Lesson 14 a - parametric equations

Mar 29, 2012Download as PPT, PDF3 likes5,059 views

Parametric equations define a curve where x and y are defined in terms of a third variable called a parameter. The graph of parametric equations consists of all points (x,y) obtained by allowing the parameter to vary over its domain. Eliminating the parameter between the two equations yields a non-parametric equation of the curve. Examples are provided of eliminating parameters between various parametric equations to obtain the curve and sketching the resulting graphs. Exercises are given to further practice eliminating parameters and sketching curves.

DEFINITION: PARAMETRIC EQUATIONS
If there are functions f and g with a common domain
T, the equations x = f(t) and y = g(t), for t in T, are
parametric equations of the curve consisting of all
points ( f(t), g(t) ), for t in T. The variable t is the
parameter.
The equations x = t + 2 and y = 3t – 1
for example are parametric equations and t is the
parameter. The equations define a graph. If t is assigned
a value, corresponding values are determined for x and
y. The pair of values for x and y constitute the
coordinates of a point of the graph. The complete graph
consists of the set of all points determined in this way

as t varies through all its chosen values. We can
eliminate t between the equations and obtain an
equation involving x and y. Thus, solving either equation
for t and substituting in the other, we get
3x – y = 7
The graph of this equation, which also the graph of
the parametric equations, is a straight line.
Example 1: Sketch the graph of the parametric
equations x = 2 + t and y = 3 – t2 .

t -3 -2 -1 0 1 2 3
x -1 0 1 2 3 4 5
y -6 -1 2 3 2 -1 -6

y

t=0
●
t=-1● ● t=1

x
t=-2 ● ● t=2

t=-3 ● ● t=3

Example 2: Eliminate the parameter between x = t + 1
and y = t2 + 3t + 2 and sketch the graph.
Solution:
Solving x = t + 1 for t, we have t = x – 1.
Substitute into y = t2 + 3t + 2, then
y = (x – 1)2 + 3(x – 1) + 2
y = x2 – 2x + 1 + 3x – 3 + 2
y = x2 + x
Reducing to the standard form,
y + ¼ = x2 + x + ¼
y + ¼ = (x + ½)2 , a parabola with V(-½,-¼)
opening upward

Example 3: Eliminate the parameter between x = sin t
and y = cos t and sketch the graph.
Solution:
Squaring both sides of the parametric equations, we
have
x2 = sin2 t and y2 = cos2 t
And adding the two equations will give us
x2 + y2 = sin2 t + cos2 t
But
sin2 t + cos2 t = 1
Therefore
x2 + y2 = 1 , a circle with C(0, 0) and r = 1

Example 4: Find the parametric representation for the
line through (1, 5) and (-2, 3).
Solution:
Letting (1, 5) and (-2, 3) be the first and second points,
respectively, of
x = x1 + r(x2 – x1)
and
y = y1 + r(y2 – y1)
We then have
x = 1 + r(-2 – 1) and y = 5 + r(3
– 5)
x = 1 – 3r y = 5 – 2r

Example 5: Eliminate the parameter between
x = sin t + cos t and y = sin t.
Solution:
Solving sin2 t + cos2 t = 1 for cos t, we have
cos t = 1 − sin2 t
Substitute into x = sin t + cos t , then
x = sin t + 1 − sin2 t
But y = sin t and y2 = sin2 t
Therefore x=y+ 1 − y2
x–y= 1 − y2
Squaring both sides
(x – y)2 = 1 – y2

Exercises:
Eliminate the parameter and sketch the curve.
• x = t2 + 1, y = t + 1
• x = t2 + t – 2 , y = t + 2
• x = cos θ , y = cos2 θ + 8 cos θ
• x = 4 cos θ , y = 7 sin θ
• x = cos θ , y = sin 2θ
• x = 1 + cos 2θ , y = 1 – sin θ

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This document provides objectives for a lesson on rotating axes and identifying conic sections. The lesson aims to teach students how to transform and simplify equations by rotating axes, identify conic sections from their equations, and graph transformed equations. Students will learn to rotate axes to transform equations, simplify the resulting equations, identify the type of conic section from the equation, and graph the transformed equations.

