Derivation of the formula for the vertex of a parabola

Question

I'm taking a course on Basic Conic Sections, and one of the ones we are discussing is of a parabola of the form

$$y = a x^2 + b x + c$$

My teacher gave me the formula:

$$x = -\frac{b}{2a}$$

as the $x$ coordinate of the vertex.

I asked her why, and she told me not to tell her how to do her job.

My smart friend mumbled something about it involving calculus, but I've always found him a rather odd fellow and I doubt I'd be able to understand a solution involving calculus, because I have no background in it. If you use something you know from calculus, explain it to someone who has no background in it. Because I sure don't.

Is there a purely algebraic or geometrical yet elegant derivation for the $x$ coordinate of a parabola of the above form?

Answer 1

Already so many answers, but I haven't seen my favorite one posted, so here's another.

The vertex occurs on the vertical line of symmetry, which is not affected by shifting up or down. So subtract $c$ to obtain the parabola $y=ax^2+bx$ having the same axis of symmetry. Factoring $y=x(ax+b)$, we see that the $x$-intercepts of this parabola occur at $x=0$ and $x=-\frac{b}{a}$, and hence the axis of symmetry lies halfway between, at $x=-\frac{b}{2a}$.

Answer 2

By the vertex I assume you mean the minimum/maximum point of the parabola. Indeed, this result can be discovered easily through a bit of calculus, but there is also a simple purely algebraic way, which I will present here.

Let's consider a generic quadratic expression:

$y = ax^2 + bx + c$

We now complete the square on this formula.

$y = a[x^2 + bx/a + c/a]$

$y = a[(x + b/2a)^2 - (b/2a)^2 + c/a]$

The expression $- (b/2a)^2 + c/a$ is a constant (it does not depend on x), so we can replace it with k for the matter of discussion.

$y = a[(x + b/2a)^2 + k]$

Now, depending on whether a is positive or negative, the parabola given by y will either have a maximum or minimum. Since a and k are fixed, this must occur when $(x + b/2a)^2$ is zero (we know it cannot be less than zero, and it can extend to infinity).

Hence, we know that for $(x + b/2a)^2$ to be zero, $x = -b/2a$. This in turn implies that the function y is at a minimum or a maximum when this is true. Q.E.D.

Answer 3

Parabolas of the form you described (y = ...) are symmetric over a vertical line through their vertex. Let's call that line x = k. This means that if the graph crosses the x-axis (meaning that $ax^2+bx+c=0$ has real solution(s)), they must be equidistant from x = k, so (k,0) must be the midpoint of the segment with endpoints at the zeros of the quadratic or k is the average of the zeros. From the quadratic formula, the two zeros of the quadratic are $\frac{-b\pm\sqrt{b^2-4ac}}{2a}$, so their sum is $-\frac{b}{a}$ and their average is $k=-\frac{b}{2a}$. So, the x-coordinate of the vertex must be $-\frac{b}{2a}$.

If the parabola does not cross the x-axis (no real solutions), there is another parabola with equation $y=ax^2+bx+c'$ for some $c'$ for which the graph is a vertical translation of the graph of the original quadratic, but crosses the x-axis. Its axis of symmetry is $x=-\frac{b}{2a}$ and because it is a vertical translation of the original, the axis of symmetry of the original is also $x=-\frac{b}{2a}$, so the vertices of both have x-coordinate $-\frac{b}{2a}$.

Answer 4

An alternative approach, which isn't necessarily the best in and of itself, but stick with me ...

Instead of using the fact that the vertex is the point where a vertical line cuts the parabola exactly in half, we'll use the fact that it's also the point where a horizontal line just brushes the curve, meeting it in a single point.

Consider how the horizontal line $y=d$ crosses the parabola as you adjust $d$. If $d$ places the line above or below the vertex, there are either 2 or 0 points of intersection (which is which depends upon whether the parabola points up or down); when $d$ places the line at the same height as the vertex, the parabola seems to sit comfortably on the line. (The line is tangent to the parabola the point they have in common, in the same way that lines can be tangent to circles.)

We can find the point of tangency (the vertex), then, by using algebra to determine when the intersection of our parabola and line equations have a single solution.

