Problem 60 - 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. \(x=2 \sin t-3, y=2 \sin t+1\)

Step-by-Step Solution

Answer

The sketch will be a straight line passing through (-3, 1). The domain of the given parametric equations is [-5, -1] and the range is [-1, 3].

Step 1: Understand the given parametric equations

The given parametric equations are \( x = 2 \sin t - 3 \) and \( y = 2 \sin t + 1 \). These equations describe a curve on the xy-plane, which is determined by the parameter 't'. In simpler terms, for each value of 't', we will get a point (x, y) on the curve.

Step 2: Sketch the plane curve

The equations are identical, which implies it's a straight line, except for the different constants subtracted or added. To sketch it, we plot a series of points respecting these parametric equations and then connect the points in the order of the parameter t. We can notice that since the amplitude of sine function is 1, after multiplying it by 2, the amplitudes of x and y both become 2. The curve will be a straight line which passes through (-3, 1)

Step 3: Determine the domain and range

The domain of a function is the set of all possible values which we can use as input, and the range is all possible output values. Since the sine function oscillates between -1 and 1, after multiplying it by 2, the oscillation amplitudes of x and y both become 2. And then subtracting 3 from x and adding 1 to y gives us the overall domain and range of the given pair of parametric equations. The domain (all possible x values) is \([2*(-1) - 3, 2*1 -3] = [-5, -1]\) and the range (all possible y values) is \([2*(-1) + 1, 2*1 +1] = [-1, 3] \).

Key Concepts

Parametric Curve Sketching

Sketching parametric curves involves understanding how each parameter affects the coordinates of points on a plane. In our example, the parametric equations are given as \(x = 2 \sin t - 3 \) and \( y = 2 \sin t + 1\). Each value of \( t \) generates a specific point on the curve described by coordinates \( (x, y) \).

To sketch the curve:

• Recognize that the equations are interconnected by the trigonometric function, sine.
• Notice how both equations have similar terms that revolve around \( 2 \sin t \).
• The expressions \(-3\) and \(+1\) modify the base curve, signifying a shift in the vertical and horizontal directions respectively.
• Plot key points by inserting various \( t \) values, connecting them in sequence.

By observing the behaviour of sine and how it manipulates the values of \( x \) and \( y \), sketching becomes less intimidating, and we see the straight-line relationship clearly, thanks to the identical structures of the equations.

Domain and Range

The domain and range in parametric equations serve to show the extent of the parameter's effect on the coordinates. In our scenario, where \( x = 2 \sin t - 3 \) and \( y = 2 \sin t + 1 \), the variation of \( \sin t \) is central.

Let's explore:

• The sine function varies between \(-1\) and \(1\). Therefore, \( 2 \sin t \) stretches this range to \(-2\) and \(2\).
• For \( x \), subtracting 3 shifts this base range to \([-5, -1]\).
• Similarly, for \( y \), adding 1 shifts it to \([-1, 3]\).

Parametric equations often pivot around the properties of their base functions, such as sine here. Therefore, you're looking at altered ranges caused by transformations to these functions within the equations.
Understanding domain and range helps outline the boundaries of what's achievable by the given parametric setup.

Sine Function Properties

Sine function properties play a pivotal role in defining our parametric equations. Here it is imperative to appreciate how this basic trigonometric function molds the traits of the curve.

Core properties include:

• The sine function is periodic, with a period of \(2\pi\). Its ability to continuously cycle makes it a powerful tool in periodic phenomena.
• With amplitudes swinging between \(-1\) and \(1\), modifications like multiplying by 2 change this swing to between \(-2\) and \(2\).
• It provides the curve with a certain fluidity, yet in our case the curve outcome flattens to a linear relationship due to its incorporation in both \( x \) and \( y \) similarly.

Recognizing these inherent properties of the sine function allows us to infer much about how it manipulates and dictates the characteristics of a given parametric equation. Understanding these properties is especially helpful for sketching and analyzing curves with trigonometric roots.