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### Course: Algebra 1>Unit 14

Lesson 1: Intro to parabolas

We can interpret what the features of a graph of a quadratic model mean in terms of a given context. Created by Sal Khan.

## Want to join the conversation?

• Why is a quadratic equation called "quadratic"? The prefix "quad-" means four, but a quadratic equation only has a degree of two. Is there something else behind it?
• I have wondered why “4” was used for a “2”. It didn't seem to make any sense at first. However, in Latin:
• Linear means a line, and having 1 dimension contains 1 solution
• Quad refers to a square, which has 2 dimensions and thus has 2 solutions
• Cubic refers to a cube, which has 3 dimensions and thus has 3 solutions
So it seems that the "quad" is focusing on the number of sides the shape has (while not yet about the number of solutions); it means a "square-like" equation. This is why the “2nd-power” is called “squared”, because for whatever value is “squared” the result produces the shape of a square. A square has 4 sides, and this is likely what “quad” or “four” is coming from.

As a matter of fact, it appears that Geometry was invented before Algebra, although we are often first taught pre-algebra and Algebra 1 before Geometry.
• Do you solve these any different if they have endpoints instead of continued lines?
• A parabola does not really have endpoints because there are no restrictions to its function's domain or range of the general x^2. Even if there is a coefficient ≠ 1 at the x^2, a coefficient for the first degree, and/or a constant, there won't be undefined aspects. The curve may be stretched or squeezed by the coefficients, and shifted up or down by the constant, but any input to the x-variable will be valid. Unlike some other functions which have restrictions either because of the operation, such as square roots, or if there is some division by 0 due to a fraction's denominator, the "x^2" does not have any values of the x variable that will-produce something undefined. So any real value: positive, negative, zero, decimal or fraction, or even an irrational value, will not have restrictions (though due to the context, it is possible for some restrictions to occur, like how Sal Khan states time cannot really be "negative".

Below are some examples for why a function may be undefined:
•If we have say 3/(x-4), while all values, positive and negative, are allowed, we cannot have 4 because 4-4 equals 0, and division by 0 is undefined.
•Without imaginary values, we can't have the square root of negative values because we will end up with a positive value by squaring either a positive or negative but whenever a value is squared,for we have either a [positive]x[positive] or [negative]x[negative], and to really produce a negative we will be needing a negative times a positive.

Hope this helps.
• This makes alot of sense.