Making Sense of Parabolas
Hi! I am Leo, Test Prep and Math tutor at The Edge Learning Center. I am here to simplify Parabolas and make them more understandable for you. Many of my students in the MYP Extended Mathematics stream struggle to make the leap from linear functions to quadratic functions: their first encounter with curve drawing.
“Leo, I need to learn about y = ax2 + bx + c.”
When a student asks me about a quadratic, the first question I always ask is, “What does it look like?”
“A parabola. A smile if it’s upward, a frown if it’s upside-down.”
To relate the leading coefficient (a in this case, which cannot be zero by definition of a quadratic function) to the orientation of the graph, we use the term positive (happy) coefficient, meaning the graph is opening upward. A negative (sad) coefficient turns its graph downward.
“What else?” I press further.
“I don’t know,” tends to be a common answer.
I can throw you a bunch of jargon to further describe the shape: the axis of symmetry, the y-intercept, the x-intercepts, the vertex, the translation from origin, the vertical or horizontal dilation, the discriminant. It is often enough to make a student’s head spin.
“How can you tell so much just by looking at that one single curve?”
Good question. The answer is… you can’t.
Taking things at face-value can only lead you so far. Just like many problems we face in life, we try to deconstruct the issue and find other ways to tackle it so that we can come up with a different approach that suits our need. Math is the same way: there is a question in front of you; can you look at it differently or manipulate it to a different form so that you can dig up new information?
“My teacher says factorize it.”
That’s a start. But now you are just following orders, not thinking for yourself. So let’s discover the reason behind this first approach.
To factorize means to resolve back into factors. And factors, as we remember from Primary Year, are whole numbers that divide other whole numbers. Better yet, if I can factorize something, I can write it as the product of two or more factors. For example, we know that 32 = 4 x 8. So 4 and 8 are factors of 32. Factorizing a quadratic fundamentally means the same thing: we are writing it as a product of “simpler” quantities. Without showing you the “how” yet, let us ask “why?”
One beautiful aspect about multiplication is the “null factor law.” In layman’s terms, when you multiply “things” to get a zero, this law of mathematics says at least one of those “things” must be zero. I say “things” because they can be numbers or expressions that vary based on the variable. For example, let’s say that h and k are two numbers or expressions. If I tell you that h multiplied by k equals to 0, the null factor law says that either h or k must be 0. So what’s the big deal about this “null factor law?” you might ask. In order to better understand, let us deconstruct the standard form of a quadratic function:
y = ax2 + bx + c (1)
Let’s assume we can somehow factorize it, as your teacher has instructed you to do, so that now it looks like this:
y = a(x – p)(x – q) (2)
At this point, the right hand side is a product of “things.” Now let us recall one of the concepts we learned from linear functions: intercepts of a graph. There are two types of intercepts: x-int and y-int. In this case, the y-int is similar to a linear function, where by letting x = 0 we are left with the last constant term, so we can identify it very easily. An x-int occurs when the graph cuts the x-axis, which can be found by letting y become 0 and solving for x. When we apply this idea to our newly formed function (2), we get:
0 = a(x – p)(x – q) (3)
Now we apply the null factor law. We can claim that at least one of these “things” must be 0. But which one? We know a cannot be zero based on the definition mentioned earlier, so we are left with two “things”: (x – p) and (x – q). Either of these could be 0, so now we have:
(x – p) = 0 or (x – q) = 0 (4)
Simple algebra tells us that x can be either p or q. Since we have been doing all these to find the x-int, we have come to the conclusion that (p, 0) and (q, 0) are the x-int (generally referred to as roots) of the quadratic.
Equation (2) is what we call the factorized (or root) form. Applying the knowledge of the orientation determined by the coefficient a, we can now connect the dots and draw a crude picture of the parabola. I say “crude” because, if you are sharp-eyed, you might recognize some problems and thus some question emerge. How deep should I draw this graph? Where does the graph turn? How wide should it be? And what happens if I can’t factorize the function?
The last question is particularly unsettling, especially since, after a student has mastered the art of factorizing quadratics, the most upsetting message they receive is that not all quadratics can be factorized. As a result of this, we might need to explore the function deeper and find out why we can’t factorize it.
We compared the factorization of quadratics to the factorization of a whole number; therefore, certain quadratics might behave like prime numbers and simply cannot be factorized. From a graphical point of view, since the factorized form shows us the x-intercepts of the function, a parabola that opens upward above the x-axis (or downward below the x-axis) should not have any of these intercepts, and thus cannot be rewritten as a factorized form. Another issue is that, unlike the factors of a whole number, the roots of a quadratic are not limited to integers. Students learn certain rules to come up with numbers that can be used to rewrite the function in a different form, but, in reality, we are not limited to using whole numbers. We can use irrational numbers, which makes the possibility exponentially greater. We need to look at another way of recognizing a parabola.
One thing that is certain is that a parabola “turns around” at a certain point. Some refer to this phenomena as the “tip,” though the proper term is “vertex.” Every parabola has a vertex, and if we are able to identify it and express the function in terms of the vertex, then we are guaranteed to come up with something crucial. And here is the second form that we need to understand to master the art of analyzing a quadratic:
y = a(x – h)2 + k (5)
Equation 5 is called the vertex form of a parabola, in which the coordinate (h, k) tells you the vertex of your parabola. I am sure you can’t wait to find out about the idea behind this form, and I can’t wait to show you! Understanding how to manipulate quadratic functions gives you a definite edge when you move on to IB Mathematics SL and HL. Come join me so I can guide you to tackle this beast.