Describing Transformations of Rational Functions
The common transformations can be applied to rational functions as usual.
Translation
By adding some number to every function value, a function graph is translated vertically. Notice that the horizontal asymptote changes, while the vertical one is unaffected.
A graph is translated horizontally by subtracting a number from the input of the function rule. Note that the number, is subtracted and not added. This is so that a positive leads to a translation to the right, which is the positive -direction. This transformation affects the vertical asymptote, but not the horizontal.
Reflection
A function is reflected in the -axis by changing the sign of all function values: Graphically, all points on the graph move to the opposite side of the -axis, while maintaining their distance to the -axis. Thus, -intercepts and vertical asymptotes are preserved.
A graph is instead reflected in the -axis by moving all points on the graph to the opposite side of the -axis. This occurs by changing the sign of the input of the function. Notice that the -intercept and horizontal asymptote are preserved.
Stretch and Shrink
A function's graph is vertically stretched or shrunk by multiplying the function rule by some constant : All vertical distances from the graph to the -axis are changed by the factor Thus, preserving any -intercepts.
By instead multiplying the input of a function rule by some constant its graph will be horizontally stretched or shrunk by the factor Since the -value of -intercepts is they are not affected by this transformation.
Describe how the function is a transformation of its parent function, Then, add the graph of to the coordinate plane below.
To transform into has to be added to the input, followed by adding to the function value: This transformation corresponds with a translation units to the left and unit upward. Thus, we can simply translate the graph of accordingly to sketch This can be done by first choosing some points on the graph of and plotting their counterpart for
Now, can be sketched by connecting the points with two curves, mimicking the appearance of Notice that the asymptotes have been translated units leftward and unit upward as well.
