Back To CourseAlgebra II: High School
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Maria has a Doctorate of Education and over 15 years of experience teaching psychology and math related courses at the university level.
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:
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:
Learning these four transformations will assist you in analyzing any polynomial graph you see. Thanks for joining me.
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Back To CourseAlgebra II: High School
23 chapters | 203 lessons