Problem 7
Question
A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. $$x=\sqrt{t}, \quad y=1-t$$
Step-by-Step Solution
Verified Answer
(a) Sketch shows a downward-opening parabola starting at (0,1). (b) The rectangular equation is \( y = 1 - x^2 \).
1Step 1: Understanding the Parameters
The given parametric equations are \( x = \sqrt{t} \) and \( y = 1 - t \). Here, \( t \) is the parameter, and we need to understand the range of \( t \). Since \( x = \sqrt{t} \), \( t \) must be non-negative (i.e., \( t \geq 0 \)).
2Step 2: Sketching the Curve
To sketch the curve, we will choose a few sample values of \( t \) to compute \( x \) and \( y \), and plot these points: - If \( t = 0 \), then \( x = \sqrt{0} = 0 \) and \( y = 1 - 0 = 1 \). The point is (0, 1).- If \( t = 1 \), then \( x = \sqrt{1} = 1 \) and \( y = 1 - 1 = 0 \). The point is (1, 0).- If \( t = 4 \), then \( x = \sqrt{4} = 2 \) and \( y = 1 - 4 = -3 \). The point is (2, -3).Plotting these points on the Cartesian plane gives an idea of the curve's shape, which moves from point (0, 1) downward to the right.
3Step 3: Eliminate the Parameter
To eliminate the parameter \( t \), we express \( t \) in terms of \( x \):\[ t = x^2 \] Substitute this expression into the equation for \( y \): \[ y = 1 - x^2 \] Thus, the rectangular (Cartesian) equation of the curve is \( y = 1 - x^2 \).
4Step 4: Confirming the Curve
The Cartesian equation \( y = 1 - x^2 \) represents a downward-opening parabola with its vertex at (0, 1). This confirms what we observed during the sketch, indicating that the parametric plot represents the upper portion of this parabola for \( x \geq 0 \).
Key Concepts
Rectangular-Coordinate EquationEliminating the ParameterCurve Sketching
Rectangular-Coordinate Equation
When dealing with parametric equations, you often want to find a related rectangular-coordinate (or Cartesian) equation. This rectangle coordinate equation allows you to describe a curve without any parameters. In this case, we have the parametric equations:
\[ x = \sqrt{t} \quad \text{and} \quad y = 1 - t \]
To transition from parametric to rectangular-coordinates, we need to eliminate the parameter. Essentially, we're looking to re-express the relationship between \( x \) and \( y \) directly. This step helps by removing the parameter \( t \) and expressing \( y \) solely in terms of \( x \). Thus, the expression \( y = 1 - x^2 \) emerges. This single equation describes the curve's shape in terms of \( x \) and \( y \) only.
\[ x = \sqrt{t} \quad \text{and} \quad y = 1 - t \]
To transition from parametric to rectangular-coordinates, we need to eliminate the parameter. Essentially, we're looking to re-express the relationship between \( x \) and \( y \) directly. This step helps by removing the parameter \( t \) and expressing \( y \) solely in terms of \( x \). Thus, the expression \( y = 1 - x^2 \) emerges. This single equation describes the curve's shape in terms of \( x \) and \( y \) only.
Eliminating the Parameter
Eliminating the parameter is a key technique when working with parametric equations. It involves removing the parameter (often \( t \)) to derive an equation in terms of just \( x \) and \( y \).
This method transforms the original parametric form into a functional form, making the analysis of the curve’s properties and behaviors, like intersections or vertices, much more straightforward.
- First, express the parameter in terms of one of the variables. For example, from the given equation \( x = \sqrt{t} \), squaring both sides gives \( t = x^2 \).
- Substitute this expression into the other parametric equation, \( y = 1 - t \). Replacing \( t \) with \( x^2 \), we arrive at \( y = 1 - x^2 \).
This method transforms the original parametric form into a functional form, making the analysis of the curve’s properties and behaviors, like intersections or vertices, much more straightforward.
Curve Sketching
Curve sketching with parametric equations offers a unique insight into a curve's trajectory. When sketching a curve from parametric equations, plot key points that reveal the path dictated by the parameter.
Connecting these points will show the curve moving in a certain direction. With this particular example, the plotted points hint at a parabola component for the curve.
In general, understand the curve's orientation and movement as the parameter changes, like whether it moves left to right or follows upward/downward motions. It makes analyzing the curve's behavior and its relationship to other coordinate systems easier.
- Pick various values for \( t \), calculate \( x \) and \( y \) using these values, and plot the resulting points.
- For instance, using \( t = 0, 1, \text{ and } 4 \) provides points (0, 1), (1, 0), and (2, -3).
Connecting these points will show the curve moving in a certain direction. With this particular example, the plotted points hint at a parabola component for the curve.
In general, understand the curve's orientation and movement as the parameter changes, like whether it moves left to right or follows upward/downward motions. It makes analyzing the curve's behavior and its relationship to other coordinate systems easier.
Other exercises in this chapter
Problem 6
Plot the point that has the given polar coordinates. $$(3,-2 \pi / 3)$$
View solution Problem 7
Graph the complex number and find its modulus. $$-2$$
View solution Problem 7
Plot the point that has the given polar coordinates. $$(-2,4 \pi / 3)$$
View solution Problem 8
A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve
View solution