Helpful Tools for Calculus, Chapter 1: Functions and Relations
Previous Section: 1.13 Implicit Equations
Parametric Equations are used to analyze motion and describe curves that would be too difficult to represent as graphs of functions or as equations. These are mostly helpful in multivariable calculus, particularly in three or more dimensions. A parametric equation can be described as a curve where each coordinate is calculated with an equation, generally with some parameter t.
The result is that there is generally some curve which is represented as c(t). The shape itself is sometimes referred to as C, and these two things are different. C could represent an image of a circle, and C is the set of points defining that circle. c(t) would be the path taken around the circle as a function of t. Thus, depending on the values of t, the circle could be traced out one time, three times, or even five and a half times.
Eliminating the Parameter
The parametric equation f(t) can be represented as f(t)=( x(t),y(t) ) and often it is helpful to attempt rewriting it in terms of x and y values as opposed to t values. This is called parameterizing the curve and ( x(t),y(t) ) is called the parametrized or parametric curve. To eliminate the parameter, we need to solve for y as a function of x, as in a normal function. This can be done in a few steps. First take x(t) and solve it for t. Substitute the equation for t into y(t) and simplify.
Example 1: Eliminate the parameter by writing in terms of x and y: c(t)=(2t-1,3-5t)
Solution: Begin by taking the equation for x(t) and solving for the variable in terms of t.
Substituting into y(t) results in
Example 2: Eliminate the parameter by writing in terms of x and y: c(t)=(t2, t sin(t4+1)
Solution: Applying the same strategy as the previous problem:
It is important to note that parameterizations are not unique. There are an infinite number of ways to parameterize a curve. For example, f(x)=x^3 can be parameterized by each of the following: (t, t3 ), (t2, t6 ), (t3, t9 ), (t4, t12 ).
Graphing a Parametric Equation
Something unique about parametric equations is that the graph has a direction. It isn’t just a line that moves left to right. A parametric curve can move right to left, back and forth, around in a loop and any direction. That is because the parameterized curve, ( x(t),y(t) ), describes the coordinate pairs and therefore the line that the curve appears as while t represents the time at which that point occurs. So, if x(t) is decreasing as t increases, the graph will move from right to left. If x(t) both increases and decreases depending on t then the curve will move back and forth. This adds the effect that a graph can have a feeling of movement, like an animation.
There are a couple of options for graphing a parametric equation. The first option is the usual suspect: make a chart. In this case, the chart will have an extra column because we need to keep track of t as well as each output. It is generally the case that t will function as the independent variable, so filling out that variable and plugging into the others will be the process.
Example 3: Calculate the values at t=0,2,5 for c(t)=( 3t+1, sin(t/2 π) )
Solution:
Example 4: Graph the curve and draw an arrow specifying the direction of the curve, as t increases by plotting some points: x(t)=t/3 , y(t)=t2+1
Solution:
Example 5: Graph the curve and draw an arrow specifying the direction of the curve, as t increases by plotting some points: c(t)=(cos(tπ), π(t-1) )
Solution:
The second method for graphing a parametric equation is to parameterize it and graph the equation representing the results. Then, choose a couple of t-values to determine the direction of the curve and the job is done!
Example 6: Graph c(t)=(t-4,t3-8t) by eliminating the parameter. Be sure to indicate the direction of the curve.
Solution:
Example 7: Graph c(t)=(1+sint,-t) by eliminating the parameter. Be sure to indicate the direction of the curve.
Solution:
This gives us a limited domain for the function, which is x∈[-π/2, π/2], however by expanding the pattern in both directions as defined by x(t) we can see that this graph is just a sideways, upside down sine curve.
Translating a Parametric Equation
Recall that translation is moving a curve horizontally or vertically along the coordinate plane. This is done by adding values to x(t) or y(t) depending on which direction the curve should be moved. Note that this is different than translating a normal curve, where horizontal translations occur by subtracting from the input of the function!
Example 8: Graph the curve representing by translating c(t)=(.5t, t2) to the right 3 units and up 2 units.
Solution: We intend to graph (.5t+3, t2+2). Let’s use the method from the most recent examples:
x=.5t+3
t=2x-6
y=(2x-6)2+2
Circles and More
From the previous chapter, we know that the graph of a circle is given by x2+y2=r2. Recall, in the unit circle, the x-coordinate can be calculated with cosθ while the y-coordinate is calculated with sinθ. For a circle of radius r, the coordinates are given by (rcosθ, rsinθ). This allows us to define c(t)=(a+rcost, b+rsint) with t∈[0, 2π] as the circle centered at coordinates (a, b).
Example 9: Write the equation of a circle centered at coordinates (3, 5) with radius 2 in parametric form.
Solution: By substituting a=3, b=5 and r=2 we get c(t)=(3+2cost, 5+2sint).
Generalizing the equations from above, we can recognize that an ellipse is a circle whose axis are of different lengths. c(t)=(a+r1cost, b+r2sint) with t∈[0, 2π] describes an ellipse centered at (a, b) with a horizontal radius of r1 and a vertical radius of r2.
Example 10: Write the equation of an ellipse centered at coordinates (-2, 1), with a horizontal radius of 2 units and a vertical radius of 1 unit, in parametric form. Then graph it demonstrating the direction of the curve.
Solution: By substitution we get c(t)=(-2+2cost, 1+sint)
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