The method of graphing functions by plotting points is effective for a computer or calculator, which can evaluate functions quickly for a lot of points. However, it is not very efficient as a way to graph functions by hand.
A more efficient way is to understand some of the common types of functions and how changes to the rule of the function affect the graph. A parent function is a function from a certain category of functions with the simplest algebraic rule. Some common parent functions include the linear function , the quadratic function , and the cubic function .
When the algebraic rule of a parent function is manipulated in certain ways, there is a change, or transformation, of the graph of the function. Some examples of transformations are translations, stretches, compressions, and reflections.
Graphs of Parent Functions
Translations of Graphs
A translation, or shift, is a transformation in which a graph is moved vertically or horizontally. Adding or subtracting a constant to the input or output of a function will translate the graph in the coordinate plane without turning, flipping, or changing its shape.
When a constant is added to or subtracted from the output of the parent function, the graph is translated vertically.
|For the graph of and :
|For the graph of and :
The graph shows the parent function , its horizontal translation to the right as , and its horizontal translation to the left as .
Vertical Stretches and Compressions
A stretch is a transformation that moves each point of a graph farther away from the - or -axis by a scale factor that is greater than 1.
A compression is a transformation that moves each point of a graph closer to the - or -axis by a scale factor that is between zero and 1.
When the output of the parent function is multiplied by a positive constant, the graph stretches or compresses vertically.
Graphing Vertical Stretches and Compressions
|Vertical Stretches and Compressions||Example|
|For the graph of and ,
Reflections of Graphs
|Reflections Across the -Axis||Example|
|For the graph of , the graph of is a reflection of the graph of across the -axis.
The graph shows the relation of the parent function and its reflection across the -axis as .
When multiple transformations are applied to a function, it is important to perform the transformations in the correct order. Think of the order of operations: If operations are not performed in the correct order, the results may be different. Also, like the order of operations, transformations involving multiplication should be performed before addition and subtraction.
Order of Transformations
1. Identify the parent function.
2. Perform any reflections.
3. Perform any stretches or compressions.
4. Perform any translations.
Identify the parent function.The given function is:
Perform any reflections.The first term of the given function contains multiplication by 3, which is positive.
Perform any stretches or compressions.The first term of the given function contains multiplication by 3, so the graph of the given function is a vertical stretch of the parent function by a factor of 3:
Perform any translations.The first term of the given function contains a subtraction of 2. The second term is an addition of 8.