TRANSFORMATIONS - LET THE COURSE BEGIN!!!
A transformation creates a new function out of an old one. There are two
basic types of transformations that we will study: Rigid and Non-Rigid. This
chapter shows how to do rigid transformations.
To start with we will transform functions form the Big 8 to create new
functions. Once you are good at this, you will be able to transform any function.
Use the abbreviation "XFORM" to represent transformation.
All transformations are discussed in one section, section 2.7, in our Lial book.
This is way too much in way too little space. It really takes about 5 or 6 sections to
discuss the transformations properly!
We begin with RIGID Vertical and Horizontal transformations on pages 266-267.
XFORM 1
RIGID VERTICAL SHIFT
Read the green box on page 266.
Vertical Transformation Definitions:
If c>0 then: f(x) + c is UP & f(x) - c is DOWN
Alternate Definition:
f(x) + c is UP if c>0 & DOWN If c<0.
XFORM 2
RIGID HORIZONTAL SHIFT
Read the green box on page 267.
Horizontal Transformation Definitions:
If c>0 then: f(x-c) is a shift RIGHT & f(x+c) is a shift LEFT.
Alternate Definition:
f(x+c) is a shift RIGHT if c<0 & LEFT if c>0.
RIGID XFORM PRACTICE:
1) Do the cool Vertical & Horizontal match exercise:
Hand out pics of transformed functions to some students and the algebraic
equivalents to others. Have them find their match and then hang the match on the
board.
Then students go around in pairs (designate experts and rotate) to make sure they
understand each f(x)-graph pair.
After the exercise point out the f(x)=x shifted left by 3 and up 3 are identical!
But do the same for f(x)=x2 and they are not identical. Why is f(x)=x so weird?
Ans: Because the exponent = 1!
2) Use f(x) = x3 + 2x2 - 3x - 5 to review Vertical and Horizontal shift combinations
on the
TI-84.
y1(x) = x3 + 2x2 - 3x - 5
y2(x) = y1(x+3)
y3(x) = (x+3)3 + 2(x+3)2 - 3(x+3) – 5
y4(x) = x3 + 11x2 + 36x + 31
Key Point: y2 = y3 = y4 (equivalent!).
Use the tracer feature of the TI-calculator to demo this fact!
Have the students expand y3 to get y4 & give a bonus point to the first one
with the correct answer!
Do this again with f(x) = x2.
y1(x) = x2
y2(x) = y1(x-1)
y3(x) = (x-1)2
y4(x) = (x-1)(x-1)
y5(x) = x2 - 2x + 1
Key Point: y2 = y3 = y4 = y5. Again, use the tracer feature of the TI-calculator
to demo this fact!
3)
XFORM 3
RIGID VERTICAL & HORIZONTAL COMBOS
Use the graph on page 271 #59 of the Lial text to graph f(x)-1; f(x-1); f(x+1); f(x2)+1. This is a great problem to have the students do. Have them use graph paper
while I use the side board with the grid.
Pop Quiz:
1) Sketch y
2) Sketch y
x 3; y
x 4; y
x 4 3
1
1
1
; y
5; y
5
x 5
x
x 5
3) Have the students work 270, #3 of the Lial text.
XFORM 4 RIGID REFLECTION IN THE X-AXIS: f(x) --> −f(x)
Consider the discrete function f(x) = {(1,5), (−3,2), (-6,−8),(4, −7)}.
In table form, f(x) and its transform the new function g(x) = –f(x) are:
x
y = f(x)
y = -f(x) = g(x)
1
5
-5
-3
2
-2
-6
-8
8
4
-7
7
Note that it is the y-values that change sign but the x-‘values’ stay the same!
XFORM 5
RIGID REFLECTION IN THE Y-AXIS: f(x) --> f(-x)
Consider again the discrete function f(x) = {(1,5), (−3,2), (-6,−8),(4, −7)}.
In table form, f(x) and its transform the new function h(x) = f(-x) are:
x
y = f(x)
x
y = f(-x) = h(x)
1
5
-1
5
-3
2
3
2
-6
-8
6
-8
4
-7
-4
-7
Note that it is the x-values that change sign, but the y-values stay the same!
Now try sketching
x 3 9 and
7 x 2 . These are tricky!
XFORM 6 RIGID REFLECTION COMBOS
Use f(x) = x3 + 2x2 - 3x - 5 on the TI84 to demo -f(x) & f(-x) & -f(-x) & f(-x),
followed by f(-(x-3))=f(3-x)=f(-x+3)
Demo how order can be important!
Around x-axis then up v.s. up then around the x-axis: different!
Around y-axis then up v.s. up then around the y-axis: same!
The Moral: Use PEMDAS and be careful with the transformation f(x+c)! Demo
x 3 2.
RIGID XFORM SUMMARY
At this point you should be able to graph any rigid transformation combo!
