1
Chapter 6: Polynomial Transformation Notes Packet
Name: ___________________________________________ Pd: _______
Lesson Book Topic
Ch.
1
2
6-1
7-8
3
4
7-8
5
8-1
6
8-3
Homework/Practice: Date
due:
Polynomial Vocabulary
Graphing Radical (Square root)
Functions
p. 4
p.6-7
Graphing Cubic Functions
Graphing Radical (cube root)
functions
Graphing Exponential Models
p.9-10
p.12-13
Graphing Logarithmic Functions (as
inverses)
Test Review
p.17-18
p.16
p.19-21
Grade:
2
6-1 Polynomial Functions: Key Vocabulary
I. Polynomial Function – single term or sum of terms
P  x   an xn  an1xn1  ...  a1x  a0 , where an ,..., a0 are coefficients and n is a nonnegative
integer
Examples:
P ( x )  8 x5  6 x 4  2 x 2  x  9
F ( x )  6 a3  2 a 2  7 a  8
II. Domain – all the _________ the function uses
Range – all the _________ the function uses
III. X-intercepts or Roots or Zeros or Solutions: where the graph________________________
a. These can be found by ______________ , graphing on a calculator, or using the
quadratic formula.
IV. Y-Intercepts – where the graph crosses/touches the ____________
a. These can be found by setting x equal to 0.
End Behavior – the direction in which the ends of the graph are pointing/moving
V.
VI. Classifying Intervals –
 Increasing Interval – as x increases f(x) increases (section of graph that is
slanting________ from left to right)

Decreasing Interval – as x increases f(x) decreases (section of graph that is slanting
______ from left to right)
Remember: you only use x-values when describing intervals.
VII. Turning Point – point where the graph changes from increasing to decreasing or viseversa
 To determine how many turning points a polynomial can have you take the degree of
the polynomial and subtract one.
VIII. Degree of a Polynomial –
Algebraically  The highest exponent
Graphically
 Even Degree: ends of graph point in ____________________
 Odd Degree = ends of graph point in __________________
IX. Functions –
Even Functions – graph is symmetric about the y- axis (if you fold graph along the yaxis the left & right side will be mirror images)
Odd Functions – graph is symmetric about the origin (if you rotate the graph
graph will look exactly the same)
180
,
Note: Functions do not have to be either Even Functions or Odd Functions; they can be neither
functions.
3
1.
EXAMPLE: Find the x- and y-intercepts of each polynomial function.
2
y x
2. y   x 2  2 x  3
3. y  10 x3  15 x 2  2 x  3
3
4.
Type of Function:
y
Degree:
5
4
End Behavior:
3
2
1
–5
–4
–3
–2
–1
–1
1
2
3
4
5
Increasing Interval:
Decreasing Interval:
Domain:
Range:
x-intercept:
y-intercept:
Type of Function:
Degree:
x
–2
–3
–4
–5
5.
y
5
End Behavior:
4
3
2
1
–5
–4
–3
–2
–1
–1
1
2
3
4
5
x
Increasing Interval:
Decreasing Interval:
Domain:
Range:
x-intercept:
y-intercept:
–2
–3
–4
–5
6.
y
Type of Function:
7
6
Degree:
5
End Behavior:
4
3
2
1
–5
–4
–3
–2
–1
–1
–2
–3
1
2
3
4
Increasing Interval:
Decreasing Interval:
Domain:
Range:
x-intercept:
y-intercept:
5 x
–4
–5
4
Classwork: Section 6-1
5
Lesson 2: Graphing the Square Root Function: f(x) =
Recall how to graph
Quadratic Functions (Parabolas):
1. Plot your vertex
2
2.  over  A  up
Ex: f(x) =
x
To Graph Square Roots:
1. Plot your vertex
2. over A  up
x (parent function)
Range: __________ 1. f ( x)  x  6
D:______ R: ______
Domain: ________
2. f ( x)  x  6
D:______ R:______
y
y
–6
–4
9
9
4
6
6
2
3
3
–2
2
4
x
6
–9
–6
–3
3
x
9
–9
–6
–3
3
–3
–3
–4
–6
–6
–6
–9
–9
y
y
9
9
9
6
6
6
3
3
3
–3
3
6
9
x
–9
–6
–3
x
9
1
x6
2
D:_____ R:_____
D:_____ R:______
y
6
5. f ( x) 
4. f ( x)  2 x  7
D:_____ R:______
–6
6
–2
3. f ( x)   x
–9
y
6
3
6
9
x
–9
–6
–3
3
–3
–3
–3
–6
–6
–6
–9
–9
–9
CONCLUSIONS:
A. What happens when values are added or subtracted ‘inside’ and ‘outside’ the parent
function?
