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Magazine Chapter 13: Problem 6
(a) Find an equation in \(x\) and \(y\) whose graph contains the points on the curve \(C .\) /b ) Sketch the graph of \(C\) and indicate the orientation. $$ x=t^{3}, \quad y=t^{2} ; \quad t \text { in } \mathbb{R} $$
Short Answer
Expert verified
The equation is \( x = y^{3/2} \) for \( y \geq 0 \), describing a parabola-like curve symmetric about the y-axis.
Step by step solution
01
Understand the Parametric Equations
The given parametric equations are \( x = t^3 \) and \( y = t^2 \). These equations describe how the \( x \) and \( y \) coordinates change as the parameter \( t \) varies in the real numbers \( \mathbb{R} \).
02
Eliminate the Parameter
To find a single equation involving \( x \) and \( y \), express \( t \) in terms of \( y \) from the equation \( y = t^2 \). Solving for \( t \), we get \( t = \pm \sqrt{y} \). Substitute this into the equation for \( x \): \( x = (\pm \sqrt{y})^3 = \pm y^{3/2} \), giving the equation \( x = y^{3/2} \).
03
Find and Simplify to Obtain the Relation
The relationship obtained from eliminating \( t \) is \( x = y^{3/2} \), which describes the curve \( C \) in terms of \( x \) and \( y \). Since \( y = t^2 \) is always non-negative, \( y \geq 0 \). Thus, the equation that describes the curve is \( x = y^{3/2} \) with \( y \geq 0 \).
04
Sketch the Graph of the Curve
To sketch the curve, notice that for positive \( t \), as \( t \) increases, \( x = t^3 \) and \( y = t^2 \) both increase. For negative \( t \), \( x = -(-t)^3 = -t^3 \) while \( y = (-t)^2 \) still increases. The curve is symmetric about the \( y \)-axis. For positive \( t \), \( t \) moves from the origin upward-right, and for negative \( t \), it moves upward-left. The graph of \( C \) resembles a parabola-like shape oriented upwards on both sides of the \( y \)-axis.
05
Indicate the Orientation
The orientation of the curve is determined by the direction of increasing \( t \). As \( t \) goes from negative to positive, \( y \) increases from 0 upwards in both the left and right direction (mirrored over the y-axis). Thus, the orientation on the right side of the y-axis goes from the origin outwards along \( x = y^{3/2} \), and similarly on the left.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Curve Sketching
Curve sketching is all about visually representing the behavior of a parametric equation on a coordinate plane. In our case, the parametric equations are given by \( x = t^3 \) and \( y = t^2 \), where \( t \) varies over real numbers. These equations tell us how the \( x \) and \( y \) coordinates behave as the parameter \( t \) changes.
To represent this behavior as a graph, you plot a series of points for values of \( t \) in a reasonable range. For positive \( t \), both \( x \) and \( y \) increase as \( t \) increases, creating a path on the graph moving upward and to the right. For negative \( t \), \( x \) becomes negative while \( y \) remains positive (since it's squared), which makes the path curve upward and to the left.
The resulting graph looks like two symmetrical halves extending outwards from the \( y \)-axis, moving upwards. To sketch such a curve:
To represent this behavior as a graph, you plot a series of points for values of \( t \) in a reasonable range. For positive \( t \), both \( x \) and \( y \) increase as \( t \) increases, creating a path on the graph moving upward and to the right. For negative \( t \), \( x \) becomes negative while \( y \) remains positive (since it's squared), which makes the path curve upward and to the left.
The resulting graph looks like two symmetrical halves extending outwards from the \( y \)-axis, moving upwards. To sketch such a curve:
- Determine important points, such as when \( t = 0 \), then both \( x \) and \( y \) equal zero.
- Mark the direction of the path using arrows for positive and negative \( t \).
- Consider the overall shape and symmetry to guide your sketch.
Equation Elimination
Equation elimination involves removing the parameter (here \( t \)) from equations to get a direct relationship between \( x \) and \( y \). Starting with the parametric equations \( x = t^3 \) and \( y = t^2 \), we aim to express one in terms of the other.
To eliminate \( t \), solve the second equation for \( t \): \( t = \pm \sqrt{y} \). Substituting into \( x = t^3 \), you get \( x = \pm y^{3/2} \).
This final equation \( x = y^{3/2} \), with \( y \geq 0 \), beautifully condenses both parametric equations into a single expression. It also confirms that the values of \( y \) are non-negative, as \( y \) represents a squared term from the original parametric definition.
To eliminate \( t \), solve the second equation for \( t \): \( t = \pm \sqrt{y} \). Substituting into \( x = t^3 \), you get \( x = \pm y^{3/2} \).
This final equation \( x = y^{3/2} \), with \( y \geq 0 \), beautifully condenses both parametric equations into a single expression. It also confirms that the values of \( y \) are non-negative, as \( y \) represents a squared term from the original parametric definition.
Mathematical Orientation
Mathematical orientation helps in understanding the direction and movement on the graph as \( t \) changes. Our parametric equations describe a curve where the parameter \( t \) determines direction.
When \( t \) starts from a negative value and moves to a positive one, the graph of our curve reflects this change. For negative \( t \), \( x = - t^3 \) and \( y = t^2 \) directs upwards to the left, while for positive values, the graph moves upwards to the right.
This dual movement results in a symmetrical curve across the \( y \)-axis. By properly indicating directions with arrows, one understands how the plot evolves from the center outward.
When \( t \) starts from a negative value and moves to a positive one, the graph of our curve reflects this change. For negative \( t \), \( x = - t^3 \) and \( y = t^2 \) directs upwards to the left, while for positive values, the graph moves upwards to the right.
This dual movement results in a symmetrical curve across the \( y \)-axis. By properly indicating directions with arrows, one understands how the plot evolves from the center outward.
Symmetry in Graphs
Symmetry in graphs refers to how a graph's features mirror around an axis. For our curve, the equations \( x = t^3 \) and \( y = t^2 \) produce a graph symmetric about the \( y \)-axis.
This symmetry happens because substituting \(-t\) into the parametric equations results in \( x = (-t)^3 = -t^3 \) and \( y = (-t)^2 = t^2 \) - one keeps its absolute value while one changes. It shows that for each point \( (x, y) \) on the curve, there exists a corresponding point \( (-x, y) \).
Understanding symmetry makes graph sketching easier as it saves you from plotting the entire graph manually; you only need half and then mirror it!
This symmetry happens because substituting \(-t\) into the parametric equations results in \( x = (-t)^3 = -t^3 \) and \( y = (-t)^2 = t^2 \) - one keeps its absolute value while one changes. It shows that for each point \( (x, y) \) on the curve, there exists a corresponding point \( (-x, y) \).
Understanding symmetry makes graph sketching easier as it saves you from plotting the entire graph manually; you only need half and then mirror it!
