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Chapter 10: Problem 6

Eliminate the parameter and then sketch the curve. $$x=t^{2}+3 t, \quad y=t-2$$

Short Answer

Expert verified
The curve is a sideways parabola: \(x = y^2 + 7y + 10\).

Step by step solution

01

Express t in Terms of y

The given equation for \(y\) is \(y = t - 2\). Solve this expression for \(t\) in terms of \(y\): \[ t = y + 2 \]
02

Substitute t in the x Equation

Substitute \(t = y + 2\) into the equation for \(x\):\[ x = (y + 2)^2 + 3(y + 2) \]
03

Expand and Simplify

Expand the expression and simplify it:\[ (y + 2)^2 = y^2 + 4y + 4 \]Now substitute this:\[ x = y^2 + 4y + 4 + 3y + 6 \]Simplify further:\[ x = y^2 + 7y + 10 \]
04

Sketch the Curve

The expression \(x = y^2 + 7y + 10\) represents a quadratic equation in \(y\) with respect to \(x\). This is a parabola opening horizontally. To sketch it, recognize its vertex form by completing the square, or plot points for several \(y\) values to find corresponding \(x\) values.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Curve Sketching
Curve sketching involves drawing a graph representing a function or equation. To sketch a curve accurately, it's essential to understand the equation type you're working with. In our exercise, after eliminating the parameter, we end up with a quadratic equation in terms of \(y\).
A quadratic equation of the form \(x = ay^2 + by + c\) represents a parabola. Depending on how the terms are arranged, the parabola may open upwards, downwards, or sideways. In this case, since \(x\) is expressed as a function of \(y\), the parabola opens horizontally.
To sketch the curve:
  • Identify the vertex: This is the point where the parabola changes direction. You can use completing the square or derivative methods to find it.
  • Determine intercepts: These are the points where the graph crosses the axes.
  • Plot additional points: Choose values for \(y\) and calculate corresponding \(x\) using the equation.
By following these steps, you'll be able to draw an accurate representation of the equation's graph.
Quadratic Equations
Quadratic equations are polynomial equations of the second degree. They typically take the form \(ax^2 + bx + c = 0\) when \(x\) is the variable, although in our case, \(x\) and \(y\) switch roles.
Key characteristics of quadratic equations include:
  • A parabolic graph: Quadratic equations always create a parabolic shape.
  • Two possible solutions: These are called roots and can be found using the quadratic formula.
  • A vertex, which represents the maximum or minimum point of the parabola.
In our exercise, the quadratic equation is \(x = y^2 + 7y + 10\). This means \(y\) determines the shape and positioning of the parabola. It's crucial to understand these principles to analyze and draw the correct graph.
Elimination of Parameter
The elimination of parameter is a process used to convert parametric equations into a single equation. This technique helps in sketching curves or simplifying the study of a motion path described by parameters.
For this technique, you must:
  • Solve the parameter equation: Express one variable in terms of the parameter.
  • Substitute: Replace the parameter in the other equation to express one variable solely in terms of the other.
By following these steps in our exercise, we eliminated the parameter \(t\) and rewrote the equation as \(x = y^2 + 7y + 10\). This equation now directly relates \(x\) and \(y\), making it much simpler to sketch or analyze. Elimination of parameter is crucial in transforming complex parametric forms into standard forms that are easier to handle and visualize.

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Most popular questions from this chapter

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