A quadratic polynomial is a polynomial whose degree (the highest power of $x$) is $2$. The general form of a quadratic polynomial is:
$$ \begin{equation}\begin{aligned} y=ax^2+bx+c\\ \end{aligned}\end{equation} $$The y-intercept
From this equation, we let $x=0$ to get the y-value of the y-intercept: $$ \begin{equation}\begin{aligned} y&=a(0)^2+b(0)+c\\ y&=c\\ \end{aligned}\end{equation} $$
Hence the y-intercept is the point $(0,c)$.
The x-intercepts a.k.a. the roots of the curve
We can also find the x-intercepts of the curve if we let $y=0$:
$$ \begin{equation}\begin{aligned} 0=ax^2+bx+c\\ \end{aligned}\end{equation} $$This can be solved by factorization or the quadratic formula:
$$ \begin{equation}\begin{aligned} x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\ \end{aligned}\end{equation} $$Knowing these points intercepts are important in sketching the curve but we still have the issue of determining where the curve is supposed to turn around.
The turning point of the curve
If we can write the curve in the form:
$$ \begin{equation}\begin{aligned} a(x+h)^2+k\\ \end{aligned}\end{equation} $$Then the coordinates of the turning point of the curve is simply $(-h,k)$.
The vertical line ($x=-h$) passing through the turning point is referred to as the axis of symmetry of the curve because the curve is symmetrical about this line.
What k means
The y coordinate of the turning point is known as the maximum/minimum value of the curve (based on the nature of the turning point).
For a maximum curve, the value $y=k$ is the maximum value. For a minimum curve, $y=k$ is the minimum value.