Relations The key question: How is a relation different from a function?  A relation is a set of ordered pairs (x, y) of real numbers.  The graph of.

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Presentation transcript:

Relations The key question: How is a relation different from a function?  A relation is a set of ordered pairs (x, y) of real numbers.  The graph of a relation is the set of points in a plane that correspond to the ordered pairs of the relation.  If x and y are functions of a third variable t, called a parameter, then we can use the parametric mode of a grapher to obtain a graph of the relation. A function is ____________ a relation. A relation is ____________ a function. always sometimes

Parametric Curve, Parametric Equations

Exploration: Graphing the Witch of Agnesi The witch of Agnesi is the curve 1. Draw the curve using the window [-5,5] by [-2,4]. What did you choose as a closed parameter interval for your grapher? In what direction is the curve traced? How far to the left and right of the origin do you think the curve extends? We used the parameter interval because our graphing calculator ignored the fact that the curve is not defined when or. The curve traced from right to left across the screen. x ranges from to.

Exploration: Graphing the Witch of Agnesi The witch of Agnesi is the curve 2. Graph the same parametric equations using the parameter intervals,, and. In each case, describe the curve you see and the direction in which it is traced by your grapher. What do you notice in each case? How does the point (0, 2) manifest in each case, and why?

Exploration: Graphing the Witch of Agnesi The witch of Agnesi is the curve 3. What happens if you replace by in the original parametrization? What happens if you use ? If you replace by, the same graph is drawn except it is traced from left to right across the screen. If you replace it by, the same graph is drawn except it is traced from left to right across the screen.

Guided Practice For each of the given parametrizations, (a) Graph the curve. What are the initial and terminal points, if any? Indicate the direction in which the curve is traced. (b) Find a Cartesian equation for a curve that contains the parametrized curve. What portion of the graph of the Cartesian equation is traced by the parametrized curve? (a) Initial point: (–2, 0) Terminal point: (2, 0)

Guided Practice For each of the given parametrizations, (a) Graph the curve. What are the initial and terminal points, if any? Indicate the direction in which the curve is traced. (b) Find a Cartesian equation for a curve that contains the parametrized curve. What portion of the graph of the Cartesian equation is traced by the parametrized curve? (b) The parametrized curve traces the lower half of the circle defined by (or all of the semicircle defined by ).

Guided Practice For each of the given parametrizations, (a) Graph the curve. What are the initial and terminal points, if any? Indicate the direction in which the curve is traced. (b) Find a Cartesian equation for a curve that contains the parametrized curve. What portion of the graph of the Cartesian equation is traced by the parametrized curve? (a) Initial and terminal point: (4, 0)

Guided Practice For each of the given parametrizations, (a) Graph the curve. What are the initial and terminal points, if any? Indicate the direction in which the curve is traced. (b) Find a Cartesian equation for a curve that contains the parametrized curve. What portion of the graph of the Cartesian equation is traced by the parametrized curve? (b) The parametrized curve traces all of the ellipse defined by

Guided Practice For each of the given parametrizations, (a) Graph the curve. What are the initial and terminal points, if any? Indicate the direction in which the curve is traced. (b) Find a Cartesian equation for a curve that contains the parametrized curve. What portion of the graph of the Cartesian equation is traced by the parametrized curve? (a) Initial point: (3, 0) Terminal point: (0, 2)

Guided Practice For each of the given parametrizations, (a) Graph the curve. What are the initial and terminal points, if any? Indicate the direction in which the curve is traced. (b) Find a Cartesian equation for a curve that contains the parametrized curve. What portion of the graph of the Cartesian equation is traced by the parametrized curve? (b) The Cartesian equation is. The portion Substitute: traced by the curve is the segment from (3, 0) to (0, 2).