Question

For the following exercises, sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve. $$ x=1+t, y=t^{2}-1,-1 \leq t \leq 1 $$

Short answer

The Cartesian equation is \(y = x^2 - 2x\). The curve runs from \((0, 0)\) to \((2, 0)\).

Step-by-step solution

Understanding the Parametric Equations

The given parametric equations are \(x = 1 + t\) and \(y = t^2 - 1\). Our task is to sketch the curve represented by these equations and eliminate the parameter \(t\) to find a single Cartesian equation in terms of \(x\) and \(y\).

Eliminating the Parameter

To eliminate \(t\), solve the first equation for \(t\): \(x = 1 + t\) becomes \(t = x - 1\). Substitute \(t = x - 1\) into the second equation \(y = t^2 - 1\): \(y = (x - 1)^2 - 1\).

Forming the Cartesian Equation

After substitution, we get \(y = (x - 1)^2 - 1\). Simplify the expression: \(y = x^2 - 2x + 1 - 1\) results in \(y = x^2 - 2x\). This is the Cartesian equation of the curve.

Sketching the Curve

To sketch the curve given the parametric boundaries \(-1 \leq t \leq 1\), calculate the points by substituting \(t\) values into \(x = 1+t\) and \(y = t^2 - 1\). For \(t = -1\), \((x, y) = (0, 0)\). For \(t = 0\), \((x, y) = (1, -1)\). For \(t = 1\), \((x, y) = (2, 0)\). Plot these points and draw the parabola \(y = x^2 - 2x\) between \(x = 0\) and \(x = 2\) as these define the range of the curve within the given \(t\) values.

Key concepts

Cartesian equation

A Cartesian equation is a way to express a relationship between two variables, typically in the form of a two-dimensional graph. Unlike parametric equations, which use a separate parameter to define the curve's points, a Cartesian equation provides a direct relation between the variables. In the context of our problem, we need to transform the parametric equations given by:

• \( x = 1 + t \)
• \( y = t^2 - 1 \)

To convert these into a Cartesian equation, we first solve for the parameter \( t \) in terms of \( x \):

• \( t = x - 1 \)

Substituting this expression into \( y = t^2 - 1 \), we find:

• \( y = (x - 1)^2 - 1 \)

Simplifying, the final Cartesian form of the curve becomes \( y = x^2 - 2x \). This equation represents the same curve as the parametric equations without involving the parameter \( t \).

parabolic curve

A parabolic curve is a U-shaped curve that can open upwards, downwards, or sideways, identified by its characteristic shape of a parabola. The Cartesian equation we derived, \( y = x^2 - 2x \), is a typical example of such a curve. This curve is specifically a standard parabola opening upwards, due to the positive coefficient in front of \( x^2 \).To understand this better, notice the general formula for a parabola, \( y = ax^2 + bx + c \), where \( a, b, \) and \( c \) determine various characteristics of the parabola.

• The coefficient \( a \) decides the direction of the parabola's opening. If \( a > 0 \), it opens upwards. If \( a < 0 \), it opens downwards.
• The vertex of the parabola, which is the peak or the lowest point of the curve, can be found using the formula \( (-b/2a, f(-b/2a)) \).

For our specific case, the equation \( y = x^2 - 2x \) simplifies to \( (a = 1, b = -2) \), indicating an upward-opening curve with its vertex at \( x = 1 \), making the full vertex coordinate \((1, -1)\). The shape and location of this parabolic curve are essential when sketching, especially to determine critical points like the vertex.

curve sketching

Curve sketching is the art of representing mathematical equations in graphical form. Understanding the behavior and properties of curves, like where they intersect axes or reach extrema, helps visualize relationships described by equations. In our exercise, the Cartesian equation \( y = x^2 - 2x \) becomes the foundation for sketching the curve, informed by the original parametric limits: \(-1 \leq t \leq 1 \).Here’s how to approach it:

• Identify key points using the parametric equations: For \( t = -1, 0, 1 \), calculate \( (x, y) \) and derive points \((0, 0), (1, -1), (2, 0)\).
• Knowing this parabola opens upwards, connect these points with a smooth curve to reflect the parabolic shape governed by the formula \( y = x^2 - 2x \).
• Recognize the vertex, \( (1, -1) \), as the lowest point of the curve, guiding the correct representation of the curve’s apex.

By placing these points on a graph and drawing the appropriate parabola shape, you achieve an accurate sketch of the curve. It's important to adhere to the parameter limits, ensuring that you only sketch the segment of the parabola that falls within the bounds \( x = [0, 2] \). This approach ensures clarity and precision in interpreting and presenting mathematical information visually.