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Holt McDougal Algebra 2: Online Textbook Help14 chapters | 233 lessons

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Lesson Transcript

Instructor:
*Artem Cheprasov*

In this lesson, you're going to learn about the definition of a reflection, quadratic equation, and how to visually and mathematically reflect a quartic equation over the x-axis and the y-axis.

If you ever stood in front of a mirror, or next to a calm pond, you would have seen a **reflection**, which in math is the flipping of a point or figure over a line of reflection (the mirror line).

Reflection can also be applied to a **quadratic equation**, which is simply an equation where the highest exponent is 2. The simplest of these is *y* = *x*^2, and the standard form of one is *y* = *ax*^2 + *bx* + *c*, where *a*, *b* and *c* are numbers and *a* cannot equal 0.

Let's see how we can reflect quadratic equations using graphs and some really easy math.

For our first example let's stick to the very simple parent graph of *y* = *x*^2.

{See video for graph}

On the screen you can see that the graph of this equation is a parabola. The reflection of such a parabola over the *x*-axis is simply written as *y* = -(*x*^2). In other words, the function of *f*(*x*) becomes -*f*(*x*) when reflected over the *x*-axis.

When reflected everything looks like an upside down version of our parent graph. Doesn't it?

{See video for graph}

In fact, remember that a positive quadratic has a smiley face, while the negative quadratic has a frown. It's really easy to recall this and important to; because, if you have an equation where the value of *a* is negative but you are coming up with a quadratic that is right-side up, then you know you are doing something wrong, and you need to double check where you've gone amiss.

More complex equations and graphs are really just as simple to reflect over the *x*-axis. Something like *y* = 2*x*^2 + 4*x* - 5, would become *y* = -(2*x*^2 + 4*x* - 5).

{See video for graph}

Now, let's go back to our simple equation of *y* = *x*^2. What if we wanted to reflect it over the *y*-axis? In this case *y* = *x*^2 would become *y* = (-*x*)^2; because, a parabola is symmetric in general, and in this case it's symmetric about the *y*-axis, the exact same graph would be created.

{See video for graph}

In general, though, you should have recognized that if we reflect a function *f*(*x*) over the *y*-axis, it becomes *f*(-*x*). That means we replace very *x* in the original function (or equation) with a -*x*. Let's try a harder example to drive this point home.

Let's say our equation is *y* = 2*x*^2 - 3*x* - 9. If we are to reflect this over the *y*-axis, what would we get? Remember, every *x* is now replaced with a -*x*, so our equation becomes *y* = 2(-*x*^2) - 3(-*x*) - 9. You didn't even need to graph the equation in order to figure out how to reflect it.

Well, that wasn't that hard, was it? Just remember the following key points when reflecting a quadratic equation. A **reflection** is the flipping of a point or figure over a line of reflection (called the mirror line), and a **quadratic equation** is an equation where the highest exponent is 2. When reflecting over the *x*-axis, the function *f*(*x*) becomes -*f*(*x*). For instance, *y* = 3*x*^2 would become *y* = -(3*x*^2). On the other hand, when we reflect the function *f*(*x*) over the *y*-axis it becomes *f*(-*x*). For instance *y* = 3*x*^2 + 2*x* would become *y* = 3(-*x*^2) + 2(-*x*).

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Holt McDougal Algebra 2: Online Textbook Help14 chapters | 233 lessons

- What is a Parabola? 4:36
- How to Reflect Quadratic Equations 3:48
- Using Quadratic Models to Find Minimum & Maximum Values: Definition, Steps & Example 9:54
- How to Factor Quadratic Equations: FOIL in Reverse 8:50
- How to Solve a Quadratic Equation by Factoring 7:53
- How to Complete the Square 8:43
- Completing the Square Practice Problems 7:31
- What is an Imaginary Number? 8:40
- How to Equate Two Complex Numbers 5:54
- How to Use the Quadratic Formula to Solve a Quadratic Equation 9:20
- How to Solve Quadratics with Complex Numbers as the Solution 5:59
- Graphing & Solving Quadratic Inequalities: Examples & Process 6:14
- Solving Quadratic Inequalities in One Variable 3:40
- How to Write Quadratic Functions 10:09
- How to Add, Subtract and Multiply Complex Numbers 5:59
- How to Graph a Complex Number on the Complex Plane 3:28
- How to Divide Complex Numbers 6:40
- How to Add Complex Numbers in the Complex Plane 3:52
- Go to Holt McDougal Algebra 2 Chapter 5: Quadratic Functions

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