So I’m in 10th grade academic math and my teacher is pretty good at explaining concepts, formulas, etc. But the one thing I can’t seem to grasp her explaining of is quadratic equations and parabola, can anyone explain it simply?
In: Mathematics
A quadratic equation is any equation of the form y = ax^2 + bx + c (or could be written this way). A parabola is the shape of the graph of that equation.
There are a lot of details to learn about but above is all there is to know to start. You’ll deal with different parts of this through calculus.
The graph is a “U” shape either with a lowest point (if right up) or highest point (if upside down). They are symmetrical left-right. The values a, b, c determine how wide and flat or tall and skinny and where that middle point is.
Good practice is just to pick some numbers a, b, c then make a table from say -10 to +10 and calculate what y= for those 21 values. Then put those 21 points on graph paper and connect them with a curve.
A parabola is a shape. There’s nuance and ‘why’s as well, but at its core that’s all that ‘parabola’ means, just like a line or a circle or a square is a shape.
Quadratic comes from the latin word for ‘square’, and it means an equation where there’s a ^2 (aka a square) involved. Lots of interesting things can be done with them, but because there’s so many things it’s hard to know where to start for ELI5 beyond this.
One of the neat things about a quadratic equation on a graph, it forms a parabola. But once again, there’s nuance involved, as well as a *ton* of supporting topics that can help make it easier to understand if you already know them. What sort of questions do you have about the two and how they relate?
Hey, you know the times table, right? Check out all the squares, like 1×1=1, 2×2=4, 3×3=9, etc.
That pattern, 1, 4, 16, 25, 36, 49, … is the pattern of “x squared” for x=1, 2, 3,…
So the ordered pairs of x and x squared are (1,1), (2,4), (3,9), etc. if you draw these dots they make the rough outline of the parabola on the graph.
Can we smooth it out? Yes by using “rational” and “real” x’s like 0.25 and sqrt(3) and pi.
Finally the equations like y=x^2 describe this pattern. Saying okay my y’s (outputs) are gonna be the squares of the x’s, whatever those are in the problem…
All the functions have a particular shape and they’re called “parent functions.”
Linear y=x (and y=mx+b) is a straight line, absolute value,y=abs(x) look like a V
quadratics, y=x^2 are U shaped (parabolic) then can open up or down. Why? Because as x increases y is squared (x=1,y=1, x=2,y=4, etc) but as x decreases (other side if zero and negative) the square is positive because a negative times a negative is positive . X=-1, y=+1, x=-2, y=+4.
Sometimes a quadratic has a negative before the square, y=-x^2. So anything squared will turn negative. That’s when the U flips upside down
Were gonna get a lot of explanations as to what exactly they are here, I’m going to instead explain why they are taught to you.
Line equations are by far the simplest equations, their simplicity makes them the most common in all sorts of things, they are applicable to anything done at a constant speed. Quadratic equations come in second after lines.
Lines represent something changing at a constant speed, quadratic equations represent when something is **accelerating** at a constant acceleration.
For example you may have learned in high school physics that gravity is a constant 9.81 meters per second squared acceleration here on earth. This constant acceleration is what makes say throwing a ball is modeled by a quadratic equation and why the shape it makes when thrown is a parabola.
Each of the numbers there represent something different, a is the acceleration constant, b is the initial speed constant and c is the initial position constant, just like in a line, mx = b, m represents speed and b represents initial position.
The more detailed reasons for why this is the case will be explained in calculus if you ever take it but this is the reason why they show up in a lot of places.
* A quadratic equation is in the form ax^2 + bx + c = 0
* When you plot it on a graph i.e. y = ax^2 + bx + c, the shape you get is a parabola. It looks like a ∪ or a ∩.
* By changing a,b,c you can stretch it and move it, but it will always look like a ∪ or a ∩, both arms in same direction.
* By looking at the parabola you can infer things about it’s equation. If it looks like ∪ then *a* is positive. If it looks like ∩ then *a* is negative.
* If both arms intersect the x-axis (i.e. y = 0), then it has 2 real solutions. If just the tip touches the x-axis then it has 1 real solution. If it doesn’t touch the x-axis at all it has no real solutions.
* The reason these zeros are important is because they tell you how to factor the equation. Finding the zeros is equivalent to factoring and vice-versa.
* You will spend a lot of time on factoring quadratic equations. E.g. turning x^2 + x – 6 into (x – 2)(x + 3). If you plot that, it will intersect the x-axis at x = 2 and x = -3.
* Sometimes there are no real factors e.g. x^2 + 1 = 0. If you plot that one, it doesn’t touch the x-axis at all.
A parabola is just a mathematical shape, and a quadratic equation is just an equation that makes a parabola.
Like a lot of things in math, it’s hard to get into detail as to what they “mean,” by themselves, because by themselves they don’t mean anything. Sort of like how a hammer can be used to hammer in nails, but it can also be used to break things, pry things apart, fight someone, or if you really want to get creative you can tie a rope to it and use it as a makeshift anchor or something. Tools in math are no different than the hammer, but tend to have _even more uses_ making it really difficult to develop an intuitive grasp on them unless you already have some kind of use for them where you can build connections yourself.
The reason we learn about parabolas and quadratic equations though is because they show up in a lot of different math. A parabola is just “a^2” after all, so you’re going to be seeing that a lot, meaning it’s super helpful to learn things like the quadratic formula and ways to solve or otherwise work with quadratics, because it will help you work with a bunch of equations you’re likely to come across in the future.
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