Transform Practice: p266#1-8all from the Larson book (make this a handout to
students)
XFORM 7
EVEN & ODD FUNCTIONS (RIGID)
'Odd' & 'Even' Functions are 'special'.
Read p 264 in the Lial book.
Odd: f(x)=-f(-x)
Even: f(x)=f(-x)
f(x)
c
x
x2
Odd
N
Y
N
Even
Y
N
Y
x3
abs(x)
sqrt(x)
1/x
[[x]]
Y
N
N
Y
N
N
Y
N
N
N
Explain that a better definition for odd is f(x) = -f(-x).
Show that -f(-x) = f(x) for f)x) = x3 and that f(-x) = f(x) for f(x) = x2
Here I did an activity where I put the following 9 questions on the board and had
them solve the problems in groups of 3:
I drew two piecewise functions on the same graph:
1. How many times will f(x) + g(x) = =. (Hint: f(X) + g(x) = 0 is EQUIVALENT TO
g(x) = −f(x), so reflect f around the x-axis & see how many times it crosses the
graph of g(x).
2. g(f(6))=? (2);
f(g(6))=? (1); g(f(-6))=? (0);
f(g(-6))=?1; f(0)+g(0)=? (2)
3. The range of f(g(x)) is ? ([0,1])
4. – 7. Lial p273 #75-76
8. Label each function as E (even), O (odd), N (neither): x3-x (O); x3-x+1 (N); x3+x
(O); x3+x+1 (N); x2 (E); x2+1 (E); x2-1 (E); x2+x (N); x2-x (N)
Hint: in each case check to see which is true: f(x)=f(-x) (E) or f(x)=-f(-x) (O) or
neither.
NON-RIGID TRANSFORMATIONS: c1f(c2x)
Read Lial page 260.
XFORM 8
NON-RIGID VERTICAL STRETCH
XFORM 9
NON-RIGID HORIZONTAL STRETCH
XFORM 10 NON-RIGID REFLECTIONS
NON-RIGID XFORM KEY POINTS
1) cf(x) changes the max/min values but keeps zeros
2) f(cx) keeps the max/min values, BUT changes their horizontal location
3) Use f(x) = x3 + 2x2 - 3x - 5 on the TI84 to demo these phenomena
4) The best way to demo non-rigid transformations is by using f(x) = sin(x) on the
TI-84
RIGID & NON-RIGID TRANSFORM REVIEW
RIGID TRANSLATIONS:
f(x) ± c, f(x ± c), -f(x), f(-x)
f(x) + c is UP if c > 0; DOWN if c < 0
f(x) - c is UP if c < 0; DOWN if c > 0
f(x - c) is a shift RIGHT if c > 0 and LEFT if c < 0
f(x + c) is a shift RIGHT if c < 0 and LEFT if c > 0
RIGID ROTATIONS:
f(-x) is a rotation around the y-axis.
-f(x) is a rotation around the x-axis.
-f(-x) is a rotation around both axes.
NON-RIGID TRANSFORMATIONS (vertical/horizontal stretching/shrinking):
Vertical: cf(x): Stretch if c > 1 & Shrink if 0 < c < 1;
Horizontal f(cx): Stretch if 0 < c < 1 & Shrink if c > 1
RIGID & NON-RIGID COMBOS: c1f(c2x-c3) + c4
Rigid & non-rigid Transform Practice: p266#9-12 all from the Larson book (make
this a handout to students)
Note to myself: Here is a good exercise to explain the potentially confusing
transformation involving c-x. First have them work the following two problems:
Larson page 225#12d first, then 7 x 2 . I will probably have to explain them in
that order. VERY IMPORTANT SIDE BENEFIT: These are a good lean-in to the
next material. These problems reinforce the idea of the meaning of equivalent
functions - all the various forms derived at during algebraic manipulations still have
the same graph - they are equivalent!! Point out that sometimes it is beneficial to
write a different form to bring out a certain aspect of a problem. ex: in one form
the vertex shows up, in another form the y-intercept is clear, in another (factored
form) the x-intercepts (zeros) show up.
Explain that Sqrt(7-x)-2=Sqrt[-(x-7)]-2 => Sqrt(x)-->Sqrt(-x)-->Sqrt[-(x-7)]-2 =>
Rotate around y-axis, followed by rt shift by 7, followed by down 2. Explain that
the order of transformation is important. There is more than one correct order of
transformations, and there are wrong orders. ¡Cuidado!
Now have them work the "Transformations of
x " Bonus Test.
A good handout: Larson page266#1-12all Review of Rigid and Non-Rigid
Transforms
A FUN & GOOD WAY TO REVIEW TRANFORMS:
Have them come to the board in pairs. I give them and equation and a shift
verbally. One side graphs the equation and the other side writes the equation.