Question

Find parametric equations for the following curves. Include an interval for the parameter values. The upper half of the parabola \(x=y^{2}\), originating at (0,0)

Short answer

Answer: The parametric equations for the upper half of the parabola \(x = y^2\) are: $$ x(t) = t^2, \quad y(t) = t, $$ with the interval for parameter values being \(t \geq 0\).

Step-by-step solution

Introduce the parameter

Let's introduce a parameter \(t\). For the upper half of the parabola, since \(x = y^2\), we can write \(y=t\) and \(x=t^2\). Therefore, the parametric equations will have the form: $$ x(t) = t^2, \quad y(t) = t $$

Determine interval for parameter values

Now, we need to find an interval for the parameter values in order to represent the upper half of the parabola. Since \(x=y^2\), the graph of the parabola will have the vertex at \((0,0)\). To represent the upper half of the parabola, we can set the interval for \(y(t)=t\) to be \(t \geq 0\), as the parabola originates at \((0,0)\) and the upper half has non-negative \(y\) values. Thus, the interval for the parameter values is \(t \geq 0\). In conclusion, the parametric equations for the upper half of the parabola \(x = y^2\) are: $$ x(t) = t^2, \quad y(t) = t, \quad t \geq 0 $$

Key concepts

Understanding Parabolas

Parabolas are one of the basic curves we encounter in algebra and calculus. They are symmetric curves, shaped like an open bowl, and are typically represented with the equation \( y = ax^2 + bx + c \). When graphed, they can open upwards or downwards depending on the sign of the coefficient \( a \). In this exercise, we're dealing with a sideways-opening parabola given by the equation \( x = y^2 \). This specific form indicates a sideways symmetry since the \( y \) variable is squared.

• Parabolas have a vertex, which in our case is at \( (0,0) \),
• The curve is symmetric around the \( y \)-axis, meaning the right half mirrors the left half,
• The focus and directrix are specific points and lines related to the geometry of the parabola, but here we focus on plotting the curve using parametric equations,

which help parameterize the parabola and provide insights into its geometric properties.

Intervals and Their Importance

Intervals play a crucial role in defining the portion of the curve we are interested in parameterizing. When we discuss parameterization, the interval specifies the range of the parameter, often represented as \( t \), that we use to define our curve.

• In this case, the parameter \( t \) represents \( y \) (\( y = t \)),
• The interval \( t \geq 0 \) ensures that we are only considering the upper half of the parabola starting from the origin, \( (0,0) \),
• By restricting \( t \) to non-negative values, we avoid the lower half and only trace half of the parabola where \( y \) is not negative,

which aligns with the problem's request to focus on representing the upper half of the parabola. Thus, properly defining the interval is integral to accurately presenting the desired portion of the graph.

Parameterization of Curves

Parameterization is a powerful mathematical technique used to express curves using a parameter, often \( t \). Instead of relating \( x \) and \( y \) coordinates directly through an equation, parameterization allows us to express these coordinates in terms of \( t \), making it easier to analyze and trace paths along curves.

• Here, we use \( y = t \) and \( x = t^2 \) to represent our parabola,
• This form is efficient as it naturally describes how \( y \) changes with \( t \), and how \( x \) is a direct square of \( y \),
• Parameterization simplifies computation, especially with complex curves where direct equations may be cumbersome to work with,

enabling a clearer understanding of the path and nature of the curve. This parametric form easily outlines the trajectory of the parabola, illustrating each part's relationship through the parameter \( t \).