B. What happens if a negative is placed in front of the equation?
C. What happens when a value is multiplied outside of the function?
D. How is the graph of y  x  2  4 transformed from the parent graph of y  x ?
6
9
x
6
The Square Root Function
Practice: Lesson #2
1.
What is the parent equation for the Square Root
Function? _____________
2. Graph the parent function for Square Root.
3. Domain: ______
Range: ______
4.
x-intercept: ________
5.
y-intercept: ________
6. What are the coordinates for the 3 major points:
_______, ________, _______
7. Based on your knowledge of transformations, describe the roles a, h, and k play
for the family of functions y  a x  h  k . (i.e what does a do, what does h do,
what does k do, and so on……)
a:___________________________________________________
h: __________________________________________________
k:__________________________________________________
8. How would each of the following graphs change in relation to the parent
graph?
a) y  x  3
_______________________________________
b) y  x  4 ________________________________________
c) y  3 x _________________________________________
d) y  x  5 _________________________________________
e) y  3 x  2  7 ______________________________________
9. Without graphing, state the Domain and Range of each function.
a) y  5 x
Domain
_________
Range
_________
b) y  x  8
_________
_________
c) y   x  7
_________
_________
d) y  x  2  3
_________
_________
e) y   x  4  1
_________
_________
7
10. Graph the following square root functions using transformations.
a) y   x
b) y  3 x  5
c) y  x  2
Domain: ________
Range:_________
Domain:________
Range:__________
Domain: _______
Range: ________
d) y   x  3
e) y  x  2  5
f) y   x  3  3
Domain: ________
Range: _______
Domain: ________
Range: ________
Range: _______
8
Lesson #3: Graphing Cubics
x3
Cubic Function: f(x) =
To Graph Cubics:
1. Plot your vertex
3
2.  over  A  up
Ex: f(x) = x 3 (parent function)
Domain: _____
Range: ______
1. f ( x)  x3  4
D:_____ R:______
y
–9
3.
–6
9
9
6
6
3
3
–3
3
6
x
9
–9
–6
6
3
3
–3
–6
–6
–9
–9
6
9
–9
x
–6
–3
3
–9
1 3
x
2
D:_____ R:_____
5. f ( x) 
y
y
9
9
9
6
6
6
3
3
3
3
6
x
9
–9
–6
–3
3
6
9
x
–9
–6
–3
3
–3
–3
–3
–6
–6
–6
–9
–9
–9
1 3
x 3
5
7. f ( x )  5( x  4)3
D:_____ R:_____
x
9
–6
D:_____ R:______
–3
6
–3
4. f ( x)  2 x3
y
6. f ( x ) 
9
–3
–3
D:_____ R:_____
–6
y
y
f ( x)   x 3
–9
2. f ( x)  ( x  4)3
D:______ R:______
6
9
x
8. f ( x)  ( x  1)3  5
D:_____ R: _____
D:_____ R:______
y
y
y
9
9
9
6
6
6
3
3
3
–9
–6
–3
3
–3
6
9
x
–9
–9
–6
–3
3
6
9
x
–6
–3
3
–3
–3
–6
–6
–6
–9
–9
–9
6
9
x
9
CONCLUSIONS:
A. What happens when values are added or subtracted ‘inside’ and ‘outside’ the parent
function?
B. What happens if a negative is placed in front of the equation?
C. What happens when a value is multiplied outside of the function?
D. What happens when a value is multiplied outside of the function?
E. How is the graph of y  ( x  1)3  5 transformed from the parent graph of y  x3 ?
Practice: Cubics Lesson #3
1.
What is the parent equation for the Cube Function?
