(Sice this to be a college level subject from your address, I would like to cover the answer analysing the perspective of the function and tracing it by the properties of the function).
f(x) = x^2-3 as a function represents parabola. It is a function of second degree.
Also we can write y = x^2-3.
Symmetry: The curve is symmetrixcal about y axis as for x and -x the function gives the same positive y value.
Vertex : The vertex of the cuve is (0, -3)
The cuve is open upwards and goes for positive ifinite values on both left and right as x becomes large and large.
The curve, obviously ,has a value of -3 lowest when x=0. And that y= -3 ids the intercept value of the cuve on the Y axis.
The curve as crosses the x axis at equal distances of sqrt3 on the right and -sqrt3 on the left.
The curve y= x^2-3 Or x^2 = y+3or parabola has Could be compares to a
X^2 = 4aX a standard parabola. So we can write x^2 = 4(1/4) (y+y). So a= 1/4 is the focal length and (0, -3+1/4) or (0, -11/4 ) are the co ordinate positions of the focus. And the equation of the directrix is y = -3-1/4 . Or y = -13/4 a parallel line to X axis .
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