Parametric Equations 85 Name Parametric

Parametric Equations 85 Name Parametric

Parametric equations are a general method for describing any curve. There are three variables for each “point”, the x direction, the y direction, and t, the time it takes to get to that point.

1. Given the equations t x = 4 yt =-+ a. Fill in the following table and graph the points. Indicate the direction of the curve/line with respect to time by using arrows. T x y .. Solution:

b. Using the two original equations, eliminate the parameter, t, to obtain an equation for y as a function of x. Does the equation you found match the function you’ve drawn?

Solution: Take any x, y points = (1,3). Find the slope of the point which is (1,3). Using another point (2,0), find the slope: m = (y2 - y1) / (x2 - x1) = (0 - 3) / (2 - 1) = -3 / 1 = -3.

The standard form equation of the line is y = mx + b. Using the point (1,3): 3 = -3(1) + b, which gives b = 6.

Therefore, the equation of the line is y = -3x + 6. This matches the points plotted, confirming that the parametric equations describe this linear relationship.

2. Find the parametric equations x(t) and y(t) for the line passing through (3,6) at t = 0 and (-4,9) at t = 2.

Solution:

- The point (3,6) corresponds to t=0.

- The point (-4,9) corresponds to t=2.

First, find the slope:

m_x = (x2 - x1) / (t2 - t1) = (-4 - 3) / (2 - 0) = -7 / 2 = -3.5

m_y = (y2 - y1) / (t2 - t1) = (9 - 6) / (2 - 0) = 3 / 2 = 1.5

Using the parametric form:

x(t) = x1 + (t)(change in x per unit t) = 3 + (-3.5)t

y(t) = y1 + (t)(change in y per unit t) = 6 + (1.5)t

which simplifies to:

x(t) = 3 - 3.5t

y(t) = 6 + 1.5t

3. Sketch a graph of x_t = sin(t), y_t = cos(2t) and write it as a Cartesian equation.

Solution:

The parametric equations are:

x = sin(t),

y = cos(2t).

Recall that cos(2t) = 1 - 2 sin²(t). Since x = sin(t), substitute:

y = 1 - 2x².

This is an equation of a parabola opening downward in the y-x plane, representing the relationship between x and y through the double-angle identity.

4. Sketch a graph of x_t = sin(2t), y_t = cos(t) and write it as a Cartesian equation.

Solution:

Given:

x = sin(2t) = 2 sin(t) cos(t),

y = cos(t).

Express x in terms of y:

x = 2 y cos(t)

But cos(t) = y, so:

x = 2 y * y = 2 y².

Thus, the Cartesian relationship between x and y is:

x = 2 y²,

which describes a parabola opening sideways.

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Paper For Above instruction

Parametric equations are invaluable tools in describing complex curves that cannot be easily represented through simple Cartesian equations. They give the flexibility to define the x and y coordinates as functions of a third parameter, usually time (t), enabling the precise modeling of dynamic systems, motion paths, and geometric shapes in mathematics and physics. This paper explores various aspects of parametric equations, from their derivation and elimination of parameters to their graphical representations, illustrating their practical applications with detailed examples and mathematical reasoning.

Understanding Parametric Equations and Their Graphs

The core idea of parametric equations involves expressing both x and y coordinates as functions of a common parameter, t. For example, in the linear case, the equations x = 4t and y = -t describe a straight line in the xy-plane. Plotting these points for varying t traces a path that can be a line or a curve, depending on the functions involved. The directionality of the curve is indicated with arrows showing the progression as t increases, providing insight into the motion along the curve.

In the example where t x = 4 and y = -t (possibly a typo in the initial instruction), assuming the intended form is x = 4t and y = -t, the points for t=0, 1, 2, etc., Land on a straight line, with the slope y/x = -1/4, conforming to the parametric equations. Graphing this, along with arrows pointing in the direction of increasing t, visually confirms the linear relationship and directionality.

Elimination of the Parameter to Obtain Cartesian Equations

One essential skill with parametric equations is eliminating the parameter t to find a Cartesian relation between x and y. In the linear example, from x=4t, solving for t gives t = x/4. Substituting into y=-t yields:

y = -x/4,

which matches the straight-line graph originally described. This process underscores the fundamental connection between parametric and Cartesian forms, as the parametric form provides a more flexible description, while the Cartesian form offers a direct relation among variables.

Line Through Two Points

Another typical application involves finding parametric equations of a line passing through two points. Given points (3,6) at t=0 and (-4,9) at t=2, the parametric form:

x(t) = x1 + (change in x) * t,

y(t) = y1 + (change in y) * t,

can be derived by calculating the slope components for x and y separately:

x(t) = 3 + (-7/2) t,

y(t) = 6 + (3/2) t.

This form allows generating the entire line segment with t varying over an interval, directly visualizing the motion from one endpoint to the other.

Graphing Trigonometric Parametric Equations

Parametric equations involving trigonometric functions often model periodic phenomena. For x = sin(t) and y=cos(2t), using identities such as cos(2t) = 1 - 2 sin²(t), we find Cartesian equations like y=1-2x². This relationship describes how the y-coordinate varies quadratically with x, producing a parabola.

Similarly, for x = sin(2t) and y=cos(t), recognizing that sin(2t) = 2 sin(t) cos(t), and substituting y = cos(t), yields the simple relation x = 2 y², describing a parabola opening sideways. These representations are essential in understanding the shapes and motion patterns described by trigonometric parametric equations.

Conclusion

Parametric equations serve as a powerful framework in mathematics, allowing detailed descriptions of curves, motions, and geometric shapes. By manipulating these equations—eliminating parameters, plotting, and transforming them into Cartesian forms—mathematicians and scientists can analyze complex phenomena with clarity and precision. The examples discussed demonstrate the versatility of parametric equations, from linear paths to sinusoidal curves, highlighting their fundamental role in both theoretical mathematics and practical applications across disciplines.

References

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