1
$\begingroup$

Consider the parabola $\rho$ given be the equation $y = ax^2 + bx + c$. Recall that varying $a$ in this equation stretches/squashes $\rho$ and that varying $c$ shifts $\rho$ vertically. The change in the parabola $\rho$ as you vary the value of $b$ is a bit harder to describe, but as you vary $b$, the vertex of $\rho$ traces out a familiar curve. What is this curve? What it the equation of this curve?

$\endgroup$
1
$\begingroup$

First let's write down the general form for the vertex of $\rho$. Through any of various means we can find that the $x$-coordinate of the vertex is $-\frac{b}{2a}$. Substituting this value into the equation for $\rho$ we get that the vertex is

$$\left(-\frac{b}{2a},\;\; c-\frac{b^2}{4a}\right).$$

Since the $x$-coordinate of the vertex is linear in $b$ and the $y$-coordinate is quadratic in $b$ (has a $b^2$), we can conclude that the vertex of $\rho$ traces out a parabola. We can reparametrize this curve to get an explicit equation to describe it. Letting $t = -\frac{b}{2a}$ we can make the substitution $b = -2at$ into the $y$-coordinate, giving us the parametric curve defined by

$$ \left(t,\;\; c-\frac{(-2at)^2}{4a}\right) \;\;=\;\; \left(t,\;\; c-at^2\right) .$$

To state this explicitly, the curve traced by the vertex of $\rho$ is given by the equation $y = c - ax^2$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.