We begin with the standard find-the-intersection strategy, setting our formulas for $y$ equal:

$d = a x^2 + b x + c$

When does this equation have a single solution for $x$? Let's see. Rearranging and invoking the Quadratic Formula, we have:

$a x^2 + b x + c - d = 0$

$x = \frac{1}{2a}\left( - b \pm \sqrt{ b^2 - 4 a (c-d) } \right)$

Now, when the stuff under the radical matters, then we have either 2 or 0 values of $x$: two values if the stuff is positive (the square root gives a quantity to add and subtract from $-b$); no roots if the stuff is negative (the square root gives an imaginary number we can't use here). We'll have our sought-after single value of $x$ when $d$ is whatever-it-has-to-be (its exact value doesn't matter) to make the stuff under the radical becomes zero, making the radical term vanish; when that happens, our value of $x$ falls out as

$x = -\frac{b}{2a}$

Just like with the other approaches.

With this approach, however, we can consider a more sophisticated situation. It may be somewhat advanced for your class, but --if you can make sense of my rambling-- you can use it to blow your teacher's mind. :)

(Before we begin: I wouldn't necessarily recommend this exploration for every student just beginning to learn about parabola equations. However, I think having it "in the archive", as it were, can be worthwhile for people who come across the question to see that the answer can lead to some interesting places.)

We ask ourselves this question:

What if we tilt the line?

That is: What can we say about the $x$-coordinate of the point of tangency where the parabola is touched by a line with slope $m$?

Let the slanted line's equation be $y = m x + d$, where $d$ is adjustable. We can make the same observations as in the horizontal case: if $d$ places the line above or below the point of tangency, the line will meet the parabola in either 2 or 0 places; for a special value of $d$, however, the line meets the parabola at the single point, the point of tangency.

As before, we set our formulas for $y$ equal to each other ...

$m x + d = a x^2 + b x + c$

... then rearrange and invoke the Quadratic Formula ...

$a x^2 + ( b - m ) x + ( c - d ) = 0$

$x = \frac{1}{2a}\left( -(b-m) \pm \sqrt{\text{stuff}} \right) = \frac{1}{2a}\left(m-b\pm \sqrt{\text{stuff}}\right)$

And as before, we don't really care about the complicated mess that "$\text{stuff}$" might be, since we're only interested in the situation where it goes away anyhow.[*] Once the radical term vanishes and the dust settles, we have this:

For the parabola described by the equation $y = a x^2 + b x + c$, the $x$-coordinate of the point where a line of slope $m$ lies tangent to the curve is given by

$x = \frac{1}{2a}(m - b)$

Here's the kicker that your smart friend[**] can appreciate: That statement really is a fact that most people don't see until Calculus. Except there, the issue tends to be phrased a little differently: It's not about finding the $x$-coordinate of the point-of-tangency based on a given slope, but about finding the slope of the line based on the $x$-coordinate for the point-of-tangency. So, we re-phrase the above equation to yield $m$:

For the parabola described by the equation $y = a x^2 + b x + c$, the slope $m$ of the line tangent to the curve at a point with $x$-coordinate $x$ is given by

$m = 2 a x + b$

You'll be way ahead of the game if you can see a neat way to get directly from the parabola equation to the slope formula. I'll give you a hint by showing how the formula fits in with a couple of facts you already know about the slopes of lines, and show you one step further, with a few unnecessary zeros thrown in:

$y = a$   slope${}=0$.
$y = a x + b$   slope${}=a + 0$.
$y = a x^2 + b x + c$   slope${}= 2 a x + b + 0$
$y = a x^3 + b x^2 + c x + d$   slope${}= 3 a x^2 + 2 b x + c + 0$
$y = a x^4 + b x^3 + c x^2 + d x + e$   slope = you tell me

Detect a pattern?

[*] Actually, you should care enough to convince yourself that you can find a value of $d$ that makes "$\text{stuff}$" go away, but for now you can trust me that such a $d$ exists.

[**] Keep in mind that "smart" doesn't always mean "having the right answers"; just as often, it means "asking the right questions". The fact that you chose not to accept a formula blindly and asked your teacher "why" --and then steadfastly continued your quest for an answer when she wouldn't provide one-- suggests to me that your smart friend has a smart friend, too.

