Lesson Objectives
• Learn how to reflect a graph across the x-axis
• Learn how to reflect a graph across the y-axis

## How to Reflect a Graph Across an Axis

Another type of graphing transformation is a reflection. When we talk about a reflection, it can be thought of as a folding or flipping of the graph over the line of reflection. This type of transformation is known as a rigid transformation since the basic shape of the graph is unchanged, we are only changing the position of the graph in the coordinate plane.

### Reflecting Across the x-axis

$$g(x)=-f(x)$$ g(x) is the graph of f(x) reflected across the x-axis. $$f(x)=\sqrt{x}$$ $$g(x)=-\sqrt{x}$$ If we look at our graph above, we can see that our function g can be obtained by flipping the graph of f across the x-axis. In other words, if a point on the graph of the function f is (a, b), then the corresponding point on the function g is (a, -b).
x f(x) g(x)
000
11-1
42-2
93-3
164-4
Another way to think about this is to say that for a given x-value, the y-value is now the opposite. For example, in the function f, we see the point (9, 3), whereas in the function g, we see the point (9, -3). Let's look at an example.
Example #1: Describe the transformation from f(x) to g(x). $$f(x)=|x|$$ $$g(x)=-|x|$$ Since g(x) = -f(x), we can state that g(x) is the graph of f(x) reflected across the x-axis.

### Reflecting Across the y-axis

$$g(x)=f(-x)$$ g(x) is the graph of f(x) reflected across the y-axis. $$f(x)=\sqrt{x}$$ $$g(x)=\sqrt{-x}$$ If we look at our graph above, we can see that our function g can be obtained by flipping the graph of f across the y-axis. In other words, if a point on the graph of the function f is (a, b), then the corresponding point on the function g is (-a, b).
x f(x) x g(x)
0000
11-11
42-42
93-93
164-164
Another way to think about this is to say that for a given y-value, the x-value is now the opposite. For example, in the function f, we see the point (9, 3), whereas in the function g, we see the point (-9, 3). Let's look at an example.
Example #2: Describe the transformation from f(x) to g(x). $$f(x)=x^3 + 2x^2 - x$$ $$g(x)=-x^3 + 2x^2 + x$$ $$f(-x)=(-x)^3 + 2(-x)^2 - (-x)$$ $$=(-1)^3 \cdot x^3 + 2 \cdot (-1)^2 \cdot x^2 + x$$ $$=-x^3 + 2x^2 + x$$ Since g(x) = f(-x), we can state that g(x) is the graph of f(x) reflected across the y-axis.

### Combining Stretching/Shrinking with Reflecting

In some cases, a function will involve more than one transformation. In the last lesson, we learned about stretching/shrinking a graph. Let's suppose we have a stretch/shrink combined with a reflection. Let's look at an example.
Example #3: Describe the transformation from f(x) to g(x). $$f(x)=\sqrt{x}$$ $$h(x)=2\sqrt{x}$$ $$g(x)=-2\sqrt{x}$$ In order to show the process, we have included an additional function h. If we think about h when compared to f, the function is vertically stretched by a factor of 2. In other words, for a given x-value, the y-value is now multiplied by 2.
When we compare g to h, we see there is a reflection across the x-axis. In other words, for a given x-value, the y-value is now changed into its opposite. Putting the two steps together, we can compare f to g. To graph g, we can start with f, vertically stretch by a factor of 2, and then reflect across the x-axis.

#### Skills Check:

Example #1

Describe the transformation from f(x) to g(x). $$f(x)=2x^2 - x + 1$$ $$g(x)=-2x^2 + x - 1$$

A
Reflected across the y-axis
B
Reflected across the x-axis

Example #2

Describe the transformation from f(x) to g(x). $$f(x)=2x^3 - 2x^2 + x$$ $$g(x)=-2x^3 - 2x^2 - x$$

A
Reflected across the y-axis
B
Reflected across the x-axis

Example #3

Describe the transformation from f(x) to g(x). $$f(x)=x^4$$ $$g(x)=-x^4$$