Question

Give a set of parametric equations that describes the following curves. Graph the curve and indicate the positive orientation. If not given, specify the interval over which the parameter varies. The segment of the parabola \(y=2 x^{2}-4,\) where \(-1 \leq x \leq 5\)

Short answer

Question: Determine the parametric equations for a segment of the parabola \(y = 2x^2 - 4\), where \(-1 \le x \le 5\), and state the interval over which the parameter t varies. Answer: The parametric equations are \(x(t) = t\) and \(y(t) = 2t^2 - 4\), with the parameter t varying over the interval \(-1 \le t \le 5\).

Step-by-step solution

Determine the parametric equations for x and y coordinates

Let \(x = t\). This choice of parameter will help keep things simple when matching the conditions \(-1 \le x \le 5\). Now that we have x in terms of t, we can express y in terms of t as well, using the given equation of the parabola \(y = 2x^2 - 4\). We substitute \(t\) for \(x\) to get the parametric equation for \(y\). \(y = 2t^2 - 4\) So the parametric equations are: \(x(t) = t\) \(y(t) = 2t^2 - 4\)

Determine the interval for the parameter t

Since we set \(x = t\), the conditions given for \(x\) also apply to \(t\). Thus, the parameter \(t\) varies over the interval \(-1 \le t \le 5\).

Graph the curve and indicate the positive orientation

To graph the curve, we can first graph the parabola given by \(y = 2x^2 - 4\) and then restrict the graph to the segment we consider when \(-1 \le x \le 5\). The graph will start at point \((-1, 6)\), which corresponds to \(t = -1\), and end at point \((5, 46)\), which corresponds to \(t = 5\). To demonstrate the positive orientation, we add an arrow in the direction of increasing \(t\) values. In summary, the parametric equations for the given curve are: \(x(t) = t\) \(y(t) = 2t^2 - 4\) And the parameter t varies over the interval \(-1 \le t \le 5\).

Key concepts

Calculus

Calculus is a branch of mathematics that provides a framework for analyzing changes in functions and sequences. It consists of two complementary sub-fields: differential calculus and integral calculus. Differential calculus focuses on the concept of the derivative, which is essentially the rate of change of a function as its input changes. Integral calculus, on the other hand, deals with the accumulation of quantities, which is represented by the integral.

When working with parametric equations in calculus, differential calculus can be used to find the slope of the tangent line to the curve at any given point. In our exercise, knowing how to differentiate the functions that define the parametric equations would enable us to find how fast the y-coordinates change with respect to the x-coordinates at any point along the curve.

Parabola

A parabola is a curve where any point is at an equal distance from a fixed point, known as the focus, and a fixed straight line, known as the directrix. The standard form of a parabolic equation is either \(y = ax^2 + bx + c\) for a vertical parabola or \(x = ay^2 + by + c\) for a horizontal one.

In the given exercise, the curve is described by the equation \(y = 2x^2 - 4\), which is a vertical parabola that opens upwards since the coefficient of \(x^2\) is positive. Parametric equations break this down into separate expressions for x and y, both in terms of the parameter, often represented as \(t\). This allows us to trace the journey of a point along the parabola, observing how both coordinates change simultaneously.

Graphing Curves

Graphing curves is a visual way to represent functions or sets of parametric equations. The graph of a function or parametric equation can display a variety of shapes and can give insight into the behavior of the equation it represents. When graphing parametric equations, each x and y coordinate pair \(\{x(t), y(t)\}\) is calculated for different values of \(t\), and then plotted on a coordinate plane.

In the context of our exercise, the parabola defined by the parametric equations is graphed over the interval \(t\) where \( -1 \leq t \leq 5\). Each value of \(t\) gives us a corresponding \(x\) and \(y\) coordinate, demonstrating the relationship between the parameter and the curve's shape.

Positive Orientation

In the context of a curve graphed in the plane, positive orientation refers to the direction along the curve that corresponds to increasing values of the parameter. This is analogous to the idea of moving forward along the curve.

For the given parabolic segment described by its parametric equations, the positive orientation is indicated by the parameter \(t\) increasing from \( -1 \leq t \leq 5\). When we graph this curve, we show its positive orientation by drawing an arrow that points from the starting point (corresponding to the smaller \(t\) value) to the ending point (corresponding to the larger \(t\) value). This provides a visual representation of the direction in which a point would travel along the curve as \(t\) increases.