Question

Consider the following parametric equations. a. Make a brief table of values of \(t, x,\) and \(y\) b. Plot the points in the table and the full parametric curve, indicating the positive orientation (the direction of increasing \(t\) ). c. Eliminate the parameter to obtain an equation in \(x\) and \(y\) d. Describe the curve. $$x=t^{3}-1, y=5 t+1 ;-3 \leq t \leq 3$$

Short answer

Question: Given the set of parametric equations: \(x = 2t + 3\) and \(y = t^2 - 1\) on the interval \(-3 \leq t \leq 3\), a. Create a table of values for \(t, x,\) and \(y\). b. Plot the points and the curve. c. Eliminate the parameter \(t\) to obtain a Cartesian equation in terms of \(x\) and \(y\). d. Describe the curve. Answer: a. The table of values for \(t, x,\) and \(y\) are: | \(t\) | \(x\) | \(y\) | |-------|-------|-------| | -3 | -3 | 8 | | -2 | -1 | 3 | | -1 | 1 | 0 | | 0 | 3 | -1 | | 1 | 5 | 0 | | 2 | 7 | 3 | | 3 | 9 | 8 | b. After plotting the points and the curve, the shape resembles a parabolic curve opening to the right. c. To eliminate the parameter t, solve for \(t\) in \(x = 2t + 3\) and substitute into \(y = t^2 - 1\): \(t = (x - 3)/2\) Substituting this expression into the equation for \(y\): \(y = [(\frac{x-3}{2})]^2 - 1\) \(y = \frac{(x - 3)^2}{4} - 1\) d. The curve is a parabola opening to the right, with a vertex at (3, -1). The orientation of the curve is positive, meaning that as the parameter \(t\) increases, the curve moves from left to right.

Step-by-step solution

(Step 1: Create a table of values)

(Start by choosing some values for the parameter \(t\) within the given interval, \(-3 \leq t \leq 3\), and calculate the corresponding \(x\) and \(y\) using the given parametric equations. Select a range of values for \(t\), for example at increments of 1, covering the entire interval.)

(Step 2: Plot the points and the curve)

(Using the table of values from step 1, plot the points on a Cartesian coordinate plane. Additionally, sketch the full parametric curve on the plane while indicating the direction of increasing \(t\) (positive orientation).)

(Step 3: Eliminate the parameter to obtain a Cartesian equation)

(To eliminate the parameter \(t\), we need to find a relationship between \(x\) and \(y\) that doesn't involve \(t\). To do this, we will solve for \(t\) in one of the parametric equations and substitute that expression into the other equation. )

(Step 4: Describe the curve)

(Based on the parametric equations and the plot obtained in step 2, describe the shape, behavior, and orientation of the curve in words, providing any relevant details.)

Key concepts

Parameter Elimination

In parametric equations, each curve is usually expressed as a set of functions that depend on a parameter, often denoted as \( t \). To eliminate this parameter and retrieve a single equation relating \( x \) and \( y \), follow these easy steps:

• Start by selecting one of the equations to solve for \( t \). For example, from \( x = t^3 - 1 \), solve for \( t \) as follows: \( t = (x+1)^{1/3} \).
• Next, substitute this expression into the other parametric equation, \( y = 5t + 1 \). Replace \( t \) with \( (x+1)^{1/3} \): \( y = 5(x+1)^{1/3} + 1 \).
• The resulting equation \( y = 5(x+1)^{1/3} + 1 \) is now a Cartesian equation, free of the parameter \( t \). This means the relationship between \( x \) and \( y \) is now described directly.

Parameter elimination simplifies the understanding of curves by focusing directly on the relationship between \( x \) and \( y \).

Cartesian Coordinates

Cartesian coordinates describe a point's location on a plane using a pair \( (x, y) \). This system, created by René Descartes, allows us to plot any geometric shape by specifying dimensions along the x and y axes.

• The \( x \)-axis runs horizontally, with values increasing from left to right.
• Meanwhile, the \( y \)-axis runs vertically, with values increasing from bottom to top.
• The intersection of these axes, known as the origin, is where \( x = 0 \) and \( y = 0 \).

When plotting from parametric equations, you calculate points for various \( t \) values, then place them within the Cartesian plane. Connecting these points offers a visual representation of the curve, vividly showcasing its behavior and trajectory. This view is instrumental in understanding how functions behave across different x-values.

Curve Plotting

Plotting a curve from parametric equations bridges the gap between mathematical expressions and visual interpretation. It's a step that transforms abstract equations into tangible graphics.

• Begin by choosing a range of \( t \) values, often within given bounds, like \([-3, 3]\).
• Calculate the corresponding \( x \) and \( y \) coordinates using your parametric equations.
• Plot these coordinates on the Cartesian plane, marking each point clearly.

By connecting these points, you trace the path of the curve. It's crucial to indicate the positive orientation, typically drawn as an arrow in the direction of increasing \( t \). This detail underlines the curve’s direction over time. Describing the plotted curve involves noting its shape—whether it’s linear, circular, or another form—and recognizing any key features or symmetries. Through careful plotting, the mathematical and graphical representation of a curve is seamlessly linked.