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Explorations in Core Math - Algebra 2: Online Textbook Help12 chapters | 146 lessons

Instructor:
*Laura Pennington*

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

Transformations of functions are useful in the analysis of functions. This lesson explains how transformations compare for different types of functions. Then, we will look at transformations of exponential functions, specifically.

Suppose you're sitting in class, and the teacher has you graph the function *f*(*x*) = *x*2. No problem! You whip up the graph in a couple of minutes.

Now, the teacher asks you to graph the function *g*(*x*) = *x*2 + 3. Guess what? You don't have to start all over!

You see, *g*(*x*) = *x*2 + 3 is what we call a transformation of the function *f*(*x*) = *x*2. **Transformations of functions** are algebraic operations on a function that correspond to moving or resizing the graph of the function. There are four types of transformations: horizontal shifts, stretching/shrinking, reflections, and vertical shifts.

Look at your functions *f*(*x*) and *g*(*x*) again. Do you recognize which type of transformation takes place to get from *f*(*x*) to *g*(*x*)? Well, we add 3 to the entire function, *f*, to get *g*. Therefore, this is a vertical shift. In other words, we can graph *g*(*x*) by shifting the graph of *f*(*x*) up 3 units.

Well, that was much easier than starting all over!

This was an example of transforming a quadratic function. Transformations for different types of functions are all the same, so they compare the same. They just look a little different for each type of function. For example, let's consider exponential functions.

**Exponential functions** are functions that contain the variable in the exponent. The most basic exponential function is *y* = *b**x*, and we call this the parent function of exponential functions. In general, a **parent function** is the most basic function of a group, or family, of functions. When *b* > 1, we have exponential growth, and the graph rises from left to right. When 0 < *b* < 1, we have exponential decay, and the graph falls from left to right.

All exponential functions can be derived from the parent function through a series of transformations. For example, consider the following function:

*h*(*x*) = -21/4*x*

This is the parent function *f*(*x*) = *b**x*, where *b* = 2, taken through the following transformations:

- Stretched horizontally by a factor of 4, because we multiply the
*x*-variable by 1/4 - Reflected over the
*x*-axis, because we multiply the whole function by a negative

Pretty neat, huh? When dealing with transformations of exponential functions, it's important to remember that horizontal shifts, horizontal stretching/shrinking, and reflections over the *y*-axis need to stick with the *x*-variable. Because the *x*-variable of an exponential function is located in the exponent, these transformations must affect the exponent part of the function. This is the biggest difference of transformations of exponential functions from other functions.

Using these transformation facts, we can find an exponential function by observing its graph and the parent function's graph, then taking the following steps:

- Identify the transformations that we take the parent function through to get to the function.
- Perform the transformations' corresponding algebraic operations on the parent function.

Let's give it a try!

The following graph displays the exponential parent function *f*(*x*) = 3*x*, which is an exponential growth model since 3 > 1, along with another exponential function, *g*(*x*). We want to find *g*(*x*).

To find *g*(*x*), we first identify the transformations that we take *f*(*x*) through to get the graph of *g*(*x*). It's obvious that we shifted the parent function's graph to the left, so we just need to figure out how many units we moved the graph to the left. We can do this by observing that the point (0,1) on *f*(*x*) ends up at the point (-4,1) on *g*(*x*). We see that the *x*-value changes from 0 to -4, so we have that we move *f*(*x*) 4 units to the left.

Now, we just identify that shifting *f*(*x*) 4 units to the left corresponds to adding 4 to the *x*-variable in the function. Therefore, *g*(*x*) = 3*x*+4.

That was kind of fun! Let's try one more. Let's consider how to change an exponential growth model, *f*(*x*) = 2*x*, into an exponential decay model.

This is an exponential growth model, because 2 > 1. We know that exponential decay models have the same shape as exponential growth models, but they fall from left to right rather than rise. Can you think of a transformation that would make this happen for the graph of *f*(*x*) = 2*x*?

If you're thinking that reflecting the graph over the *y*-axis would do it, you're correct! This transformation corresponds to multiplying the *x*-variable by a negative, so this is the graph of the function *g*(*x*) = 2-*x*. Ultimately, the reason this works is because of properties of negative exponents. That is,

- 2-
*x*= 1/(2*x*) = (1/2)*x*

We see that multiplying the *x*-variable by a negative in the exponent turns the function into an exponential decay model by changing the *b*-value of the parent function to a number between 0 and 1.

So neat!

**Transformations of functions** are algebraic operations on a function that correspond to moving or resizing the graph of the function. There are four types of transformations: horizontal shifts, stretching/shrinking, reflections, and vertical shifts.

When we compare transformations of different types of functions, we find that they work exactly the same for each type of function. However, remember that horizontal shifts, horizontal stretching/shrinking, and reflection over the *y*-axis need to take place in the *x*-variable exponent portion of the function.

A **parent function** is the most basic function of a group, or family, of functions. We can identify functions by observing their graphs along with the parent function's graph then following these steps:

- Identify the transformations that we take the parent function's graph through to get to our graph.
- Identify the transformations' corresponding algebraic operations, and perform them on the parent function.

Being familiar with transformations of functions is extremely useful when analyzing or graphing functions, so it's great that we're now more familiar with this concept.

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Explorations in Core Math - Algebra 2: Online Textbook Help12 chapters | 146 lessons

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