Do Now - #18 on p.558 Graph the set of points whose polar coordinates satisfy the given equations and inequalities.

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Do Now - #18 on p.558 Graph the set of points whose polar coordinates satisfy the given equations and inequalities.

Relating Polar and Cartesian Coordinates Section 10.5b

Relating Polar and Cartesian Coordinates Coordinate Conversion Equations: Ray Common origin Initial Ray

Relating Polar and Cartesian Coordinates Some curves are easier to work with in polar coordinates, others in Cartesian coordinates… Observe: Polar Equation Cartesian Equivalent

Relating Polar and Cartesian Coordinates Find a polar equation for the circle Support graphically. Expand and simplify: Conversion equations: Algebra: or Check the graph!

Relating Polar and Cartesian Coordinates Find a Cartesian equivalent for the polar equation. Identify the graph. (a) Conversion equations Completing the square The graph of the equivalent Cartesian equation is a circle with radius 2 and center (2, 0).

Relating Polar and Cartesian Coordinates Find a Cartesian equivalent for the polar equation. Identify the graph. (b) Conversion equations The graph of the equivalent Cartesian equation is a line with slope 2 and y-intercept –4.

Exploration 2 The polar curves and , where n is an integer and , are rose curves. 1. Graph for . Describe the curves. Graph window: [–4.7, 4.7] by [–3.1, 3.1] The graphs are rose curves with 4 petals when , 8 petals when , and 12 petals when . 2. What is the shortest length a -interval can have and still produce the graphs in (1)? Shortest interval:

Exploration 2 The polar curves and , where n is an integer and , are rose curves. 3. Based on your observations in (1), describe the graph of when n is a nonzero even integer. The graph is a rose curve with petals. 4. Graph for . Describe the curves. Graph window: [–4.7, 4.7] by [–3.1, 3.1] The graphs are rose curves with 3 petals when , 5 petals when , and 7 petals when .

Exploration 2 The polar curves and , where n is an integer and , are rose curves. 5. What is the shortest length a -interval can have and still produce the graphs in (4)? Shortest interval: 6. Based on your observations in (4), describe the graph of when n is a nonzero odd integer different from . The graph is a rose curve with petals.

Guided Practice Replace the polar equation by an equivalent Cartesian equation. Then identify or describe the graph. A parabola that opens to the right

Guided Practice Replace the polar equation by an equivalent Cartesian equation. Then identify or describe the graph. The union of two lines

Guided Practice Replace the polar equation by an equivalent Cartesian equation. Then identify or describe the graph. A circle with center (0, 4) and radius 4

Guided Practice Replace the Cartesian equation by an equivalent polar equation. Support graphically. How about the graph?

Guided Practice Replace the Cartesian equation by an equivalent polar equation. Support graphically. Graph: How about the graph?