Section 2.5 part 1: Odd/Even Functions, Vertical Changes
A function f is even if f (−x) = f (x) for all x. (“The − inside is canceled”.) For example, even powers are
even functions: (−x)2 = x2 , (−x)4 = x4 , etc. (but not even roots).
A function f is odd if√f (−x) = √
−f (x) for all x. (“The − comes out of f ”.) Odd powers and roots are odd
functions: (−x)3 = −x3 , 3 −x = − 3 x, etc.
Remark: An even function has y-axis symmetry, and an odd function has origin symmetry!
Ex 1: Determine if the following functions are even, odd, or neither.
(a) f (x) =
√
3
x−x
(b) g(x) =
x2
1 + x4
(c) h(x) =
x3
x+1
Graphing Absolute Values |f (x)|
A very important even function is the absolute value function
(
(
x
if x ≥ 0
f (x)
if f (x) is not negative
so |f (x)| =
|x| =
−x if x < 0
−f (x) if f (x) < 0
Think: The absolute value leaves positive numbers alone and “flips” negative numbers to the opposite sign.
When graphing |f (x)|, flip the negative parts over the x-axis!
Domain and Range of y = |f (x)|: The domain of y = |f (x)| is the same as for y = f (x); the graph doesn’t
move left or right. The range, though, usually does change! I suggest you draw a sample picture for y = f (x)
and then flip its negative part over to see the new range.
Ex 2: If y = f (x) has domain [−2, 3] and range [−4, 1], what is the domain and range of y = |f (x)|?
Suggestion: Start with any sketch with x going from −2 to 3 (the domain) and y from −4 to 1 (the range).
Vertical Transformations
We will next study transformations that change how a graph looks. Today, we’ll look at vertical transformations: they change y-values of points, and they affect the range of a function.
Vertical transforms are located on the outside of the function f (x). There are several types:
Type
Shift up by k
Shift down by k
Stretch vertically by c > 0
Compress vertically by c > 0
Reflect over x-axis
Samples:
y = 2f (x)
y = f (x) − 4
Stretches vertically by 2
Moves 4 units down
Equation
y = f (x) + k
y = f (x) − k
y = c · f (x)
y = f (x)/c
y = −f (x)
Effect
Moves the graph k units up
Opposite of shifting up
Multiplies all y-values by factor of c
Opposite of stretching
Flips the entire graph upside-down
This can be combined with a stretch.
y = −f (x)/3
y = 2f (x) + 1
Flips graph over, vertically compresses by 3
Stretches by 2, then moves up 1
Remarks:
• These transformations follow the order of operations. Stretches and reflections happen before shifts.
• A stretch by 1/c is the same as a compression by c.
• Vertical transforms change both ends of the range (think: all values in the range are y-values!).
– Example: Say y = f (x) has range [1, 2]. y = f (x) + 1 has range [1 + 1, 2 + 1], and y = −f (x) has
range [−2, −1]. (Why is it not [−1, −2]??)
Ex 3: Graph y = 2|x| − 5. If P (−1, 1) is on the graph of y = |x|, what’s the corresponding point on
y = 2|x| − 5?
Getting started: Sketch the basic (untransformed) function y = |x| and the point P . Apply each transformation one at a time to both the graph and the point’s coordinates.
Ex 4: If a function y = f (x) has range [−2, 4], then what is the range of
(a) y = f (x)/2 + 1?
(b) y = −f (x)/2 + 1?
Next class, we will look at horizontal transformations as well.