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Families of Functions

Functions that are related in some way are often called a family.

For example, linear functions that have the same slope form a family. The graphs of these functions are parallel lines.

Let’s look at the graphs of the functions:

f(x) = -2x

f(x) = -2x - 3

f(x) = -2x + 4

Here are some points that satisfy each of these functions:

x f(x) = -2x f(x) = -2x - 3 f(x) = -2x + 4
-2

0

2

(-2, 4)

(0, 0)

(2, -4)

(-2, 1)

(0, -3)

(2, -7)

(-2, 8)

(0, 4)

(2, 0)

 

The graphs of the functions are shown. Notice the relationship between the three graphs.

• The top graph is f(x) = -2x.

• The second graph is f(x) = -2x - 3. Notice that for each input x, the output is 3 less than f(x) = -2x. This means that the graph of f(x) = -2x - 3 is the same as that of f(x)The graph of f(x) 2 3 x is shown. Line A and Line B are parallel to f(x). a. Find the equation of the function for Line A.x except that it is shifted down 3 units.

• The third graph is f(x) = -2x + 4. Notice that for each input x, the output is 4 more than f(x) = -2x. This means that the graph of f(x) = -2x + 4 is the same as that of f(x) = -2x except that it is shifted up 4 units.

This means that we could graph all three functions by first graphing f(x) = -2x. Then, we could draw the graph of f(x) = -2x - 3 by shifting the graph of f(x) = -2x down 3 units. Likewise, we could draw the graph of f(x) = -2x + 4 by shifting the graph of f(x) = -2x up 4 units.

 

Example

The graph of is shown. Line A and Line B are parallel to f(x).

a. Find the equation of the function for Line A.

b. Find the equation of the function for Line B.

Solution

a. Line A is the function shifted up 5 units. So, the function for Line A is

b. Line B is the function shifted down 4 units. So, the function for Line B is

 

 
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