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{{ courseTrack.displayTitle }} {{ printedBook.courseTrack.name }} {{ printedBook.name }} # Transformations of Exponential Functions

Like many other function families, exponential functions can be subject to transformations such as translations, reflections, stretches and shrinks. However, as opposed to linear functions and absolute value functions, when an exponential function is transformed, it's possible that the result is something other than an exponential function.
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, $g(x) = f(x) + k,$ a function graph is translated vertically. Notice that since the resulting function has a constant term, it can't be written in the form $g(x) = a \cdot b^x.$ Thus, $g$ 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. Translate graph upward

A graph is translated horizontally by subtracting a number from the input of the function rule. $g(x) = f(x - h)$ Note that the number, $h,$ is subtracted and not added — a positive $h$ translates the graph to the right. Translate graph to the right

Rule

### Reflection

A function is reflected in the $x$-axis by changing the sign of all function values: $g(x) = \text{-} f(x).$ Graphically, all points on the graph move to the opposite side of the $x$-axis, while maintaining their distance to the $x$-axis. Reflect graph in $x$-axis

A graph is instead reflected in the $y$-axis, moving all points on the graph to the opposite side of the $y$-axis, by changing the sign of the input of the function. $g(x) = f(\text{-} x)$ Note that the $y$-intercept is preserved. Reflect graph in $y$-axis

Rule

### Stretch and Shrink

A function graph is vertically stretched or shrunk by multiplying the function rule by some constant $a > 0$: $g(x) = a \cdot f(x).$ All vertical distances from the graph to the $x$-axis are changed by the factor $a.$ Thus, preserving any $x$-intercepts. Stretch graph vertically

By instead multiplying the input of a function rule by some constant $a > 0,$ $g(x) = f(a \cdot x),$ its graph will be horizontally stretched or shrunk by the factor $\frac 1 a.$ Since the $x$-value of $y$-intercepts is $0,$ they are not affected by this transformation. Stretch graph horizontally

Exercise

Graph the following functions. $f(x) = 0.5 \cdot 2^x \qquad g(x) = f(x) - 2$

Solution

The function $g$ is a downward translation of $f$ by $2$ units. Therefore, we can begin by graphing $f,$ and then translate that graph downward in order to graph $g.$ The initial value of $f(x)$ is $0.5,$ and the function value doubles every time $x$ is increases by $1.$ Using this, we can graph $f$, starting by plotting a few of its points and connecting them with a smooth curve. We can now translate a copy of this graph $2$ units downward, to get the graph of $g.$ We have now graphed both $f$ and $g.$

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Exercise Reflecting $f$ in the $y$-axis, and then translating the resulting graph $1$ unit to left, gives $g.$ Determine which graph, I or II, is the graph of $g.$

Solution

Let's transform the graph of $f$ to find the graph of $g.$ That way, we'll be able to tell which of I and II is the graph of $g.$ The reflection in the $y$-axis moves every point on the graph to the other side of the $y$-axis, while maintaining their distance to it. If we want to, we can choose a certain point to follow. Translating this resulting graph $1$ unit to the left gives us the graph of $g.$ Looking at the graph, we can see that it will then coincide with I. Thus, I is the graph of $g.$

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