Linear Functions and Modeling

Transforming Linear Graphs

Translations of Linear Functions

A translation of a linear function shifts its graph vertically or horizontally.

The parent function, or most basic function in the linear family, is the linear function f(x)=xf(x) = x. Its graph is a line that passes through the origin and has a slope of m=1m=1. Other linear functions can be graphed as a transformation of the parent function f(x)=xf(x) = x.

A translation, or shift, is a transformation in which a graph is moved vertically or horizontally. For any function f(x)f(x), the function can be translated vertically kk units or horizontally hh units.

For h>0h\gt0 and k>0k\gt0:

  • The graph of f(x)+kf(x)+k is the graph of f(x)f(x) translated up kk units.
  • The graph of f(x)kf(x)-k is the graph of f(x)f(x) translated down kk units.
  • The graph of f(xh)f(x-h) is the graph of f(x)f(x) translated right hh units.
  • The graph of f(x+h)f(x+h) is the graph of f(x)f(x) translated left hh units.

A translation of the parent linear function f(x)=xf(x) = x can be viewed as either a horizontal or vertical translation.

For example, the parent function of a line is:
f(x)=xf(x) = x
Its yy-intercept is zero. Its xx-intercept is also zero. One translation of the parent function is:
f(x)=x+4f(x)=x+4
It is a vertical translation of 4 units up or a horizontal translation 4 units to the left. The yy-intercept is 4. The xx-intercept is –4. Another translation is:
f(x)=x5f(x)=x-5
It is a vertical translation 5 units down or a horizontal translation 5 units to the right. The yy-intercept is –5. The xx-intercept is 5. All functions have a slope of 1.
Translations of a parent function, such as f(x)=xf(x)=x, consists of a shift up, down, left, or right. For instance, a shift of the parent function 4 units up or to the left is represented by f(x)=x+4f(x)=x+4. A shift 5 units down or to the right is represented by f(x)=x5f(x)=x-5.

Stretches, Compressions, and Reflections of Linear Functions

A stretch or compression of the graph of a linear function is a pull away from an axis or a push toward an axis. A reflection of the graph of a linear function is a flip across a line.

For any function f(x)f(x), the function can be stretched, compressed, or reflected across an axis. A reflection is a transformation in which a figure is flipped across a line.

For a>0a>0, where aa is a scaling factor:

  • The graph of af(x)af(x) is a vertical stretch or compression of the graph of f(x)f(x) by a factor of aa. For a>1a \gt 1, the graph of f(x)f(x) is stretched. For 0<a<10\lt a\lt 1, the graph of f(x)f(x) is compressed.
  • The graph of f(x)-f(x) is the reflection of the graph of f(x)f(x) across the xx-axis.
For example, the parent function of a line is:
f(x)=xf(x) = x
Its slope is 1. The function that shows the vertical stretch of the parent function by a factor of 4 is:
f(x)=4xf(x)=4x
The slope of the vertical stretch function is 4. The function that shows a vertical compression of the parent function by a factor of 14\frac{1}{4} is:
f(x)=14xf(x)=\frac{1}{4}x
The slope of the vertical compression function is 14\frac{1}{4}.
The function that shows a reflection across the xx-axis is:
f(x)=xf(x)=-x
The slope of the reflection function is –1. All three functions have a yy-intercept of zero.
Stretches, compressions, and reflections of parent functions largely depend on the slope. For a parent function, such as f(x)=xf(x) = x, a vertical stretch of 4 is represented by f(x)=4xf(x)=4x. A vertical compression of 14\frac{1}{4} is represented by f(x)=14xf(x)=\frac{1}{4}x. A reflection of the parent function over the xx-axis is represented by f(x)=xf(x)=-x.

Combined Transformations of Linear Functions

Some linear functions are the result of multiple transformations of the linear parent function.
Sometimes, a parent function may be transformed in more than one way. In general, the order of transformations is like the order of operations: Perform stretches, compressions, and reflections first (multiplication and division), followed by translations (addition and subtraction).
Step-By-Step Example
Graphing a Linear Function with Combined Transformations
Graph the given linear function by using transformations of its parent function f(x)=xf(x)=x:
f(x)=2x+3f(x)=2x+3
Step 1
Identify the types of transformations of the parent function f(x)=xf(x)=x that are used in the given function:
f(x)=2x+3f(x)=2x+3
The first term in the linear function is 2x2x. It shows that the parent function is multiplied by 2, so the graph will be stretched vertically by a factor of 2.

The next term in the given linear function is 3. It shows that 2 is added to the vertically stretched function f(x)=2x. So, the vertically stretched graph will be translated 3 units up.

Step 2
Graph the first transformation:
f(x)=2xf(x)=2x
The graph of the parent function is vertically stretched by a factor of 2.
Solution
Then graph the second transformation. The function f(x)=2xf(x)=2x is translated up 3 units. This is represented by the graph of the function:
f(x)=2x+3f(x)=2x+3