A family of curves is a set of curves, such as y = ax?, where we consider the appearance of this graph for every possible value of a. Each value of a causes the graph of y = ax? to behave slightly differently. For example, a = 1 gives an upward-facing parabola, a = 2 gives an upward-facing parabola that has been vertically stretched by a factor of 2, a = -1 gives a downward facing parabola and a = 0 gives the y-axis (a rather sad "parabola"). Each of the curves shares the basic formula y = of a produces a different curve. As a collection, they are called a family of curves. a.x², but each value Two families of curves are orthogonal trajectories if EVERY member of the first family is orthogonal to EVERY member of the second family. A simple example is the family of horizontal lines y = c and the family of vertical lines x = and EVERY value of d, the resulting curves are orthogonal. d. For EVERY value of c Note: this question requires content covered in Section 3.5. Complete the square (you may need to research or review this concept!) (a) and then explain in your own words the appearance of each member of the family of curves x2 + y? = = ax. Complete the square and then explain in your own words the appear- (b) ance of each member of the family of curves x2 + y? = by. %3D Show that the families of curves x2 + y2 = ax and x? + y² = by are (c) orthogonal trajectories.
A family of curves is a set of curves, such as y = ax?, where we consider the appearance of this graph for every possible value of a. Each value of a causes the graph of y = ax? to behave slightly differently. For example, a = 1 gives an upward-facing parabola, a = 2 gives an upward-facing parabola that has been vertically stretched by a factor of 2, a = -1 gives a downward facing parabola and a = 0 gives the y-axis (a rather sad "parabola"). Each of the curves shares the basic formula y = of a produces a different curve. As a collection, they are called a family of curves. a.x², but each value Two families of curves are orthogonal trajectories if EVERY member of the first family is orthogonal to EVERY member of the second family. A simple example is the family of horizontal lines y = c and the family of vertical lines x = and EVERY value of d, the resulting curves are orthogonal. d. For EVERY value of c Note: this question requires content covered in Section 3.5. Complete the square (you may need to research or review this concept!) (a) and then explain in your own words the appearance of each member of the family of curves x2 + y? = = ax. Complete the square and then explain in your own words the appear- (b) ance of each member of the family of curves x2 + y? = by. %3D Show that the families of curves x2 + y2 = ax and x? + y² = by are (c) orthogonal trajectories.
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Transcribed Image Text:6.
Two curves are orthogonal if their tangent lines are perpendicular at each
point of intersection.
A family of curves is a set of curves, such as y = ax2, where we consider the appearance
of this graph for every possible value of a. Each value of a causes the graph of y = ax?
to behave slightly differently. For example, a = 1 gives an upward-facing parabola,
a = 2 gives an upward-facing parabola that has been vertically stretched by a factor
of 2, a = -1 gives a downward facing parabola and a = 0 gives the y-axis (a rather
sad "parabola"). Each of the curves shares the basic formula y = ax2, but each value
of a produces a different curve. As a collection, they are called a family of curves.
%3D
Two families of curves are orthogonal trajectories if EVERY member of the first family
is orthogonal to EVERY member of the second family. A simple example is the family
of horizontal lines y = c and the family of vertical lines x = d. For EVERY value of c
and EVERY value of d, the resulting curves are orthogonal.
%3D
Note: this question requires content covered in Section 3.5.
Complete the square (you may need to research or review this concept!)
(a)
and then explain in your own words the appearance of each member of the family
of curves x² + y?
= ax.
Complete the square and then explain in your own words the appear-
(b)
ance of each member of the family of curves x2 + y? = by.
Show that the families of curves x2 + y?
= ax and + y? = by are
(c)
orthogonal trajectories.
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