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Big Ideas Math Algebra 2: Online Textbook Help11 chapters | 145 lessons

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Lesson Transcript

Instructor:
*Stephanie Matalone*

Stephanie taught high school science and math and has a Master's Degree in Secondary Education.

In this lesson, we will not only go over the basic definition of a quadratic function, we will also talk about transformations of those functions. In other words, we will discuss how to move the graph around by changing the formula.

You stand in your backyard and throw a ball into the air. It makes a nice arc and then comes back down to the ground. That pretty shape you just made looks exactly like the graph of a quadratic function! **Quadratic functions** are second order functions, which means the highest exponent for a variable is two. They're usually in this form: *f(x) = ax2 + bx + c*. One thing to note about that equation is that the coefficient *a* cannot be equal to zero.

Quadratic functions can be graphed just like any other function. The neat thing about these is that they will always graph into a curved shape called a **parabola**. Parabolas are u-shaped and can be upside down depending on the numbers in the equation.

Let's look at the parent function of a quadratic: *f(x) = x2*. If we compare this to the usual form of *f(x) = ax2 + bx + c*, we can see that *a* = 1, *b* = 0, and *c* = 0. When we graph this parent function, we get our typical parabola in an u-shape.

Let's say you took a step to the left and threw the ball higher in your backyard. You just transformed your parabola! We can transform graphs by shifting them, flipping them, stretching them, or shrinking them. We can do this by changing the equation of the graph.

The first type of transformations we will deal with are called **shifts**. This means we are moving the graph horizontally to the left or right or vertically up or down. Let's say we want to move our parent graph of *f(x) = x2* to the right five units. To do this, we have to subtract five from the *x* value inside parentheses like so: *f(x) = (x - 5)2*. Any shifts to the right will be completed through subtracting number inside the parentheses, while any shifts to the left will done be by adding a number inside the parentheses.

If you want to shift the graph up five, you will add five to *x*, but this time, you do not need parentheses, or you can go outside of them: *f(x) = x2 + 5* or *f(x) = (x2) + 5*. Any vertical shifts up will be done by adding a number outside of the parentheses, while any vertical shifts down will come from subtracting a number outside of the parentheses.

If you want to change the width of your graph, you can do so in the vertical or horizontal direction. To compress or stretch vertically, you will multiply the entire equation by a number. If that number is greater than one, the graph will stretch. If that number is between 0 and 1, that graph will compress. Think about the graph being pushed on from above and below and being compressed towards the *x*-axis. For example, the function *f(x) = 1/4(x2)* will compress vertically. In other words, the graph will get wider.

You can also change the width of the graph by compressing or stretching the graph in the horizontal direction. This time, you will multiply just *x* by a number. If that number is greater than one, the graph will be compressed. If the number is between 0 and 1, the graph will be stretched. This time, think about the graph being compressed toward the *y*-axis because it it being pushed from the left and right. For this example, we will look at *f(x) = (1/4x)2*. This graph is being stretched horizontally, which means it will get wider.

Lastly, graphs can be flipped. This means the u-shape of the parabola will turn upside down. To do this, we simply make the entire function negative. The new graph will look like an upside down U. For example, *f(x) = -(x2)* will be the same in all regards except it opens downward.

Let's put it all together now! What if you want your graph to have multiple transformations? It's easy, just follow the instructions. Let's shift our graph to the left 10, down 5, and flip it. We would write the equation like this: *f(x) = -(x + 10)2 - 5*.

**Quadratic functions** are second order functions, meaning the highest exponent for a variable is two. They're usually in this form: *f(x) = ax2 + bx + c*. They will always graph into a curved shape called a **parabola**, which is a u-shape. We can transform graphs by **shifting** them (moving graphs up/down or left/right), flipping them, stretching them, or shrinking them.

So you want to transform your quadratic graph? It's simple! Change your equation around according to the following table and you are good to go!

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Big Ideas Math Algebra 2: Online Textbook Help11 chapters | 145 lessons

- Transformations of Quadratic Functions 4:33
- Find the Axis Of Symmetry: Equation, Formula & Vertex 6:11
- Using Quadratic Models to Find Minimum & Maximum Values: Definition, Steps & Example 9:54
- Parabola Intercept Form: Definition & Explanation 3:46
- The Focus and Directrix of a Parabola 4:47
- Writing Quadratic Equations for Given Points
- Using Quadratic Functions to Model a Given Data Set or Situation 5:53
- Go to Big Ideas Math Algebra 2 - Chapter 2: Quadratic Functions

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