Lesson 11 translation of axesJean Leano

This document discusses translating coordinate axes. It defines translating axes as shifting the old axes by h units horizontally and k units vertically, keeping their directions the same. When translating axes, the new coordinates (x', y') of any point P can be determined by its old coordinates (x, y) using the formulas x = x' + h and y = y' + k, where h and k are the horizontal and vertical shifts. The objectives are to translate axes, find new coordinates after translation, find new equations if the origin is translated, simplify equations by translation, and graph transformed equations.

Lesson 10 conic sections - hyperbolaJean Leano

The document discusses properties of hyperbolas including: - A hyperbola is defined as the set of points where the difference between distances to two fixed points (foci) is constant. - The standard equation of a horizontal hyperbola is (x-h)2/a2 - (y-k)2/b2 = 1 and a vertical hyperbola is (y-k)2/a2 - (x-h)2/b2 = 1. - Examples are given to find the equation of hyperbolas given properties like the center, foci, vertices or asymptotes and to draw the graph of hyperbolas given their equation.

Lesson 9 conic sections - ellipseJean Leano

An ellipse is defined as the set of all points where the sum of the distances from two fixed points (foci) is a constant (the length of the major axis). Key properties include: - The vertices are the endpoints of the major axis. - The distance from the center to each focus is the eccentricity. - The general equation of an ellipse with center at (h,k) is (x-h)^2/a^2 + (y-k)^2/b^2 = 1. - Examples are provided to illustrate finding the equation of an ellipse given properties like the foci, vertices, or axes.

Lesson 6 straight lineJean Leano

This document discusses different types of straight lines and their equations in a plane. It covers lines parallel to coordinate axes, lines through a given point with a given slope, lines with a given slope and y-intercept, and lines passing through two given points. It also discusses finding the equation of the line perpendicular to a given line and passing through a given point, as well as finding the distance from a point to a line. Examples of finding various line equations are provided.

Lesson 5 locus of a pointJean Leano

The document discusses how to find the equation of a locus by considering an arbitrary point (x,y) on the curve and relating x and y based on the description of the curve. It provides examples of locus problems, such as finding the equation for all points equidistant from two given points, where the sum of distances to two given points is a constant, where the sum of squared distances to two given points is a constant, all points on a line with a given slope through a given point, and where the difference of distances to two given points is a constant.

Lesson 4 division of a line segmentJean Leano

The document discusses dividing a line segment into parts at given ratios. It defines the point of division as the point that divides a line segment into a given ratio. It then presents the formula to find the point of division using similar triangles. The document provides examples of finding the point of division for line segments divided into equal parts or divided at specific ratios. It gives examples of finding the midpoint and trisection points of line segments.

Lesson 3 angle between two intersecting linesJean Leano

The document discusses how to find the angle between two lines that intersect. It provides examples of finding the angle between lines defined by two points each and discusses how to determine the angle when additional information is given, such as one line making a 45 degree angle and one having a specific slope. It also gives an example of finding the interior angles of a triangle given the coordinates of its vertices.

Lesson 2 inclination and slope of a lineJean Leano

The document defines inclination and slope of a line. It states that inclination is the smallest positive angle measured counterclockwise from the positive x-axis to the line. Slope is defined as the tangent of the inclination angle. It also discusses properties of parallel and perpendicular lines, including that parallel lines have equal slopes and perpendicular lines have slopes that are negative reciprocals of each other. Several example problems are provided relating to finding slopes and angles of inclination of lines, determining if lines or triangles are perpendicular or right, and identifying if points lie on a straight line.

Lesson 1: distance between two pointsJean Leano

This document provides definitions and concepts related to analytic geometry. It discusses the Cartesian coordinate system, ordered pairs, axes, and coordinates. It defines distance formulas for horizontal, vertical, and slant line segments. Sample problems are provided to calculate distances between points and to determine geometric properties related to triangles and rectangles on a coordinate plane. The objectives are to familiarize students with the coordinate system and to determine distances, slopes, angles of inclination for lines and line segments.

