Transformations of Exponential Functions
Rule
Transformations of Exponential Functions
The types of transformations that are used to transform linear and absolute value functions, can also be applied to exponential functions in the same way. The way transformations affect graphs is independent of what type of graph is being transformed. Thus, even though exponential functions are fundamentally different from linear ones, they are affected similarly by these transformations.
Rule
Translation
By adding some number to every function value, a function graph is translated vertically. Notice that since the resulting function has a constant term, it can't be written in the form Thus, is not an exponential function. This is the only of the following transformations that changes an exponential function into something other than an exponential function.
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 — a positive translates the graph to the right.
Rule
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.
A graph is instead reflected in the -axis, moving all points on the graph to the opposite side of the -axis, by changing the sign of the input of the function. Note that the -intercept is preserved.
Rule
Stretch and Shrink
A function 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.
The function is a downward translation of by units. Therefore, we can begin by graphing and then translate that graph downward in order to graph The initial value of is and the function value doubles every time is increases by Using this, we can graph , starting by plotting a few of its points and connecting them with a smooth curve.
We can now translate a copy of this graph units downward, to get the graph of
We have now graphed both and
Reflecting in the -axis, and then translating the resulting graph unit to left, gives Determine which graph, I or II, is the graph of
Let's transform the graph of to find the graph of That way, we'll be able to tell which of I and II is the graph of The reflection in the -axis moves every point on the graph to the other side of the -axis, while maintaining their distance to it. If we want to, we can choose a certain point to follow.
Translating this resulting graph unit to the left gives us the graph of Looking at the graph, we can see that it will then coincide with I.
Thus, I is the graph of