_____________
2. Graph the parent function for Cube.
3. Domain: ______ Range: ______
4. x-intercept: _______ y-intercept: _____
5. How would each of the following graphs change in relation to the parent graph?
a)
y  ( x  3)3
_____________________________________________
b)
y  ( x  4)3
_____________________________________________
c)
y  3( x)3 _____________________________________________
d)
y  x3  5
_____________________________________________
e)
y  x3  6 ____________________________________________
10
9. Graph the following Cube functions using transformations.
a) y   x3
b) y  3x3  5
Domain: ________
Range:_________
Domain: ________
Range:__________
Domain: ________
Range: _______
Domain: ________
Range: ________
d) y  ( x  3)3
e) y  ( x  2)3  5
c) y  x3  2
Domain: _______
Range: ________
f) y  ( x  3)3  3
Domain:________
Range: _______
11
Lesson #4: Cube Root Families of graphsCube Root Function model:
y  a3 x h  k
To Graph Cube Roots:
1. Plot your vertex
2. 3 over  A  up
Ex: f(x) =
3
x (parent function)
Range: __________ 1. f ( x)  3 x  6
Domain: ________
2. f ( x)  3 x  6
D: ______ R:______
y
–9
–6
D: _______ R:_______
y
y
9
9
9
6
6
6
3
3
3
–3
3
6
x
9
–9
–6
–3
3
6
9
x
–9
–6
–3
3
–3
–3
–3
–6
–6
–6
–9
–9
–9
3. f ( x)   3 x
1
f ( x)  3 x  6  2
2
4. f ( x)  2 3 x  7
5.
D: _______ R:_______
D: ______ R: ______
D:______ R: ______
y
6
y
y
9
9
6
6
3
3
9
6
3
–9
–6
–3
3
6
9
–3
–6
–9
CONCLUSIONS: use the model
x
–9
–6
–3
3
6
9
x
–9
–6
–3
3
–3
–3
–6
–6
–9
–9
y  a 3 x  h  k to answer each question:
A. What happens when values are added or subtracted ‘inside’ and ‘outside’ the parent function?
B. What happens if a negative is placed in front of the equation?
C. What happens when a value is multiplied outside of the function?
D. How is the graph of
y  3 x  2  4 transformed from the parent graph of y  3 x ?
6
9
x
9
x
12
Practice Lesson #4: The Cube Root Function
1.
What is the parent equation for the Cube Root Function? _____________
2. Graph the parent function for Cube Root.
3. Domain: ______
Range: ______
4. x-intercept: ________
y-intercept: ________
5. What are the coordinates for the 3 major points:
_______, ________, _______
6. Based on your knowledge of transformations, describe the roles a, h, and k play
for the family of functions y  a 3 x  h  k . (i.e what does a do, what does h do,
what does k do, and so on……)
a: _________________________________________________
h: _________________________________________________
k:__ ________________________________________________
7. How would each of the following graphs change in relation to the parent graph?
f) y  3 x  3
_____________________________________________
g) y  3 x  4
_____________________________________________
h) y  3 3 x
_____________________________________________
i)
y  3 x 5
_____________________________________________
j) y  3 x  6
_____________________________________________
k) y  3 3 x  2  7
_____________________________________________
13
9. Graph the following cube root functions using transformations.
a) y   3 x
b) y  3 3 x  5
Domain: ________
Range:_________
Domain: ________
Range:__________
d) y   3 x  3
e) y  3 x  2  5
Domain: ________
Range: _______
Domain: ________
Range: ________
c) y  3 x  2
Domain: _______
Range: ________
f) y   3 x  3  3
Domain:________
Range: _______
14
Lesson #5: Exponential Functions
Introduction: Exponential Functions
Example – The temperature of a can of soda is 60◦C. It is placed in the refrigerator, which
reduces the temperature by 20% every hour. The following graph, 𝑦 = 60(0.8)𝑥 , models the
temperature of the soda can.
1. What does 60 represent in the
equation?
2. What does 0.8 represent in the
equation?
3. Will the temperature ever
reach 0◦C?
4. State the domain and range
for this graph.
5. When will the soda be ready to drink?
Definition of the Exponential Function
The example above is an exponential function. It is defined by 𝑓(𝑥) = 𝑏 𝑥 , where b is a positive
constant other than one and x is any real number. Other examples of exponential functions are:
𝑓(𝑥) = 2𝑥
General form:
ℎ(𝑥) = 10𝑥
𝑔(𝑥) = 3𝑥+2
f ( x)  ab x h  k .
