Translations of Linear Functions
The parent function, or most basic function in the linear family, is the linear function . Its graph is a line that passes through the origin and has a slope of . Other linear functions can be graphed as a transformation of the parent function .
A translation, or shift, is a transformation in which a graph is moved vertically or horizontally. For any function , the function can be translated vertically units or horizontally units.
For and :
- The graph of is the graph of translated up units.
- The graph of is the graph of translated down units.
- The graph of is the graph of translated right units.
- The graph of is the graph of translated left units.
A translation of the parent linear function can be viewed as either a horizontal or vertical translation.For example, the parent function of a line is:
Stretches, Compressions, and Reflections of Linear Functions
For any function , 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 , where is a scaling factor:
- The graph of is a vertical stretch or compression of the graph of by a factor of . For , the graph of is stretched. For , the graph of is compressed.
- The graph of is the reflection of the graph of across the -axis.
The function that shows a reflection across the -axis is:
Combined Transformations of Linear Functions
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.