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.

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.

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.

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.

### 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) |
---|---|---|

0 | 0 | 0 |

1 | 1 | -1 |

4 | 2 | -2 |

9 | 3 | -3 |

16 | 4 | -4 |

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) |
---|---|---|---|

0 | 0 | 0 | 0 |

1 | 1 | -1 | 1 |

4 | 2 | -4 | 2 |

9 | 3 | -9 | 3 |

16 | 4 | -16 | 4 |

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). $$f(x)=2x^2 - x + 1$$ $$g(x)=-2x^2 + x - 1$$

Please choose the best answer.

A

Reflected across the y-axis

B

Reflected across the x-axis

Example #2

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

Please choose the best answer.

A

Reflected across the y-axis

B

Reflected across the x-axis

Example #3

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

Please choose the best answer.

A

Reflected across the y-axis

B

Reflected across the x-axis

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