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Basic Transformations of Polynomial Graphs

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  • 0:03 Types of Transformations
  • 1:52 Up and Down Transfromation
  • 2:52 Left and Right Transformation
  • 4:28 Pinching and Stretching
  • 5:22 Flipping the Graph
  • 5:56 Example Transfromation
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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.

Types of 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).


Polynomial graph
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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 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.


Transformation up
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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.


Transformation down
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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.

Left and Right Transformation

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.


Transformation left
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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).


Transformation right
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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).

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