c
Math 150, Fall 2008, Benjamin
Aurispa
Chapter 2, Continued
2.4 Transformations of Functions
This section tells how transformations change the graph of a function. It is very useful in graphing functions,
when you know its general shape.
1. Vertical Shifts
Suppose we are given y = f (x) and c > 0.
(a) To graph y = f (x) + c, shift the graph of y = f (x) up by c.
(b) To graph y = f (x) − c, shift the graph of y = f (x) down by c.
Examples: Sketch the graphs of the following functions.
g(x) = x2 + 4
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
h(x) = x3 − 3
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
2. Horizontal Shifts
Suppose we are given y = f (x) and c > 0.
(a) To graph y = f (x − c), shift the graph of y = f (x) to the right by c.
(b) To graph y = f (x + c), shift the graph of y = f (x) to the left by c.
Examples: Sketch the graphs of the following functions.
g(x) = (x − 1)2
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
h(x) =
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
1
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
√
x+2
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
c
Math 150, Fall 2008, Benjamin
Aurispa
3. Reflections
(a) To graph y = −f (x), reflect the graph of y = f (x) across the x-axis.
(b) To graph y = f (−x), reflect the graph of y = f (x) across the y-axis.
Examples:
g(x) = −|x|
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
h(x) =
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
√
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
−x
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
4. Vertical Stretching and Shrinking
To graph y = cf (x):
(a) If c > 1, stretch the graph of y = f (x) vertically by a factor of c.
(b) If 0 < c < 1, shrink the graph of y = f (x) vertically by a factor of c.
Examples:
g(x) = 21 x2
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
h(x) = 2x3
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
5. Horizontal Scaling
To graph y = f (cx):
(a) If c > 1, shrink the graph of y = f (x) horizontally by a factor of 1/c.
(b) If 0 < c < 1, stretch the graph of y = f (x) horizontally by a factor of 1/c.
Examples:
g(x) = |3x|
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
2
h(x) =
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
q
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
1
4x
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
c
Math 150, Fall 2008, Benjamin
Aurispa
You can also combine many of these transformations on a function.
Examples:
• g(x) = −2(x + 1)2 + 3
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
• h(x) =
p
p
p
p
p
p
p
√
−3x − 2
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
Even and Odd Functions
Let f be a function.
1. f is even if f (−x) = f (x) for all x in the domain.
The graph of an even function is symmetric about the y-axis.
2. f is odd if f (−x) = −f (x) for all x in the domain.
The graph of an odd function is symmetric about the origin.
Determine whether the following functions are even, odd, or neither.
f (x) = x +
1
x
f (x) = x3 − x2 + 1
f (x) = x4 − 4x2
3
c
Math 150, Fall 2008, Benjamin
Aurispa
2.5 Quadratic Functions; Maxima and Minima
A quadratic function is a function of the form f (x) = ax2 + bx + c where a 6= 0.
A quadratic function always has a parabola as its graph. To determine information about the parabola, we
must write the quadratic function in standard form, f (x) = a(x − h)2 + k, by completing the square.
Once we have written the quadratic function in standard form, then the graph of f is a parabola with vertex
(h, k). The parabola opens upward if a > 0 and opens downward if a < 0. We can see why this is by using
transformations.
Example: Write the following quadratic function in standard form, find the vertex of the parabola, and
sketch the graph.
f (x) = 3x2 − 6x + 1
Let f be a quadratic function written in standard form, f (x) = a(x − h)2 + k. The maximum or minimum
value of f occurs at the vertex.
If a > 0, then the minimum value of f is k and it occurs when x = h.
If a < 0, then the maximum value of f is k and it occurs when x = h.
Find the maximum or minimum value of f (x) = −2x2 − 12x − 10.
There is another way of determining the vertex and maximum/minimum of a quadratic function.
b
The maximum or minimum value of a quadratic function f (x) = ax2 + bx + c occurs at x = − 2a
.
b
If a > 0, then the minimum value is f (− 2a
).
b
If a < 0, then the maximum value is f (− 2a
).
b
b
b
b
So, essentially, h = − 2a
and k = f (− 2a
) so that the vertex is − 2a
, f (− 2a
) .
4
c
Math 150, Fall 2008, Benjamin
Aurispa
Example: Find the maximum or minimum value of the quadratic function f (x) = −4x2 − 24x − 41.
The domain of a quadratic function is always all real numbers: (−∞, ∞).
However, the range of a parabola depends on where the vertex of the parabola is and whether it opens
upward or downward.
If a > 0, then the range of f is [k, ∞).
If a < 0, then the range of f is (−∞, k].
Example: What is the range of the quadratic function, f (x) = x2 − 5x − 9?
If a function has many maxima or minima, depending on what part of the domain we are looking at, then
these maxima/minima are called local maxima/minima. You can find local maxima/minima using your
calculator.
Example: Find the local maxima/minima of the function x4 − 5x3 + 6x + 5.
5