Finding points for F(x)=2x^2

Cbrock

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This was in an algebra II book, it looks a bit simple to me but I thought I'd err on the side of putting something simple in the advanced catagory instead of putting something difficult in the begining category. :) Please let me know if I should move it.

This is what the book says. I don't need anyone to do the questions for me, I'm just unsure of how to get the answer.
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Here's what they say the answer is for number one
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I know what a parabola is but I don't know how they got those points.Do I substitute any number I want for x or is it specific? Thanks in advance!
 
Finding points for F(x)=2x^2 … Do I substitute any number I want for x or is it specific? …
Hi Cbrock. First, a note about notation. Names in math are case-sensitive, so F and f are two different names. It's best to use names and symbols as given in the exercise, so write f(x) instead of F(x) because those two symbols represent different functions.

Yes, you're free to choose any values for x, but, when graphing the behavior of a function, we generally want to show the interesting part(s), so the x-values must be chosen accordingly. For the exercises you posted, using the book's suggestion of x={-3,-2,-1,0,1,2,3} for the seven points will work because they will show each curve in the vicinity of the parabola's vertex.

Here's a different function, where the book's x-values would not be a good choice (because we wouldn't see the vertex, if we used them):

g(x) = 2x^2 - 64x + 512

In this case, we would want to pick three x-values around each side of the vertex, like x={10,12,14,16,18,20,22}. We could first use the formula x=-B/(2A) to find the x-value at the vertex (16) and then pick other values on each side.



In case you're not familiar with function notation f(x), I'll talk about it. The symbol f(x) is a different way to write the dependent variable y. This function notation is more helpful than simply writing y. The letter f in the symbol f(x) is the function's name, so, if we happen to work with three functions at once, it's nice to name them f, g and h instead of y, y and y (which would be confusing). Also, the independent variable (or it's value) is shown inside the parentheses, so function notation provides more information than writing just y. For example:

y = 2x^2

Now, if I tell you that y=18, then you know that the y-coordinate is 18 for some point on the graph, but the associated x-coordinate is not immediately clear (that is, a value of x which makes 2x^2 equal to 18). However, if I write f(3)=18 instead, then the notation shows that x=3 leads to y=18. So, writing function notation for y, the equation f(3)=18 tells us the value of both x and y.

In the machine model of a function, symbol x represents the input and symbol f(x) represents the output. The input appears inside the parentheses, and the entire symbol -- like f(3) or f(x) -- represents the function's output.

Symbol f(x) is function notation for the variable y (the output variable) when the input variable is x.

Symbol f(3) for example is function notation for the constant 18 (that is, it represents the specific number output, when the input is 3).

So this is why we say y = f(x). The symbol y and symbols like f(x) or f(3) represent the same thing (a variable or a constant, depending on whether the x-value is specified or not).

If you're still unsure about any of this, you can google keywords function notation, or ask if you can't find a clear explanation. Perhaps we can find some online lessons for you about function notation and the machine model of a function, in general. Otherwise, it will become clear with more practice. Cheers

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They just said, "Find seven points ...", so you don't have to choose the particular points they did. But it is perhaps most natural to choose (a) integers, which are easy to work with, and (b) numbers near zero; so the values {-3, -2, -1, 0, 1, 2, 3} are an obvious set of 7 to use. You could also go with {-1.5, -1, -0.5, 0, 0.5, 1, 1.5} if you wanted. For other functions (e.g. 1000x^2 or x^2/1000), you might want larger or smaller numbers.

You could, if you were adventurous, take x = -5.2, -4.3, -pi, 1.12334, and so on -- but your teacher would be worried about you.

Then, of course, whatever values you chose for x, you put them into the function to find y, and plot.
 
Another way to write f(x) is y, that is y=f(x).
So f(x) = 2x^2 is the same as y = 2x^2.
Now if the right hand side equal 11, then y=11. This is because the right hand side and the left hand side are equal. The right hand side says to square x and then multiply by 2. The result is called y.
So for example, if x=5, then we square it to get 25 and then multiply by 2 to get 50. So when x=5, y = 50. This is one point.
Now pick any 6 others x values, square then and then multiply by 2 and call the results y values.
 
Of course! But for other reasons.

And my son would absolutely make "adventurous" choices like those. I worried about him, too.
 
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