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Math 105: Precalculus Algebra14 chapters | 114 lessons | 12 flashcard sets

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Lesson Transcript

Instructor:
*Cathryn Jackson*

Cat has taught a variety of subjects, including communications, mathematics, and technology. Cat has a master's degree in education and is currently working on her Ph.D.

Symmetry can occur in many different ways in mathematics. In this lesson, learn how to identify unique forms of symmetry by finding the line of symmetry.

You've probably seen symmetry often in real life. Every time you look in a mirror, glass, and even water, you can see a reflection of yourself. This is called symmetry. **Symmetry** is created when there is an exact replica or reflection of a shape or a line. You can also see symmetry in everyday objects. For example, you can take a cup and imagine a line going down the center of that cup. On each side of that line, the cup has an exact replica, or reflection, of itself.

Flowers will also often have symmetry.

There are some shapes that are symmetrical only when you draw a line in the right place. Look at this pencil.

In mathematics, we would call that horizontal line on the pencil the axis or line of symmetry. We call it that because that is the line that shows the reflection of the pencil; no other line would create a reflection on the pencil.

The axis or **line of symmetry** is an imaginary line that runs through the center of a line or shape creating two perfectly identical halves. In higher level mathematics, you will be asked to find the axis of symmetry of a parabola.

This is a **parabola**, a u-shaped line on the graph.

We can identify the line of symmetry graphically by simply finding the farthest point of the curve of the parabola. This is called the **vertex**, the point where two lines connect. If the parabola were a hill, the very highest point on that hill would represent the vertex of the parabola, or if the parabola were a valley, the very lowest spot in the valley would represent the vertex of the parabola.

Take a look at this graph. Do you see the vertex? It's at the point (2, 3).

Okay, now look at this graph. Can you identify the line of symmetry? The vertex is located at (-2, -4). Again, on this graph, the vertex is the very lowest point of the parabola. We can draw an imaginary line through this point to find the line of symmetry. So we would write this as *y* = -4.

You will most often see lines of symmetry on quadratic functions. Quadratic functions are often written in the standard form *y = ax^2 + bx + c*, where *a*, *b*, and *c* equal all real numbers. All quadratic functions create a parabola that opens up or opens down.

You can identify the line of symmetry of a quadratic function in standard form by using the formula *x* = -*b* / 2*a*. Let's take a look at an example.

To find the line of symmetry for this equation, you need to plug in the correct numbers into the formula. So our equation is *y* = 2*x*^2 - 4*x* - 3, and standard form is *y = ax^2 + bx + c*. So that means that *a* = 2 and *b* = -4. Let's plug that into our formula *x* = -*b* / 2*a*, which gives us *x* = -(-4/2(2)). Now evaluate the equation. 2 x 2 = 4 and -(-4)/4 = 4/4 = 1. So now we have 1, which gives us *x* = 1. If we were to graph this equation, it would look like this.

You can see the line of symmetry goes through the vertex (1, -5 ).

Quadratic functions can also be written in vertex form: *y* = (*x* - *h*)^2 + *k*, where *h = x* and *k = y*. That means that *h* is equal to the *x*-coordinate and *k* is equal to the *y*-coordinate of the vertex. Take a look at this example.

*y* = (*x* - 4)^2 + 8

These are pretty simple. We know that in the vertex formula *x = h*, and in this equation *h* = 4, so our vertex is *x* = 4. Why is the answer a positive 4 and not a negative 4? Remember that *h* has a negative sign in front of it in the vertex equation. To avoid any sign errors, remember this quick trick: Think of that number and whatever sign happens to be in front of it as -*h*. You can graph the equation to see where the vertex sits on the point (4, 8).

Try this equation. *y* = (*x* + 3)^2 + 4

First, identify -*h*. We can see in this equation that *-h* = 3. That's a positive 3, so now take the opposite of -*h*, which is -3, and you have your line of symmetry: *x* = -3.

Take a look at the graph of this equation. See the line of symmetry going through the vertex (-3, 4)?

Today, you have learned about the line of symmetry, which is the imaginary line that runs through the center of a line or shape creating two perfectly identical halves. You can find the line of symmetry graphically or algebraically. To find the line of symmetry graphically, find the vertex, or the farthest point where two lines connect, and write either *x =* or *y =* and then insert the *x-* or *y-*coordinate.

To find the line of symmetry algebraically, you need to identify if the equation is written in standard form or vertex form. Standard form is *y = ax^2 + bx + c*, where *a*, *b*, and *c* equal all real numbers. You can use the formula *x* = -*b* / 2*a* to find the line of symmetry. Vertex form is *y* = (*x* - *h*)^2 + *k*. where *h = x* and *k = y*. Identify which number is *-h* in the equation, and then write the opposite of *-h* for your line of symmetry.

By the end of this lesson you should be able to:

- Identify a line of symmetry in real world examples and in graphs
- Calculate the line of symmetry in a quadratic equation

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