Question

Find parametric equations for the parabola \(y=x^{2}.\)

Short answer

Question: Find the parametric equations for the given parabola \(y = x^2\). Answer: The parametric equations for the given parabola are \(x = t\) and \(y = t^2\).

Step-by-step solution

Choose a parameter and find the relationship with \(x\)

Let's choose the parameter \(t\) and relate it to \(x\). We can write \(x=t\). This creates an easy starting point, as there are no restrictions on our parameter \(t\).

Find a relationship for \(y\)

Now, we have \(x = t\). We can substitute this into the equation for the given parabola \(y = x^2\). This will give us a relationship for \(y\) in terms of our parameter \(t\). Doing the substitution, we have: \(y = t^2\)

Write the parametric equations

Now that we have found a relationship for both \(x\) and \(y\) in terms of the parameter \(t\), we can write the parametric equations for the given parabola: \(x = t\) \(y = t^2\) These are the parametric equations for the given parabola \(y = x^2\) in terms of the parameter \(t\).

Key concepts

Understanding the Parabola

A parabola is a fascinating geometric shape often encountered in mathematics. It is a symmetric open curve formed by the locus of points which are equidistant from a fixed point known as the focus and a fixed straight line called the directrix. In the coordinate plane, the most common form of a parabola is given by the quadratic equation \(y = x^2\), where the vertex is at the origin (0,0). This type of parabola opens upwards.

Key features of a parabola include:

• The vertex: The point where the parabola changes direction, located at the lowest or highest point on the graph, for \(y=x^2\), this is at (0,0).
• The axis of symmetry: A vertical line that runs through the vertex, dividing the parabola into two mirror-image halves. For \(y = x^2\), this line is \(x=0\).
• The focus and directrix: These help define the parabola’s shape according to its geometrical properties.

Understanding these features is crucial for studying parabolas in coordinate geometry, as it helps in graphing and solving problems related to parabolic shapes.

The Art of Parameterization

Parameterization is a crucial concept in mathematics, especially when dealing with curves like the parabola. It involves representing a set of equations using parameters, which allows you to express the coordinates of any point on the curve in terms of these parameters.

To create parametric equations for a curve like a parabola \(y = x^2\), we introduce a parameter \(t\). A parameter acts like a slider that moves you along the curve, providing a dynamic way to describe the motion or position on a graph.

Here's how parameterization works in our example:

• Start by setting \(x = t\). This is a simple choice as \(t\) can take any real number.
• Next, use the original equation to substitute for \(y\). Since \(y = x^2\), and \(x = t\), we have \(y = t^2\).
• The parametric equations become \(x(t) = t\) and \(y(t) = t^2\), effectively expressing the entire parabola in terms of \(t\).

This kind of representation offers powerful insights and flexibility, particularly when analyzing the movement along the curve or when applying calculus for computations such as finding tangents and areas.

Coordinate Geometry and its Application

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebraic equations to represent geometric figures. This fusion of algebra and geometry allows us to explore shapes and their properties in a coordinate system.

In the realm of parabolas, coordinate geometry is incredibly useful. It enables us to visualize how parabolas, like \(y = x^2\), are laid out in a plane and aids in finding intersections, calculating distances, and determining shapes' orientation.

With the help of the coordinate system, we can:

• Plot points on a graph using coordinates \((x, y)\), making it easier to understand the structure of a parabola.
• Use equations, such as parametric forms, to describe and manipulate curves.
• Apply geometric transformations like shifts, stretches, and rotations to analyze how parabolas and other figures change under different conditions.

Through coordinate geometry, the abstract notion of shapes becomes more tangible and manageable, leading to deeper insights in both pure and applied mathematics.