Answer 5

Here is the simplest way I know, using a bit of algebra. Starting with Isaac's observation that the vertex lies on the axis of symmetry, the line x = k. Question is, what's k? That's your answer.

if f(x) = ax² + bx + c

Then the symmetry around x = k means that, for any h

f(k+h) = f(k-h)

From that, just multiply it all out and some simple additive cancellation gives

-2akh - bh = 2akh + bh

h cancels out, and a bit more algebra gives

k = -$\frac{b}{2a}$

To really complete this, however, we should show that x = k = -$\frac{b}{2a}$ is a maximum or minimum point. Again, with more algebra:

f(k+h) - f(k) boils down to ah²

So when a > 0, then f(k+h) > f(k) for any h != 0, establishing that x = k is a minimum point

And likewise when a < 0, x = k is a maximum. This also shows the symmetry around x = k, and how the coefficient a determines whether the curve is concave upward or downward.

Answer 6

Complete the square \begin{align} y = ax^2 + bx + c & = a\left(x^2 + \frac b a x \right) + c \\[10pt] & = a\left( x^2 + \frac b a x + \frac {b^2}{4a^2} \right) + c - \frac {b^2}{4a} \\[10pt] & = a\ \underbrace{\left( x + \frac b {2a} \right)^2}_\text{a square} + \underbrace{\frac{4ac-b^2}{4a}}_{\text{ No $x$ appears here.}}. \end{align}

The square is $0$ when $x = -\frac{b}{2a}$ and is positive when $x\neq -\frac{b}{2a}$. From that you can conclude things about the shape of the graph, including the location of the vertex.

In particular, this shows you why the shape of the graph is the same regardless of the values of $a,b,c$ as long as $a\ne0$.

Answer 7

I like Americo's answer using translation of y = ax^2. But to finish it, we should at least note that ax^2 has a maximum or minimum at <0,0>, which is quite obvious, and that translation preserves order and therefore maxima and minima, which is pretty easy.

Answer 8

The vertex of a parabola is the point on the axis of symmetry that intersects it (Wikipedia).

The point $(x,y)=(0,0) $ is the vertex of the parabola given by $y=ax^2$. By making the change of coordinates

$$X=x+\frac{b}{2a},Y=y+\frac{b^{2}-4ac}{4a},$$

equation $y=ax^{2}+bx+c$ is transformed into $Y=aX^{2}$, the vertex of which is the point $(X,Y)=(0,0)$, i.e. $(x,y)=\left(-\dfrac{b}{2a},-\dfrac{b^{2}-4ac}{4a}\right)$.

(This change of coordinates is a translation of both axes $x$, $y$, respectively, by $-\dfrac{b}{2a},-\dfrac{b^{2}-4ac}{4a}$.)

Hence, the $x$-coordinate of the vertex of the parabola is $-\dfrac{b}{2a}$.

---

Edit. Example. Plot of $y=2x^2-x+4$; blue axes: $X=x-1/4,Y=y-31/8$

Answer 9

Echoing Americo's and Isaac's answers, but without appealing to the quadratic equation:

First treat the special case f(x)=ax²+d (i.e. no "x" term); by remembering that the line that divides the parabola symmetrically passes through the vertex, and is in fact the y-axis (x=0), we see that 0 is the abscissa of the vertex.

To treat the more general f(x)=ax²+bx+c, we try a substitution of the form x=x'-h (geometrically, this corresponds to translating your coordinate system horizontally by h units). Expanding this, you will get something like a(x')²+(2ah+b)x'+(constant term). To go back to the special case we first treated, we find the value of h such that the coefficient of x' is 0. Thus you obtain the expression for the vertex's x-coordinate.

Answer 10

There are nice answers already but I would like to add another way to look at this:

The quadratic equation $a(x-r_1)(x-r_2) = ax^2 - a(r_1 + r_2)x + r_1 r_2 = 0$ defines a parabola, $r_1, r_2$ are the roots (intersections with the x-axis).

The middle point is between the two roots, $\frac{r_1 + r_2}{2} = -\frac{- a(r_1 + r_2)}{2a}$.

This leads to an interesting question about what happens when the parabola does not have real roots.

Answer 11

It could be argued that the parabola is symetric respect to the vertical line passing through the vertex point. So, the vertex is located in the midpoint between the two roots of $y=0$.

As pointed out before, by completing the square, the parabola equation can be written as $$ y = a(x - x_+)(x - x_-)\,, $$

where $x_+$ and $x_-$ are the roots mentioned above, given by $$ x_\pm = \dfrac{ -b \pm \sqrt{ b^2 - 4 a c} }{2a} \,.$$

Then, the midpoint between $x_+$ and $x_-$ is $$ x_0 = \dfrac12 (x_+ + x_- ) = -\frac{b}{2a} \,.$$

Note that this demonstration is also valid even when there is no real roots for $y=0$ (which is the case when $b^2 < 4a c$), since the complex terms in the last equation are mutually canceled.

