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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.

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.

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 $g(x)=a⋅b_{x}.$ Thus, g is Translate graph upward

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

g(x)=-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

g(x)=f(-x)

Note that the y-intercept is preserved.
Reflect graph in y-axis

g(x)=a⋅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

g(x)=f(a⋅x),

its graph will be horizontally stretched or shrunk by the factor $a1 .$ Since the x-value of y-intercepts is 0, they are not affected by this transformation.
Stretch graph horizontally

Graph the following functions.

Show 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.

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.

Show 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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