Question

Sketch the following sets of points in the \(x-y\) plane. $$ \left\\{\left(x, x^{2}\right): x \in \mathbb{R}\right\\} $$

Short answer

The set of points \(\{(x, x^{2}): x \in \mathbb{R} \}\) can be represented in the \(x-y\) plane as an upwards-opening parabola.

Step-by-step solution

Understand the Conditions

The given set of points is \(\{(x, x^{2}): x \in \mathbb{R} \}\). Here, each point is formed by an x-coordinate and its square as the y-coordinate, for all real numbers x.

Identify the Curve

The relationship \(y = x^2\) between the x and y coordinates describes a parabolic curve in the x-y plane, which opens upwards. This curve is symmetric about the y-axis.

Sketch the Curve

Drawing the parabola involves marking a point at the origin (0, 0), and noting that as x moves away from 0 in either the positive or negative direction, \(x^2\) (and thus y) gets larger. Due to the symmetry, points at \(x\) and \(-x\) will have the same positive y-coordinate.