Sketch of the parabola:

Answer 12

_{(source: hotmath.com)}

$y = ax^2 + bx + c$

for parabola written in Vertex Form vertex is (h,k).

$y = m(x-h)^2 + k = \underline{m}x^2 + \underline{(-2mh)}x+ \underline{mh^2 +k}$

$\rightarrow$

$a = m$

$b = -2mh = -2ah \rightarrow h = -b/2a$

$c = mh^2 + k = ah^2 + k \rightarrow k = c - b^2/4a$

source

Answer 13

Lots of interesting answers here. Might as well make it an even dozen. :-)

Actually, this can be derived directly from the definition of a parabola. Using the definition and the distance formula, you can derive the Standard Format for the equation of a parabola:

$$4p\left(y-k\right) = \left(x-h\right)^2$$

where $V=(h,k)$ is the vertex of the parabola and $p$ is the distance from the vertex to the focus (and also from the vertex to the directrix). If $p > 0$, the parabola opens upward; if $p < 0$, the parabola opens downward. I assume you've run across this standard form already and are familiar with it (there are plenty of online pages you can reference if not).

Now just expand the right-hand-side and solve for y. You end up with:

$$y = \left.\frac1{4p}\right.x^2 - \left.\frac{h}{2p}\right.x + \frac{h^2}{4p} + k$$

which is an equation of the form $y = ax^2 + bx + c$ with

$$a = \frac1{4p},\qquad b = -\frac{h}{2p},\qquad c = \frac{h^2}{4p} + k$$

Now just go right down the line and solve for $p$, $h$, and $k$.

$$\begin{align} a & = \frac1{4p} \\ p & = \frac1{4a} \\ \end{align}$$

[Here's the one you're interested in...] $$\begin{align} b & = -\frac{h}{2p} \\ b & = -\frac{h}{\frac1{2a}}\qquad \text{substituting}\; p = \frac1{4a}\; \text{above} \\ -\frac{b}{2a} & = h \\ \end{align}$$

and lastly,

$$\begin{align} c & = \frac{h^2}{4p} + k \\ c & = \frac{\left(-\frac{b}{2a}\right)^2}{\frac1a} + k\qquad \text{substituting for}\; p\;\text{&}\;h\;\text{above} \\ c & = \frac{\frac{b^2}{4a^2}}{\frac1a} + k \\ c & = \frac{b^2}{4a} + k \\ c - \frac{b^2}{4a} & = k \\ \end{align}$$

Now you can easily convert back & forth between Standard Format and the other.

Answer 14

I'll add to David Lewis' answer in more detail (for those who are looking for the full derivation!):

Let the point $x_0$ define the axis of symmetry. By definition, $x_0$ has the property that any deviation $\delta x$ from $x_0$, irrespective of whether it is positive or negative, will give you the same value of $y$. That is \begin{equation} y (x + \delta x) = y (x - \delta x) \end{equation} A positive deviation, $\delta x$, from $x_0$ can be expressed as \begin{equation} y^{+} = a (x_0 + \delta x)^2 + b (x_0 + \delta x) + c \end{equation} Similarly, a negative deviation, $- \delta x$, from $x_0$ can be expressed as \begin{equation} y^{-} = a (x_0 - \delta x)^2 + b (x_0 - \delta x) + c \end{equation} In order to find the value of $x_o$ for the axis of symmetry, we set these two expressions equal to one another \begin{equation} \begin{aligned} a (x_0 + \delta x)^2 + b (x_0 + \delta x) + c & = a (x_0 - \delta x)^2 + b (x_0 - \delta x) + c \nonumber \\ a (x_0^2 + 2 x_0 \delta + \delta^2) + b (x_0 + \delta ) + c & = a (x_0^2 - 2 x_0 \delta + \delta^2) + b (x_0 - \delta ) + c \end{aligned} \end{equation} Now we just simplify to get \begin{equation} \begin{aligned} 2 a x_0 \delta + b \delta & = - 2 a x_0 \delta - b \delta \\ 4 a x_0 \delta + 2 b \delta & = 0 \\ 4 a x_0 \delta = - 2 b \delta \\ x_0 = - \dfrac{2b}{4a} \nonumber \end{aligned} \end{equation} or \begin{equation} x_0 = - \dfrac{b}{2a} \label{eq:axis_of_symmetry_parabola} \end{equation}