Problem 59 - Algebra and Trigonometry

Question

Sketch the plane curve represented by the given parametric equations. Then use interval notation to give each relation’s domain and range. \(x=4 \cos t+2, y=4 \cos t-1\)

Step-by-Step Solution

Answer

The graph of the parametric equations forms an ellipse around the points (2, -1). The domain of the function is \(-2 \leq x \leq 6\) and the range is \(-5 \leq y \leq 3\).

Step 1: Sketch the Plane Curve

Observe that from the given equations \(x = 4 \cos t + 2\) and \(y = 4 \cos t - 1\), both x and y are parametrized with the same cosine function, demonstrating that the plane curve is symmetric. To visualize this, plot some values of t, from -2π to 2π, compute the corresponding x and y, and sketch the results on a plane. It should form an ellipse around the points (2, -1).

Step 2: Determine the Domain and Range

The domain is the set of all possible x-values. Because the cosine function ranges from -1 to 1, we can solve for the range of the x-coordinate as follows: Minimize \(x = 4 \cos t + 2\) by substituting -1 for \(\cos t\) to get \(x_{min} = 4*(-1) + 2 = -2\). Maximize x by replacing \(\cos t\) with 1 to result in \(x_{max} = 4*1 + 2 = 6\). Thus the domain is \(-2 \leq x \leq 6\). Similarly, for the range, by replacing -1 and 1 in \(y = 4\cos t - 1\), it gives the minimum and maximum y-values respectively. So, the range is \(-5 \leq y \leq 3\).

Key Concepts

Plane Curve

A plane curve is essentially a curve that lies completely on a flat, two-dimensional plane. When dealing with parametric equations, a plane curve can be visualized by plotting points generated by the equations for various values of the parameter. In the equation given: \(x = 4 \cos t + 2\) and \(y = 4 \cos t - 1\), the curve doesn't involve any z-component, confirming that it is a plane curve.
To sketch this plane curve, you take the parameter \(t\) across its common interval, let's say from \(-2\pi\) to \(2\pi\). For each value of \(t\), calculate \(x\) and \(y\) and plot these points on a graph. This visualization will show a symmetrical shape due to the repetitive nature of the cosine function, often revealing an ellipse.

Domain and Range

The domain and range are crucial concepts in understanding the behavior of functions. The domain refers to all possible values of \(x\), whereas the range is the set of all possible values of \(y\).
In parametric equations, these sets are determined partly by the trigonometric functions involved, in this case, cosine. Since \( \cos t \) varies between \(-1\) and \(1\), you can ascertain the overall possible values for \(x\) and \(y\) by evaluating the parametric equations at these extremes.

• Domain: Substituting \(-1\) and \(1\) into \(x = 4 \cos t + 2\) gives: - \(x_{min} = -2\) and \(x_{max} = 6\). Thus, the domain is \([-2, 6]\).
• Range: Similarly for \(y = 4 \cos t - 1\), we get: - \(y_{min} = -5\) and \(y_{max} = 3\). Therefore, the range is \([-5, 3]\).

Cosine Function

The cosine function is a fundamental part of trigonometry, representing the x-coordinate of a point on the unit circle as that point moves around the circle. Its values fluctuate in a cyclic manner between \(-1\) and \(1\).
In our parametric equations, \( \cos t \) simultaneously influences both \(x\) and \(y\), providing a consistent cyclic influence that contributes to the symmetry of the plane curve. As you examine \(x = 4 \cos t + 2\) and \(y = 4 \cos t - 1\), notice:

• The amplitude of \(4\) stretches the cosine values from \(\pm 1\) to \(\pm 4\).
• The horizontal shifts \(+2\) and \(-1\) then adjust these outputs to create the plane curve we sketch — an ellipse centered at \((2, -1)\).

The harmony in how cosine functions manage repetition is why both \(x\) and \(y\) share similar behavior.

Ellipse

An ellipse is a type of plane curve that looks like a squashed circle. It's the locus of points such that the sum of the distances to two focal points is constant. The influence of the cosine functions in the parametric equations is key here.
In typical parametric ellipse equations, you usually have:

• \( x = a \cos t \)
• \( y = b \sin t \)

But in this case, our provided parametric equations \(x = 4 \cos t + 2\) and \(y = 4 \cos t - 1\) use cosine for both \(x\) and \(y\). This setup uses horizontal and vertical cosine components to maintain symmetry and form an ellipse.
The given curve naturally aligns parallel to the axes, with the center at point \((2, -1)\), as both formulas share the same influencing cosine component. Each value of \(\cos t\) corresponding to some \(t\) has an immediate impact on coordinating \(x\) and \(y\).

Parametrization

Parametrization provides a way to define a mathematical object by expressing its coordinates as functions of one or more parameters. Here, instead of expressing \(y\) directly as a function of \(x\), we use a parameter \(t\) in parametric equations to define both \(x(t)\) and \(y(t)\).
This approach is especially useful for curves that are hard to express with a single equation, like an ellipse, or whose shape or form might change with time or another variable.
With the parametric form \(x = 4 \cos t + 2\) and \(y = 4 \cos t - 1\), both \(x\) and \(y\) are defined by the same trigonometric cycle, which allows us to identify dramatic behavior patterns easily and explore more complex geometrical shapes.

• This makes calculating transformation movements or other dynamic shifts a lot easier.
• Parametric equations often simplify the calculations needed for plotting different types of curves, as shown here.