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Magazine Chapter 9: Problem 5
Graph the curve defined by the parametric equations. $$x=t^{2}, y=t^{3}, t \text { in }[-2,2]$$
Short Answer
Expert verified
The curve is a symmetric cubic-like shape starting at (4,-8) and ending at (4,8), passing through (0,0).
Step by step solution
01
Understanding the Parametric Equations
We have two parametric equations: - For the x-coordinate, the equation is \( x = t^2 \), which is a parabola opening to the right.- For the y-coordinate, the equation is \( y = t^3 \), which is a cubic function that passes through the origin \((0, 0)\) and has both negative and positive values for negative and positive \( t \), respectively.
02
Identifying Key Points
Determine the points on the curve for specific t-values within the range \( t = [-2, 2] \):- When \( t = -2 \), \( x = (-2)^2 = 4 \) and \( y = (-2)^3 = -8 \). The point is \( (4, -8) \).- When \( t = -1 \), \( x = (-1)^2 = 1 \) and \( y = (-1)^3 = -1 \). The point is \( (1, -1) \).- When \( t = 0 \), \( x = 0^2 = 0 \) and \( y = 0^3 = 0 \). The point is \( (0, 0) \).- When \( t = 1 \), \( x = 1^2 = 1 \) and \( y = 1^3 = 1 \). The point is \( (1, 1) \).- When \( t = 2 \), \( x = 2^2 = 4 \) and \( y = 2^3 = 8 \). The point is \( (4, 8) \).
03
Plotting the Points
Plot the points calculated in Step 2 on a Cartesian coordinate system:- (4, -8) when \( t = -2 \)- (1, -1) when \( t = -1 \)- (0, 0) when \( t = 0 \)- (1, 1) when \( t = 1 \)- (4, 8) when \( t = 2 \)These points should be plotted in the corresponding coordinates, joined smoothly, as the parameter \( t \) moves from -2 to 2.
04
Analyzing the Graph
The graph starts at \( t = -2 \) at the point \( (4, -8) \), moves through \( (1, -1) \), passes through the origin \( (0, 0) \), moves to \( (1, 1) \), and ends at \( (4, 8) \) when \( t = 2 \). The curve moves symmetrically about the x-axis and is smooth through the plotted points, illustrating the parametric equations' relationship.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
parabola
A parabola is a distinct curve that results from quadratic functions, typically taking the form of \( y = ax^2 + bx + c \). In the case of parametric equations, the parabola can appear in different orientations based on the parameters given. For this exercise, the equation for the x-coordinate is given as \( x = t^2 \). This reveals a horizontally oriented parabola, as the variable \( t \) is squared to produce values for \( x \).
Here are some essential characteristics of parabolas in general:
Here are some essential characteristics of parabolas in general:
- Symmetry: Parabolas are symmetric, meaning one side mirrors the other across the axis of symmetry.
- Vertex: The point where the parabola changes direction is called the vertex.
- Direction: A parabola can open upward, downward, left, or right, determined by the sign of the squared term's coefficient in its equation.
cubic function
Cubic functions are equations of the form \( y = ax^3 + bx^2 + cx + d \), which produce curves that can have up to two hills or valleys (known as inflection points). In parametric form, as seen in the equation \( y = t^3 \), the cubic nature of \( y \) is dependent solely on \( t \).
Some key aspects of cubic functions are:
Some key aspects of cubic functions are:
- Shape: These functions can produce an "S"-shaped curve.
- Intercept: Cubic functions cross the y-axis at the origin if the constant term is zero.
- End behavior: As \( t \) becomes positive or negative, \( y \) will either rise to infinity or fall to negative infinity, reflecting the curve's behavior in opposite directions.
Cartesian coordinate system
The Cartesian coordinate system is a grid framework that helps us visualize mathematical functions and their relationships. It uses two perpendicular axes, typically the x-axis and y-axis, to plot points and create graphs.
Key elements of the Cartesian coordinate system include:
Key elements of the Cartesian coordinate system include:
- Axes: The horizontal axis is known as the x-axis, while the vertical axis is the y-axis. Points are plotted based on their coordinate pair \((x, y)\).
- Quadrants: The plane is divided into four quadrants, defined by the positive and negative values of \( x \) and \( y \).
- Origin: The point \((0, 0)\) where the x-axis and y-axis intersect is the center of the system.
parameter range
The parameter in parametric equations refers to a variable, often denoted by \( t \), which both x and y rely on for their values. By varying this parameter, different points on a curve are determined. The range of the parameter \( t \) specifies how far along the path of the curve we calculate.
Important aspects of parameter range include:
Important aspects of parameter range include:
- Scope: The parameter range limits the section of the curve we observe, defining its beginning and end.
- Continuity: This range ensures a continuous curve as we smoothly transition between points.
- Interpretation: It's vital to analyze how changes in \( t \) affect \( x \) and \( y \) to understand the curve's behavior.