What is an asymptote? _____________________
What is the asymptote for the general graph of
y  bx
1 𝑥−4
𝑦 = (2)
15
Exploration: Transformations of Functions
Objective: To discover how to shift the graph of the exponential function horizontally or
vertically, or to stretch the graph, or to reflect the graph about the x- or y- axis.
Directions:
1. Create a new Word document and save it as YourLastNamePeriod#FT.docx
2. Open up the Graph Program: Start – All Programs – HCPS Software – Graph
3. Graph a new function by going to Function, Insert Function. Type in the equation in the top
line.
HINT: 𝑓(𝑥) = 2𝑥 is f(x) = 2^x
𝑔(𝑥) = 3𝑥+2 is g(x) = 3^(x+2)
4. Graph more than one function on a graph. Change the color of each function under “Graph
Properties” at the bottom of the Insert Function menu.
5. Copy the graph into your Word Document by selecting Edit, Copy Image in the Graph Program,
then selecting Paste or Ctrl-V in the Word Document. You will have to shrink the graph down.
6. Graph the following functions and paste them into your word document.
Graph 1
f ( x)  2 x
f ( x)  2 x  3
7. Write 5 complete sentences, one for
each graph, that describes how changing the
equation affects the graph.
f ( x)  2 x  1
f ( x)  2 x  5
f ( x)  2  4
x
Graph 2
f ( x)  2 x
f ( x )  2( x 2)
f ( x )  2( x 3)
f ( x )  2( x 5)
f ( x )  2( x 6)
Graph 3
f ( x)  2 x
f ( x )  2(  x )
Graph 4
f ( x)  2 x
f ( x )  2 x
Graph 5
f ( x)  2 x
f ( x )   0.5 2 x
f ( x)   2 2 x
f ( x )   0.25 2 x
f ( x )   3 2 x
Example: “Graph 1: Adding a number to the
outside of the function causes the graph to
___________.”
Be specific and include details – e.g. does it
matter if the number is positive or negative?
A whole number or decimal?
8. Save your work and drop it in my Virtual
Share folder: RALAWREN. This is worth a
classwork grade.
16
Lesson #5 Practice
Use the graphing calculator’s Table feature (or do a t-table by hand) to graph points for each function
below. Show horizontal asymptotes with a dotted line.
1. f ( x)  2 x 3
2. f ( x)  2 x  3
3. f ( x)  2 x 1  3
Domain:
Domain:
Domain:
Range:
Range:
Range:
Asymptote:
Asymptote:
Asymptote:
4. f ( x)  3 x 2  1
5. f ( x)  3x 1  2
6. f ( x)  3 x
Domain:
Domain:
Domain:
Range:
Range:
Range:
Asymptote:
Asymptote:
Asymptote:
17
Lesson #6: Graphing Log Functions
Logarithmic Functions are inverses of exponential functions. An example of this is the Richter
scale that measures the intensity of earthquakes. For each increase in one unit on this scale,
there is a tenfold increase in the intensity of the earthquake. For example, the Chilean
earthquake in 2010 measured 8.8 and the Japanese earthquake in 2011 measured 9.0 on the
Richter scale.
The Japanese earthquake was 100.2 or 1.58 times more intense than the Chilean earthquake.
Parent function:
9
8
7
6
5
4
3
2
1
-9 -8 -7 -6 -5 -4
-3 -2 -1
0
1
-1
-2
-3
-4
2
3
4
5
6
7
8
9
Work with graphing calculator to do the following tasks:
1.
Graph parent function: y = log(x).
2.
What is the asymptote? ______________
3.
What is the Domain? _____________
4.
What is the Range ? _____________
5.
What are 2 points on this graph? ___________, _________
-5
-6
-7
-8
-9
Graph the parent and each of the following functions on your calculator. Describe the changes.
Part A:
1.
y = log(x) + 3 ________________________________________________________
2.
y = log(x) - 5 ________________________________________________________
What can you say about values added/subtracted outside of the function?
______________________________________________________________________________________
Has the asymptote changed? ________________________________
Part B:
3.
y = log(x + 2)
____________________________________________________________
4.
y = log(x – 8)
____________________________________________________________
What can you say about values added/subtracted inside the function?
_________________________________________________________________________________________
Has the asymptote changed? If so, how? _______________________________________________________
Part C:
Now describe the changes if Part A and B are put together. Also, list the asymptote equations.
