Question

Consider the family of curves \(x=\left(2+\frac{1}{2} \sin a t\right) \cos \left(t+\frac{\sin b t}{c}\right), y=\left(2+\frac{1}{2} \sin a t\right)\) \(\sin \left(t+\frac{\sin b t}{c}\right) .\) Plot the curve for the given values of \(a, b,\) and \(c\) with \(0 \leq t \leq 2 \pi\) $$a=7, b=4, c=1$$

Short answer

Answer: The parameter values \(a\), \(b\), and \(c\) in the given parametric equations determine the shape and size of the curve. Changing these parameters will affect the appearance of the curve in the plane. Specifically, \(a\) affects the frequency of oscillations in the curve, \(b\) influences the deviation from the basic shape, and \(c\) determines the scaling of the deviation.

Step-by-step solution

1. Understand the parametric equations

The parametric equations for this family of curves are given as: $$x=\left(2+\frac{1}{2} \sin a t\right) \cos \left(t+\frac{\sin b t}{c}\right),$$ $$y=\left(2+\frac{1}{2} \sin a t\right) \sin \left(t+\frac{\sin b t}{c}\right).$$ These equations describe the \(x\) and \(y\) coordinates of a point in the plane at a certain value of the parameter \(t\). The parameters \(a, b\) and \(c\) determine the shape and size of the curve. Changing these parameters will change the appearance of the curve in the plane.

2. Set up the parametric equations with given parameter values

Now we need to plug in the given values of \(a=7, b=4, c=1\) into the parametric equations: $$x=\left(2+\frac{1}{2} \sin 7 t\right) \cos \left(t+\frac{\sin 4 t}{1}\right),$$ $$y=\left(2+\frac{1}{2} \sin 7 t\right) \sin \left(t+\frac{\sin 4 t}{1}\right).$$ These are the parametric equations we will use to plot the curve.

3. Plot the curve with given parameter values for \(0 \leq t \leq 2\pi\)

To plot the curve, we will need to calculate the \(x\) and \(y\) coordinates of points on the curve for various values of \(t\) in the range \(0 \leq t \leq 2\pi\). We can use a graphing calculator or a suitable software to plot the curve with these parametric equations: - Plug in the parametric equations with the given values of \(a, b,\) and \(c\). - Set the range for the parameter \(t\) as \(0 \leq t \leq 2\pi\). - Generate the plot. You should obtain a curve illustrating the relationship between the \(x\) and \(y\) coordinates as described by the parametric equations with the given values of parameters \(a, b\) and \(c\).

Key concepts

Curves

Curves in mathematics represent continuous and smooth paths that can be defined in a plane or space. In the context of parametric equations, curves are described by a set of equations that express the coordinates of points on the path as functions of a parameter, typically denoted as \( t \). This parameter allows us to trace the curve by varying its values over a specific interval.

When dealing with parametric curves, you often encounter two functions: \( x(t) \) and \( y(t) \), which represent the \( x \)-coordinate and \( y \)-coordinate of a curve point at parameter \( t \). The parameter \( t \) can be thought of as a time variable that traces the progression of a point along the curve as it changes value.

• Curves can take many shapes, including circles, ellipses, spirals, and more complex forms.
• The family of parametric equations can generate intricate curves that reveal different properties depending on the parameters used.
• By exploring how the parameter affects the equation, students can better understand how curves transition in a pattern and form.

Parametric Plotting

Parametric plotting involves graphing the equations that control the path and shape of the curve, using parametric coordinates like \( x(t) \) and \( y(t) \). This method contrasts with standard plotting, where \( y \) is typically expressed as a function of \( x \). Parametric plots are especially useful when the relation between \( x \) and \( y \) is difficult to express as a single function.

• In parametric plotting, the equations are evaluated for a range of \( t \) values, building the curve one point at a time.
• Software tools or graphing calculators can automate this process, making it easier to visualize complex curves.
• The parameter \( t \) does not need to have any physical meaning itself but serves as a tool to explore the curve.

In the exercise provided, the particular values of \( a = 7, b = 4, c = 1 \) and \( 0 \leq t \leq 2 \pi \) use trigonometric functions to generate a visually appealing curve. Adjusting any parameter slightly can dramatically reshape the plotted path, emphasizing the flexibility of parametric equations.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, facilitates the exploration and representation of geometric figures using a coordinate system. In the context of parametric equations, coordinate geometry is employed to analyze the relationships between the geometrical shape and its algebraic representation.

The essence of coordinate geometry with parametric equations is in transforming algebraic expressions into visual plots that define curves. By understanding these equations, students can deduce crucial properties of the curves, such as symmetry, periodicity, and intersection with axes.

• Coordinate axes allow points defined by equations to be plotted, giving a visual representation of an abstract concept.
• One can discern the significance of parameters and constants in shaping the geometric properties of the curve.
• This branch of mathematics also allows for calculations such as tangents, normals, and arc lengths based on the parametric form.

Utilizing coordinate geometry with parametric plotting helps students link algebraic skills with visual intuition, enhancing their understanding of the relationship between mathematics and real-world shapes and motions.