Graphs of Functions (Part 1)
2.5(1)
Even/ odd functions
Shifts and Scale
Changes
POD
Simplify the following difference quotient,
for f(x) = x3.
f ( x  h)  f ( x )
h
What does this expression become as h
approaches 0?
POD
Simplify the following difference quotient,
for f(x) = x3.
3x  3xh  h
2
2
What does this expression become as h
approaches 0?
Even/ odd Functions
Even functions are reflected over an axis-- which one?
Odd functions are reflected over something else-- what?
The acid test:
For even functions, f(-x) = f(x)
For odd functions, f(-x) = -f(x)
(What does that mean in English?)
Write which is which on the parent function handout.
At each table, graph two of the following to
see if they are even, odd, or neither
As you graph each one, test it algebraically.
f ( x)  x 3
p( x)  3x 4  2 x 2  5
g ( x)  x 2  4
q ( x)  2 x 5  7 x 3  4 x
h( x)  ( x  4) 2
k ( x)  x 3  x 2
m( x )  x 3  1
t ( x)  x 2  6
At each table, graph two of the following to
see if they are even, odd, or neither
Notice how p(-x) = p(x), once everything is simplified–
it is an even function. The graph matches this.
p( x)  3( x) 4  2( x) 2  5
 3x 4  2 x 2  5
 p ( x)
At each table, graph two of the following to
see if they are even, odd, or neither
Notice how q(-x) = -q(x), once everything is simplified– it
is an odd function. The graph matches this.
q( x)  2( x) 5  7( x) 3  4( x)
 2( x 5 )  7( x 3 )  4( x)
 2 x 5  7 x 3  4 x
 ( 2 x 5  7 x 3  4 x )
 q( x)
At each table, graph two of the following to
see if they are even, odd, or neither
Notice how k(-x) ≠ k(x) or -k(x), once everything is
simplified– it is neither even nor odd. The graph matches this.
k ( x)  ( x) 3  ( x) 2
 x3  x 2
Vertical/ horizontal shifts
Start by graphing y = x2 and then y = x2 + 5.
What do you notice about the relationship between
the graphs? How does that compare to the
relationship between the equations?
Now, graph y = (x + 5)2.
How do the graph and equation compare to y = x2?
Vertical/ horizontal shifts
y = x2
y = x2 + 5
(y – 5 = x2)
y = (x + 5)2
Shifts and equations in general
A vertical shift of c:
y = f(x) + c
y - c = f(x)
A horizontal shift of c:
y = f(x-c)
How would each of these graphs compare to y = x2
y = x2 + 6x + 9
y = x2 - 6x + 9
y = x2 +3?
Vertical/ horizontal scale changes
(Vertical/ horizontal stretching and compressing)
Start by graphing
y = sin x
Then, at tables graph one of these:
y = 3sin x
y = 1/3 sin (x)
y = sin (3x)
y = -3sin(x)
What do you notice about the relationship
between the graphs? Between the equations?
Vertical/ horizontal scale changes
(Vertical/ horizontal stretching and compressing)
y = sin x
y = 3sin x
What do you notice about the relationship between
the graphs? Between the equations?
Vertical/ horizontal scale changes
(Vertical/ horizontal stretching and compressing)
y = sin x
y = sin (3x)
What do you notice about the relationship between
the graphs? Between the equations?
Vertical/ horizontal scale changes
(Vertical/ horizontal stretching and compressing)
y = sin x
y = (1/3)sin x
What do you notice about the relationship between
the graphs? Between the equations?
Vertical/ horizontal scale changes
(Vertical/ horizontal stretching and compressing)
y = sin x
y = -3sin x
What do you notice about the relationship between
the graphs? Between the equations?
Scale changes and equations in general
A vertical stretch of c:
y = cf(x)
y/c = f(x)
A horizontal stretch of c: y = f(x/c)
We’re used to thinking of expansion with values of c
greater than 1. How would we achieve a
compression?
Scale changes and equations in general
If c is negative, the graph will
reflect over the y- axis when multiplied by x.
reflect over the x-axis when multiplied by y.
The bottom line…
If you change the equation, you do the
opposite to the graph.
When you change a graph, everything
changes: intercepts, asymptotes, holes,
domain, and range.
Scale change and translation
Many graphs have a combination of
scale change and translation
(multiplication and addition). In that
case, just like PEMDAS, you multiply
then add– scale change then
translation– in each direction.
Scale change and translation
Try it with a point. Give the
coordinates of the point (3, -1) after
the function undergoes the
transformation
y = 2f(x+5) – 5. Remember, scale
change first.
Scale change and translation
Try it with a point. Give the
coordinates of the point (3, -1) after
the function undergoes the
transformation
y = 2f(x+5) – 5.
(-2, -7)
Try it, if there’s time
In groups, find a parent (tool kit) function that is even, or one
that is odd. Graph it on your calculators.
Shift it 3 units up and 4 units to the left. What is the new
equation? Graph that to test your work.
Stretch your original graph horizontally by a factor of 2, and
reflect it over the x-axis. What is the new equation? Graph
that to test your work.
For a Take a Chance Award, come up to demonstrate your work.