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Chapter 4: Problem 23

The parametric equations describe the motion of a particle. Find an equation of the curve along which the particle moves. $$\begin{aligned}&x=t+4\\\&y=t^{2}-3\end{aligned}$$

Short Answer

Expert verified
The particle moves along the curve \(y = x^2 - 8x + 13\).

Step by step solution

01

Identify the parametric equations

We are given the parametric equations \(x = t + 4\) and \(y = t^2 - 3\). Our goal is to find a relationship between \(x\) and \(y\) that eliminates \(t\).
02

Solve for t from x

Take the first equation \(x = t + 4\) and solve for \(t\): \[ t = x - 4 \]
03

Substitute t into the equation for y

Substitute the expression for \(t\) from Step 2 into the second parametric equation \(y = t^2 - 3\):\[ y = (x - 4)^2 - 3 \]
04

Simplify the equation

Expand and simplify the expression obtained in Step 3:\[ y = (x - 4)^2 - 3 = x^2 - 8x + 16 - 3 \]\[ y = x^2 - 8x + 13 \]
05

Write the equation of the curve

The equation of the curve in terms of \(x\) and \(y\) is:\[ y = x^2 - 8x + 13 \]

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Key Concepts

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

Equation of Curve
In mathematics, an equation of a curve is crucial for simplifying complex parametric forms into more usable expressions. For a given particle's path described by parametric equations, we aim to find a relationship between coordinates that does not involve the parameter. Here, we derived the curve equation from the parametric forms by eliminating the parameter. This results in a functional relationship between the dependent variable, in this case, \( y \), and the independent variable, \( x \).
  • The initial equation \( x = t + 4 \) informed us how \( t \) can be expressed in terms of \( x \).
  • Replacing \( t \) in the second equation \( y = t^2 - 3 \) allowed us to solve for \( y \) purely in terms of \( x \).
The efficiency of this method allows you to study and utilize the curve without needing to refer to the original parameter, simplifying many mathematical and real-world applications.
Particle Motion
When examining particle motion through parametric equations, each equation typically represents one dimension of movement. In our exercise:- The first equation, \( x = t + 4 \), depicts horizontal motion.- The second equation, \( y = t^2 - 3 \), depicts vertical motion. Utilizing both, we can track the particle's path over time. This approach is incredibly beneficial because it allows:
  • Visualization of complicated trajectories without initial plotting.
  • Detailed insight into spatial dynamics, predicting where the particle will be at any given time.
In essence, parametric equations provide a clear view of an object's position in a plane, factoring in time or another parameter, which is critical for dynamic analysis.
Coordinate Geometry
Coordinate Geometry, or co-geom for short, involves examining curves, lines, and other geometrical figures using algebraic equations in a coordinate plane. The essence of coordinate geometry shines in exercises where you need to transform parametric forms into rectangular equations.Our given task involved such a transformation:
  • From parametric equations, transferring the data into a single \( y = f(x) \) format provides insights into properties like intercepts and symmetries.
  • This conversion helps in graphing the curve quickly on a Cartesian plane, enabling easier analysis.
With both mathematical rigor and geometric intuition, coordinate geometry becomes a potent tool for conceptual understanding and practical application in various mathematical problems.
Quadratic Equation
A Quadratic Equation is a polynomial equation of the form \( ax^2 + bx + c = 0 \). In the context of our problem, converting parametric equations into a quadratic helps link particle motion to a specific trajectory.The resultant equation \( y = x^2 - 8x + 13 \) is quadratic since the highest degree of \( x \) is 2. Properties of quadratic equations, such as:
  • The shape of the graph being a parabola, open upwards in this case due to the positive leading coefficient.
  • The vertex, axis of symmetry, and direction of opening can all be discerned from the standard form.
These properties help study motion dynamics, offering quantitative insight which is richly depicted by simple algebraic manipulation, pivotal for fields like physics and engineering.

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Most popular questions from this chapter

Give an example of: A parameterization of a quarter circle centered at the origin of radius 2 in the first quadrant.

Give an example of: A family of quadratic functions which has zeros at \(x=\) 0 and \(x=b\)

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Evaluate the limit using the fact that $$\lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{n}=e$$ $$\lim _{x \rightarrow 0^{+}}(1+x)^{1 / x}$$

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