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Algebra II: High School23 chapters | 203 lessons

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Lesson Transcript

Instructor:
*Maria Airth*

Maria has a Doctorate of Education and over 15 years of experience teaching psychology and math related courses at the university level.

The basic transformations for a graph are movement up and down, left and right, pinched or stretched graphs, and flipped graphs. This lesson will review how to accomplish each of these transformations.

Hello, and welcome to this lesson on basic transformations of polynomial graphs. To begin, it is probably a good idea to know what a polynomial is and what a basic polynomial graph looks like.

Well, polynomial is short for **polynomial function**, and it refers to algebraic functions which can have many terms. It is normally presented with an *f* of *x* notation like this: *f*(*x*) = *x*^2.

*f*(*x*) = *x*^2 is a **parent function** because it is a standard unit function of that form. The graph looks like a smooth parabola with both ends up above the *x*-axis and the graph going through the points (1,1) and (-1, 1).

What do you think of when you think of the word 'transformation?' I normally think of something changing. Maybe the color red, morphing or changing into the color orange when yellow is added. Or, who could forget the classic car to a robot transformation? Any sort of change from one thing or look to another can be considered transformation.

This lesson focuses on **transformations of polynomial graphs**, which are, simply put, changes to the graphs of polynomial functions. For a much more in-depth discussion of polynomial graphs, please refer to the lesson on understanding polynomial graphs.

So, returning to the image of the basic parent graph of *f*(*x*) = *x*^2, what types of transformations could this graph go through? It could move up and down, left and right; it could get pinched in or stretched out, or it could even flip over. Let's look at how to accomplish each of these types of transformations separately, then we'll conclude with putting them together for a large transformation.

Up and down transformations for functions are caused by the addition or subtraction of a number outside the original function. Thus, *f*(*x*) + 2 would move the graph 2 places up. In our example, we are using the parent function of *f*(*x*) = *x*^2, so to move this up, we would graph *f*(*x*) = *x*^2 + 2.

Moving a graph down is the same principle, except we subtract instead. *f*(*x*) = *x*^2 - 2 moves the parent graph down two places. Notice how the turning point on the graph is at -2 in the transformed graph instead of zero as in the parent graph.

So, up and down movement requires addition or subtraction (respectively) outside of the function. This is the easiest transformation because it makes logical sense - when you add, you increase (or move) up; when you subtract, you decrease (or move) down.

To move left and right, we have to add or subtract inside the function like this: *f*(*x* + 2) = (*x* + 2)^2. Moving up and down was logical. Moving left and right actually does the opposite of what would be logical. If you add 2 into the function, you actually move the graph backwards, to the left 2 places.

Thinking mathematically, the logic is still there, just in reverse. Remember, to graph by hand, we are always looking for what *y* equals when *x* is a certain value. Normally, we try to figure out what *x* we need to make *y* = 0. If your function is *f*(*x*) = (*x* + 2)^2, it is actually pretty easy to see that we need *x* = -2 for *y* to be 0. So, *x* = -2 would clearly move the graph to the left. So, even though at first it seems illogical that adding inside the function would move to the left, when you look closer, it is clearly correct.

The reverse is true as well. *f*(*x* - 2) = (*x* - 2)^2 means that we need *x* to be positive 2 in order for *y* to be 0 . This will move the graph to the right (in the positive direction for the *x*-axis).

When moving left and right, the addition or subtraction is done within the function, and the movement is reverse logic (addition moves left, subtraction moves right).

Now we move to multiplying the function. These transformations depend on whether the multiplier is larger or smaller than 1 - multipliers smaller than 1 stretch the graph, while multipliers larger than 1 pinch the graph. It makes logical sense that multiplying a function by a fraction (smaller than 1) would stretch the graph because each *y*-value result is only a fraction of the chosen *x*-value.

Similarly, since multiplying by a whole number greater than 1 would increase the *y* result in any function, it is easy to see that this would pinch the graph of the parent function (the *y*-value increases faster than the *x* because of the multiplier). Multiplying by a number smaller than 1 stretches the graph, just as a number larger than 1 pinches the graph.

The last form of transformation is a flip. When you flip a graph, you are basically displaying the opposite form of that graph. What is opposite in algebra? That's right, negative! So, the only thing you need to do to flip a graph is make it negative.

Our parent function example is *f*(*x*) = *x*^2, so the opposite (or negative) version of that is *f*(*x*) = -*x*^2 and here is the transformation:

To flip the graph, all you have to do is make the function negative.

So, what if we incorporate all of these transformations into a single graph? What would the graph of *f*(*x*) = -3(*x* + 2)^2 - 2 look like? Well, let's first review the function:

- We have a -2 outside the function, so we will be moving down two places.
- There is the addition of 2 within the function, so the graph will move to the left two places.
- There is a whole number multiplier of 3, so the graph will be pinched.
- It is negative, so the entire graph will be flipped upside down.

And there you have it, the graph transforms exactly as we predicted it would. Don't forget that these rules work for all polynomial functions, not just *f*(*x*) = *x*^2.

In this lesson, we covered the four types of transformations of a polynomial graph. The most basic **parent function** was used to illustrate the transformations. As a review:

- Up and down transformations occur when adding or subtracting outside the function.
- Left and right transformations occur when adding or subtracting inside the function.
- The graph is pinched when multiplying by numbers greater than 1 and stretched when multiplying by numbers less than 1.
- To flip the graph, make the function negative.

Learning these four transformations will assist you in analyzing any polynomial graph you see. Thanks for joining me.

During this lesson, you may develop the ability to:

- Define polynomial and parent function
- Create four transformations: up/down, left/right, pinching/stretching, and flipping

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Algebra II: High School23 chapters | 203 lessons

- How to Evaluate a Polynomial in Function Notation 8:22
- Understanding Basic Polynomial Graphs 9:15
- Basic Transformations of Polynomial Graphs 7:37
- How to Add, Subtract and Multiply Polynomials 6:53
- Pascal's Triangle: Definition and Use with Polynomials 7:26
- The Binomial Theorem: Defining Expressions 13:35
- How to Divide Polynomials with Long Division 8:05
- How to Use Synthetic Division to Divide Polynomials 6:51
- Dividing Polynomials with Long and Synthetic Division: Practice Problems 10:11
- Remainder Theorem & Factor Theorem: Definition & Examples 7:00
- Operations with Polynomials in Several Variables 6:09
- Go to Algebra II: Polynomials

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