Consider the function

$$
y=x^{2}-2 \lambda x+a^{2}
$$

Here, $a$ is a given positive value, while $\lambda$ can take any value from $-\infty$ to $+\infty$.

$1^{0}$. Characterize the shape of the function curve!

$2^{0}$. Determine the position of the curve relative to the $Y$-axis as $\lambda \leqq 0$. What is the relative position of two curves for which $\lambda$ has opposite signs?

$3^{0}$. For which values of $\lambda$ does the curve intersect the $X$-axis, and on which side of the $Y$-axis?

$4^{0}$. Which line does the vertex of the curve trace as $\lambda$ varies within the specified limits?
$1^{0}$. $y$ is a quadratic function of $x$, where the coefficient of $x^{2}$ is positive; thus, a parabola $G_{\lambda}$ corresponds to it, whose vertex is the lowest point of the curve, and its axis of symmetry is parallel to the $Y$-axis. If we write the function in the form

$$
y=(x-\lambda)^{2}+a^{2}-\lambda^{2}
$$

it is clear that the vertex of the curve corresponds to the coordinates $x=\lambda, y=a^{2}-\lambda^{2}$.

$2^{0}$. If $\lambda=0$, then the coordinates of the vertex of the curve are $x=0, y=a^{2}$.

The vertex is a fixed point on the $Y$-axis; the axis of symmetry of the curve is the $Y$-axis. If $\lambda>0$, then the vertex is to the right of the $Y$-axis, and if $\lambda<0$, it is to the left of the $Y$-axis.

If $|\lambda|<a$, the curve intersects the $X$-axis at two points, and the $y$-coordinate of the vertex is positive, $y=a^{2}-\lambda^{2}>0$.

If $|\lambda|=a$, the curve touches the $X$-axis. (The vertex lies on the $X$-axis.)

If $|\lambda|>a$, the curve does not intersect the $X$-axis, and the $y$-coordinate of the vertex is positive, $y=a^{2}-\lambda^{2}>0$.

![](https://cdn.mathpix.com/cropped/2024_05_02_c1c30546ee03baa26e84g-1.jpg?height=539&width=382&top_left_y=744&top_left_x=861)

Since the product of the roots of the equation $x^{2}-2 \lambda x+a^{2}=0$ is positive, the roots have the same sign, so the points of intersection are on the same side of the $Y$-axis; if $\lambda>0$, they are to the right of the $Y$-axis, and if $\lambda<0$, they are to the left of the $Y$-axis.

If $|\lambda|=a$, the curve touches the $X$-axis. (The vertex lies on the $X$-axis.)

$4^{0}$. The vertex is determined by the coordinates $x=\lambda, y=a^{2}-\lambda^{2}$, so the relationship

$$
y=a^{2}-x^{2}
$$

holds between them. Thus, the vertices of the parabolas in question trace out another parabola $G$, whose vertex is at the point $\left(x=0, y=a^{2}\right)$, and this curve has its highest point at this vertex.

András Deák (Érseki rg. V. o. Bp. II.)