Lesson 14 b - parametric-1Jean Leano

Lesson 12 rotation of axesJean Leano

Lesson 11 translation of axesJean Leano

Lesson 10 conic sections - hyperbolaJean Leano

Lesson 9 conic sections - ellipseJean Leano

Lesson 6 straight lineJean Leano

Lesson 5 locus of a pointJean Leano

Lesson 4 division of a line segmentJean Leano

Lesson 3 angle between two intersecting linesJean Leano

Lesson 2 inclination and slope of a lineJean Leano

Lesson 1: distance between two pointsJean Leano

Lesson 14 a - parametric equations

1. PARAMETRIC EQUATIONS

2. DEFINITION: PARAMETRIC EQUATIONS If there are functions f and g with a common domain T, the equations x = f(t) and y = g(t), for t in T, are parametric equations of the curve consisting of all points ( f(t), g(t) ), for t in T. The variable t is the parameter. The equations x = t + 2 and y = 3t – 1 for example are parametric equations and t is the parameter. The equations define a graph. If t is assigned a value, corresponding values are determined for x and y. The pair of values for x and y constitute the coordinates of a point of the graph. The complete graph consists of the set of all points determined in this way

3. as t varies through all its chosen values. We can eliminate t between the equations and obtain an equation involving x and y. Thus, solving either equation for t and substituting in the other, we get 3x – y = 7 The graph of this equation, which also the graph of the parametric equations, is a straight line. Example 1: Sketch the graph of the parametric equations x = 2 + t and y = 3 – t2 . t -3 -2 -1 0 1 2 3 x -1 0 1 2 3 4 5 y -6 -1 2 3 2 -1 -6

4. y t=0 ● t=-1● ● t=1 x t=-2 ● ● t=2 t=-3 ● ● t=3

5. Example 2: Eliminate the parameter between x = t + 1 and y = t2 + 3t + 2 and sketch the graph. Solution: Solving x = t + 1 for t, we have t = x – 1. Substitute into y = t2 + 3t + 2, then y = (x – 1)2 + 3(x – 1) + 2 y = x2 – 2x + 1 + 3x – 3 + 2 y = x2 + x Reducing to the standard form, y + ¼ = x2 + x + ¼ y + ¼ = (x + ½)2 , a parabola with V(-½,-¼) opening upward

6. y 2 1 0 1 2 x -2 -1 -1 -2

7. Example 3: Eliminate the parameter between x = sin t and y = cos t and sketch the graph. Solution: Squaring both sides of the parametric equations, we have x2 = sin2 t and y2 = cos2 t And adding the two equations will give us x2 + y2 = sin2 t + cos2 t But sin2 t + cos2 t = 1 Therefore x2 + y2 = 1 , a circle with C(0, 0) and r = 1

8. y 2 1 0 1 2 x -2 -1 -1 -2

9. Example 4: Find the parametric representation for the line through (1, 5) and (-2, 3). Solution: Letting (1, 5) and (-2, 3) be the first and second points, respectively, of x = x1 + r(x2 – x1) and y = y1 + r(y2 – y1) We then have x = 1 + r(-2 – 1) and y = 5 + r(3 – 5) x = 1 – 3r y = 5 – 2r

10. Example 5: Eliminate the parameter between x = sin t + cos t and y = sin t. Solution: Solving sin2 t + cos2 t = 1 for cos t, we have cos t = 1 − sin2 t Substitute into x = sin t + cos t , then x = sin t + 1 − sin2 t But y = sin t and y2 = sin2 t Therefore x=y+ 1 − y2 x–y= 1 − y2 Squaring both sides (x – y)2 = 1 – y2

11. Exercises: Eliminate the parameter and sketch the curve. • x = t2 + 1, y = t + 1 • x = t2 + t – 2 , y = t + 2 • x = cos θ , y = cos2 θ + 8 cos θ • x = 4 cos θ , y = 7 sin θ • x = cos θ , y = sin 2θ • x = 1 + cos 2θ , y = 1 – sin θ

Lesson 14 a - parametric equations

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Lesson 14 a - parametric equations