Question

Describe the similarities and differences between the parametric equations \(x=t, y=t^{2}\) and \(x=-t, y=t^{2},\) where \(t \geq 0\) in each case.

Short answer

Answer: The similarities are that both sets of parametric equations represent the same parabola with a vertical axis of symmetry and vertex at the origin, and both share the same parametric expression for y. The differences are that the first set represents the right half of the parabola, while the second set represents the left half, and their orientation along the x-axis is different, with the first set opening towards the positive x-axis and the second set opening towards the negative x-axis.

Step-by-step solution

Understand the given parametric equations

The first set of parametric equations is \(x=t,\, y=t^{2}\) with \(t \geq 0\). This represents the right half of the parabola \(y=x^2\), as the \(x\)-coordinate increases for \(t \geq 0\). The second set of parametric equations is \(x=-t,\, y=t^{2}\) with \(t \geq 0\). This represents the left half of the parabola \(y=x^2\), but moving in the opposite direction for the x-coordinate, as the \(x\)-coordinate decreases for \(t \geq 0\). Now, we will investigate further similarities and differences in the two sets of parametric equations.

Analyze similarities

The similarities between the two sets of parametric equations are: 1. Both sets of parametric equations represent a parabola with a vertical axis of symmetry and vertex at the origin \((0, 0)\). 2. Both parametric equations share the same parametric expression for \(y\), i.e., \(y=t^{2}\). This means that the height of the parabola will be the same for both sets of parametric equations.

Analyze differences

The differences between the two sets of parametric equations are: 1. The first set of parametric equations, \(x=t\), represents the right half of the parabola, while the second set of parametric equations, \(x=-t\), represents the left half of the parabola. 2. The orientation of the two parabolas is different. For the first set of parametric equations, the parabola opens towards the positive \(x\)-axis, while for the second set of parametric equations, the parabola opens towards the negative \(x\)-axis. In conclusion, the parametric equations \(x=t, y=t^{2}\) and \(x=-t, y=t^{2}\) represent the right and left halves of the same parabola \(y=x^2\), with the only difference being their orientation along the \(x\)-axis.

Key concepts

parabola

A parabola is a U-shaped curve that can open upwards, downwards, to the left, or to the right. In our exercise, we are dealing with a parabolic curve defined by the equation \(y = x^2\). This particular form is known for its vertical symmetry and its vertex at the origin, \((0, 0)\).
In a parabola, all the points are equidistant from a fixed point called the focus, and a straight line called the directrix. This relationship ensures its symmetrical shape. Parabolas have a variety of applications, from satellite dishes to headlights on cars, because they focus signals to a single point.
When defining a parabola in terms of parametric equations, the equations \(x = t\) and \(y = t^2\) describe the right-hand side of \(y = x^2\), whereas \(x = -t\) and \(y = t^2\) describe the left-hand side. These equations encapsulate the same parabolic shape seen in the standard \(y = x^2\) equation.

parametric representation

Parametric representation provides a method to express sets of equations to depict geometric shapes, such as lines, circles, and parabolas, in terms of a parameter. Here, both halves of the parabola \(y = x^2\) are represented using 't' as a parameter.
Through parametric equations, one can represent complex curves by separate expressions for each axis, such as \(x = f(t)\) and \(y = g(t)\). This approach is beneficial because it allows flexibility and can convey more complex behaviors than standard Cartesian equations.
For instance, the parametric set \(x = t\), \(y = t^2\) for \(t \geq 0\), enables us to trace the right side of the parabola. Reversely, using \(x = -t\), \(y = t^2\) for the same parameter range traces the left side. This tool helps in breaking down symmetric structures into more usable forms for analysis.

axis of symmetry

The axis of symmetry in a parabola is a vital feature that allows it to have a mirrored appearance along a certain line. For the parabola described by the equation \(y = x^2\), this line is the y-axis. This is because any point \((x, y)\) on one side of the parabola has a corresponding point \((-x, y)\) on the opposite side.
In terms of parametric equations, the symmetry is apparent as both \(x = t\) and \(x = -t\) produce identical \(y\) values given \(y = t^2\). This axis acts as a divider, with identical shapes mirrored on either side.
The importance of the axis of symmetry is evident in its role in many mathematical and physical systems. It simplifies the calculation and understanding of parabolic motion, beams in engineering, and even in optics, where light's focus through lenses and mirrors often relates to symmetric properties.