do all Functions have Inverses?
In order for a function to have an inverse, a fucntion must be 1 to 1. That is, no two elements in the domain can have the same y value.
Ex: is not a 1-1 function.
We can however restrict the domain to find partial inverses. i.e has an inverse.
1-1 Fucntions
)4,2( and )4,2( )( 2 xxf
]0|[ )( 2 xxxxf
To find an inverse:◦ 1. Check if the function is 1-1. Restrict the
domain if the original function is not 1-1.◦ 2. Write f(x) as y.◦ 3. Switch all x and y in the equation.◦ 4. Solve for the “new” y.◦ 5. Rewrite the domain if necessary.◦ 6. Check that or graph on
the TI and check for symmetry about y=x.
Finding Inverses
xxfgxgf ))(())((
Find the inverses to the following functions:
Examples
34)( xxf
3)( xxf
1)( 2 xxf
pg 203 (# 4,5,7,9-12,14,15,19,23,29,31,35)
Homework
Finding Inverses WS (# 1-7)
Homework
2.6 VariationOne Day
A variation or proportion is used to describe relationships between variable quantities.
k is a nonzero real number called a constant of variation or constant of proportionality.
Variation
Direct Variation◦ y varies directly with x◦ y is directly proportional to x
Inverse Variation◦ y varies inversely with x◦ y is inversely proportional to x
Combined or Joint Variation◦ z varies jointly with x and y◦ z varies directly with x and inversely
with y
Types of Variation
kxy
x
ky
kxyz
y
kxz
V varies jointly as B and H
P varies directly as the square of V and inversely as R
The volume, V, of a gas varies directly as the temperature, T, and inversely as the pressure, P
The distance, D, that a free-falling object falls varies directly as the square of the time, T, that it falls
Write a Variation Equation for the Following:
1. Write the general formula that variables and a constant of variation.
2. Substitute the initial conditions for the variables and solve for the constant of variation k.
3. Substitute the constant of variation k into the general formula from your first step.
4. Use you general formula to solve the problem.
Solving Variation Problems
The price, P, of a diamond is directly proportional to the square of the weight, W. If a 1 carat diamond costs $2000, find the price of a 0.7 carat diamond.
Example
The electrical resistance R of a wire varies directly as its length l and inversely as the square of its diameter d. A wire 100ft long of diameter 0.01in has a resistance of 25ohms. Find the resistance of a wire with a diameter of 0.015in and 50ft of length.
Example
pg 209 (# 2-4,6,8,10,13,14,16,17,20,21,23)
Homework