5.
y = log(x + 2) - 5
____________________________________________________________
6.
y = log(x – 3) + 7
____________________________________________________________
Part D: Now put in these functions and describe the changes
7.
y = 5 log(x)
____________________________________________________________
18
8.
y = 1/3 log(x)
____________________________________________________________
9.
y = - log(x)
____________________________________________________________
What can you say about values multiplied outside the function? What about the asymptote?
_______________________________________________________________________________________
Part E: Describe the changes that will take place from the parent graph. List any asymptotes.
10.
y = 5 log(x) + 3
____________________________________________________________
11.
y = -2 log(x-4) ________________________________________________________________
19
Unit 9 Test Review – Transformations and Graphs of Functions
Graph the following functions and answer the questions below.
1. y   ( x  2  2
Name of graph: ___________
Parent graph: ____________
Domain: ________________
Range: _________________
Asymptote: _______________
Pivot Point: _______________
4. f ( x)  2( x  1) 3  3
Name of graph: ___________
Parent graph: ____________
Domain: ________________
Range: _________________
Asymptote: _______________
Pivot Point: _______________
2. y  3 x  4
1
3. y   ( x  2)3
3
Name if graph: ____________
Parent graph: _____________
Domain: _________________
Range: __________________
Asymptote: _______________
Pivot Point: ________________
Name of graph: _________
Parent graph: ___________
Domain: ________________
Range: _______________
Asymptote: ____________
Pivot Point: _____________
5. y 
13
x
2
Name if graph: ____________
Parent graph: _____________
Domain: _________________
Range: __________________
Asymptote: _______________
Pivot Point: ________________
6. f ( x)  2 3 x  1  3
Name of graph: _________
Parent graph: ___________
Domain: ________________
Range: _______________
Asymptote: ____________
Pivot Point: _____________
20
7. y  3x 2  1
8. y  log x
Name of graph: ___________
Parent graph: ____________
Domain: ________________
Range: _________________
Asymptote: _______________
Pivot Point: _______________
Name if graph: ____________
Parent graph: _____________
Domain: _________________
Range: __________________
Asymptote: _______________
Pivot Point: ________________
10. f ( x)  4x  2
11. y  2 x  4
Name of graph: ___________
Parent graph: ____________
Domain: ________________
Range: _________________
Asymptote: _______________
Pivot Point: _______________
Name if graph: ____________
Parent graph: _____________
Domain: _________________
Range: __________________
Asymptote: _______________
Pivot Point: ________________
9. y  ( x  2)2  5
Name of graph: _________
Parent graph: ___________
Domain: ________________
Range: _______________
Asymptote: ____________
Pivot Point: _____________
12. f ( x)  2( x  3)3  2
Name of graph: _________
Parent graph: ___________
Domain: ________________
Range: _______________
Asymptote: ____________
Pivot Point: _____________
Describe how each of the following graphs transform in relation to the parent graph.
13. y   x  8  9 _________________________________________________________________
14. y  3 x  2
_________________________________________________________________
21
1
15. y  log( x  2)  3 _______________________________________________________________
4
16. y  3x2  3
__________________________________________________________________
1
17. y  ( x)3  2 __________________________________________________________________
3
18. y  2( x  1)3 ___________________________________________________________________
19. y   log x  5 __________________________________________________________________
2
20. y   x 2  3 __________________________________________________________________
3
21.
y  2 3 x  4 ___________________________________________________________________
22. y  4 x  1 _____________________________________________________________________
Write an equation of each of the translated graphs.
23. If the graph of y  3 3 x is shifted left 5 units and up 2 units, what is the equation of the
translated graph?
1 3
x is flipped over the x-axis and moved down 3 units, what is the
2
equation of the translated graph?
24. If the graph of y 
25. If the graph of y  log x is shifted left 4 units, up 5 units, and is vertically stretched by 3
what is the equation of the translated graph?
26. If the graph of y  x is flipped over the x-axis and moved down 6 units, what is the
equation of the translated graph?
27. If the graph of y  2 x is shifted left 1 units and up 7 units, what is the equation of the
translated graph?
28. If the graph of y  x  3 is flipped over the x-axis and moved down 7 units, what is the
equation of the translated graph?