Quadratics: Equations & Graphs

Instructor: Glenda Boozer
We will look at quadratic equations: what they are, some forms in which we might find them, how we can put them into standard form, and how their graphs look, including some features of the graphs.


Quadratic equations: they sound like a big deal; even the Major General mentions them in the Pirates of Penzance: 'I understand equations, both the simple and quadratical, . . . .' So what is the big deal?

A quadratic equation is any equation where the variable is raised to the second power (squared), but not to any higher power. It can always be written in the form ax^2 + bx + c = 0, which we refer to as standard form, although there are other forms. Either b or c can be equal to 0, but a can't be 0, because if it were, we wouldn't have any x^2 to talk about.

Standard Form

You could see an equation like 2x^2 + 3x = 5, but if we simply subtract 5 from both sides, we get 2x^2 + 3x - 5 = 0, and we're back in the standard form. If we see x^2 = 9, that can be written as 1x^2 + 0x - 9 = 0, and there we are again. If you are accustomed to working with equations in general, you'll be fine with these. To put them into standard form, we can: first, combine any like terms; second, get everything on the left side, leaving 0 on the right side of the = sign; third, put the squared term first, then the one with the variable that isn't squared, if there is one, and then any constant that you might have.

Graphing and Vertex Form

When we graph a quadratic equation, we need a y variable as well as an x, so we use y = ax^2 + bx + c. We might also write it as y = a(x + h)^2 + k; you'll see why in a little bit.

Let's start with the simplest kind of quadratic equation: y = x^2. Here's a graph:

Simplest Quadratic Graph
graph of y = x^2

Now let's just change it up a little: how about y = x^2 + 1?

Add 1
graph of y = x^2 + 1

Look! It's shaped the same, but the whole thing is moved up one number. Ok, then, what if we go back and change y = x^2 to y = 2x^2?

Multiply by 2
y = 2x^2

This graph looks skinnier than the other one. Ok, one more variation to cover the basics, and this one is a little more involved. Let's graph y = (x + 1)^2, which would be the same as y = x^2 + 2x + 1:

Add 1 Before Squaring
y = (x + 1)^2

Now this one just made the whole graph scoot over by one number. These are the basic things that can happen to the graph of a quadratic equation. The shape is called a parabola, and we can add a number to the result of our calculation in order to move the graph up or down; we can multiply x^2 by a number to make the graph skinnier or fatter; we can add a number to what we are squaring in order to move the parabola to the left or right; or we can do more than one of these, in combination. This is why, for graphing purposes, we have the vertex form of the quadratic equation, y = a(x - h)^2 + k. The a tells us how fat or skinny the graph is; the h tells us how much to the right or left the parabola moves; the k tells us how far up and down it moves, all from the basic parabola graph of y = x^2. Notice that it says -h, because if we add 1, the graph moves so that the bottom is at -1 instead of 0. That bottom point is called the vertex, and if a is a negative number, the parabola flips over and the vertex is the top point, like this:

Change the sign
y = -x^2

The other feature of the graph of a quadratic equation that we often talk about is the axis of symmetry. This is an imaginary line that runs right through the middle of the parabola, up and down, and through the vertex. If we had the graph printed on paper and we folded the paper along the axis of symmetry, then the two halves of the parabola would be together and all matched up. That's where the symmetry comes in!

If our quadratic equation is in vertex form y = a(x - h)^2 + k, then the vertex will be at the point (h,k) and the axis of symmetry will be the vertical line x = h. If our quadratic equation is in the form y = ax^2 + bx + c, it turns out that the vertex will be (-b/2a,a(-b/2a)^2 + b(-b/2a) + c). How messy! What it boils down to, though, is that the x coordinate will be -b/2a and the y coordinate will be the answer if you take the x coordinate and use it to replace the x in ax^2 + bx + c. Let's try it out on, say, y = 3x^2 + 6x + 1:

Add 1 Before Squaring, Multiply by 3, Then Subtract 2
y = 3x^2 + 6x + 1

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