Introduction
You can change a function’s position or shape by adding
or multiplying a constant to that function. This is called a
transformation. When adding a constant, you can
transform a function in two distinct ways. The first is a
transformation on the independent variable of the
function; that is, given a function f(x), we add some
constant k to x: f(x) becomes f(x + k). The second is a
transformation on the dependent variable; given a
function f(x), we add some constant k to f(x): f(x)
becomes f(x) + k.
1
5.8.1: Replacing f(x) with f(x) + k and f(x + k)
Introduction, continued
In this lesson, we consider the transformation on a
function by a constant k, either when k is added to the
independent variable, x, or when k is added to the
dependent variable, f(x). Given f(x) and a constant k, we
will observe the transformations f(x) + k and f(x + k), and
examine how transformations affect the vertex of a
quadratic equation.
2
5.8.1: Replacing f(x) with f(x) + k and f(x + k)
Key Concepts
• To determine the effects of the constant on a graph,
compare the vertex of the original function to the
vertex of the transformed function.
• Neither f(x + k) nor f(x) + k will change the shape of
the function so long as k is a constant.
• Transformations that do not change the shape or size
of the function but move it horizontally and/or
vertically are called translations.
• Translations are performed by adding a constant to
the independent or dependent variable.
3
5.8.1: Replacing f(x) with f(x) + k and f(x + k)
Key Concepts, continued
Vertical Translations—Adding a Constant to the
Dependent Variable, f(x) + k
• f(x) + k moves the graph of the function k units up or
down depending on whether k is greater than or less
than 0.
• If k is positive in f(x) + k, the graph of the function will
be moved up.
• If k is negative in f(x) + k, the graph of the function will
be moved down.
4
5.8.1: Replacing f(x) with f(x) + k and f(x + k)
Key Concepts, continued
Vertical translations: f(x) + k
When k is positive, k > 0, the When k is negative, k < 0,
graph moves up:
the graph moves down:
5
5.8.1: Replacing f(x) with f(x) + k and f(x + k)
Key Concepts, continued
Horizontal Translations—Adding a Constant to the
Independent Variable, f (x + k)
• f(x + k) moves the graph of the function k units to the
right or left depending on whether k is greater than or
less than 0.
• If k is positive in f(x + k), the function will be moved to
the left.
• If k is negative in f(x + k), the function will be moved to
the right.
6
5.8.1: Replacing f(x) with f(x) + k and f(x + k)
Key Concepts, continued
Horizontal translations: f(x + k)
When k is positive, k > 0, the When k is negative, k < 0,
graph moves left:
the graph moves right:
7
5.8.1: Replacing f(x) with f(x) + k and f(x + k)
Common Errors/Misconceptions
• incorrectly moving the graph in the direction opposite
that indicated by k, especially in horizontal shifts; for
example, moving the graph left when it should be
moved right
• incorrectly moving the graph left and right versus up
and down (and vice versa) when operating with f(x + k)
and f(x) + k
8
5.8.1: Replacing f(x) with f(x) + k and f(x + k)
Guided Practice
Example 1
Consider the function f(x) = x2 and the constant k = 2.
What is f(x) + k? How are the graphs of f(x) and f(x) + k
different?
9
5.8.1: Replacing f(x) with f(x) + k and f(x + k)
Guided Practice: Example 1, continued
1. Substitute the value of k into the function.
If f(x) = x2 and k = 2, then f(x) + k = x2 + 2.
10
5.8.1: Replacing f(x) with f(x) + k and f(x + k)
Guided Practice: Example 1, continued
2. Use a table of values to graph the
functions on the same coordinate plane.
x
f(x)
f(x) + 2
–2
4
6
–1
1
3
0
0
2
1
1
3
2
4
6
11
5.8.1: Replacing f(x) with f(x) + k and f(x + k)
Guided Practice: Example 1, continued
12
5.8.1: Replacing f(x) with f(x) + k and f(x + k)
Guided Practice: Example 1, continued
3. Compare the graphs of the functions.
Notice the shape and horizontal position of the two
graphs are the same. The only difference between
the two graphs is that the value of f(x) + 2 is 2 more
than f(x) for all values of x. In other words, the
transformed graph is 2 units up from the original
graph.
✔
13
5.8.1: Replacing f(x) with f(x) + k and f(x + k)
Guided Practice: Example 1, continued
14
5.8.1: Replacing f(x) with f(x) + k and f(x + k)
Guided Practice
Example 3
Consider the function f(x) = x2, its graph, and the
constant k = 4. What is f(x + k)? How are the graphs of
f(x) and f(x + k) different?
15
5.8.1: Replacing f(x) with f(x) + k and f(x + k)
Guided Practice: Example 3, continued
1. Substitute the value of k into the function.
If f(x) = x2 and k = 4, then f(x + k) = f(x + 4) = (x + 4)2.
16
5.8.1: Replacing f(x) with f(x) + k and f(x + k)
Guided Practice: Example 3, continued
2. Use a table of values to graph the
functions on the same coordinate plane.
x
f (x)
f (x + 4)
–6
36
4
–4
16
0
–2
4
4
0
0
16
2
4
36
4
16
64
5.8.1: Replacing f(x) with f(x) + k and f(x + k)
17
Guided Practice: Example 3, continued
18
5.8.1: Replacing f(x) with f(x) + k and f(x + k)
Guided Practice: Example 3, continued
3. Compare the graphs of the functions.
Notice the shape and vertical position of the two
graphs are the same. The only difference between
the two graphs is that every point on the curve of
f(x) has been shifted 4 units to the left in the graph
of f(x + 4).
✔
19
5.8.1: Replacing f(x) with f(x) + k and f(x + k)
Guided Practice: Example 3, continued
20
5.8.1: Replacing f(x) with f(x) + k and f(x + k)