Section 11 1-notes_2
- 1. Section 11.1 Polar Coordinates and Graphs Objectives: 1. Graph polar functions and equations. 2. Use tests to determine if polar functions and equations exhibit symmetry. Who is this man? He invented the rectangular coordinate system. r E N E DE S C Ar T ES
- 2. Polar Coordinates We know how to locate points on the plane by using a rectangular coordinate system. Graph (–4, 4!3). How many different ways can we name this point? Polar Coordinates There is another way we can locate points on the plane, and that is by using the polar coordinate system. O is a fixed point called the pole. The polar axis is a horizontal ray emanating to the right from the pole. Show Show polar rectangular system grid
- 3. Polar Coordinates The position of a point P can be described by its distance from O, and Polar the measure of the Axis angle formed by øp and the polar axis. Pole (O) Show polar grid Polar Coordinates Plot the points (3, 20°) (3, 380°) What do you notice?
- 4. Polar Coordinates How would you represent (4, –30°)? How would you represent (-2, 50°)? Polar Coordinates How could we find the rectangular coordinates of the point (4, 60°)? Show Rectangular Grid Show Polar Grid Hide Rectangular Grid Hide Polar Grid
- 5. Polar Coordinates Find the rectangular coordinates of the point (3, -135°). Show Rectangular Grid Show Polar Grid Hide Rectangular Grid Hide Polar Grid Polar Coordinates In general, if we have the polar coordinates of a point (r, !), how do we find the rectangular coordinates (x, y)? NQ
- 6. Polar Coordinates Now suppose we have the rectangular coordinates of a point (x, y), and we wish to change to the polar coordinates (r, !). ?? NQ Polar Coordinates Find the polar coordinates of the point (6, -2). Answers Show Rectangular Grid Show Polar Grid Hide Rectangular Grid Hide Polar Grid
- 7. Polar Coordinates Points aren't very interesting by themselves. Descartes invented his coordinate so that he would have an algebraic method to describe curves. We can also describe curves in the polar coordinate system. Write a polar equation for the line x – 2y = 4. Solve your equation for r. Polar Coordinates This is way more complicated than the rectangular equation was! We will, however, see curves where the polar representation is much simpler than the rectangular representation. Answer
- 8. Polar Graphs Sketch the graphs: y=0 r=0 Polar Graphs Sketch the graphs: y=3 r=3
- 9. Polar Graphs Make a conjecture about the graphs of y = constant and r = constant, and the relationship between the graphs. Use the Sketchpad file to help you with your conjectures. Polar Graphs Sketch the graphs: y=x r=!
- 10. Polar Graphs Sketch the graphs: y = -x r = -! Polar Graphs Make a conjecture about the graphs of y = a·x and r = a·!, and the relationship between the graphs. Use the Sketchpad file to help you with your conjectures.
- 11. Polar Graphs Let's explore the graphs of r = a·cos! and r = a·sin!. Polar Graphs These graphs certainly look like circles. Convert the equations r = a·cos! and r = a·sin! into rectangular coordinates.
- 12. Polar Graphs What would these equations look like in standard form? Polar Graphs Let's look at the graphs of these equations.
- 13. Polar Graphs Do you see any symmetry in the graph of r = acos !? Polar Graphs It's easy to see why, since cos! = cos(-!). Symmetry Test #1
- 14. Polar Graphs Do you see any symmetry in the graph of r = asin!? Polar Graphs There are two tests for !=!/2 symmetry: Symmetry Test #2
- 15. Polar Graphs There are two tests for !=!/2 symmetry: Symmetry Test #3 Polar Graphs The other type of symmetry we will be concerned with is polar (origin) symmetry. We'll see examples of graphs that exhibit this symmetry a little later. For now, the test is: Symmetry Test #4
- 16. Polar Graphs Polar Graphs It is also possible to define a function in polar coordinates. Recall our formal definition for a function in rectangular coordinates: Define a function in polar coordinates:
- 17. Polar Graphs Based on that definition, are r = acos! and r = asin! functions on the polar plane? Are the rectangular equations that give the same graphs on the rectangular plane functions? Polar Graphs Now let's look at this family of equations: r = a + bsin!. Are these equations functions? What symmetries do these graphs have?
- 18. Polar Graphs Sketch the graphs: y = 3 + 2sinx r = 3 + 2sin! Polar Graphs Sketch the graphs: y = 2 + 2sinx r = 2 + 2sin!
- 19. Polar Graphs Sketch the graphs: y = 1 + 2sinx r = 1 + 2sin! Polar Graphs So, for the graphs of r = a + bsin!, what happens if: a>b a=b a<b
- 20. Polar Graphs Let's verify our conjectures about the graphs of y = a + bsinx and r = a + bsin!. Polar Graphs Sketch the graph of r = -2 + 3sin!.
- 21. Polar Graphs Compare and contrast the graphs of the following equations: r = 2 + 3sin! r = -2 + 3sin! r = 2 - 3sin! r = -2 - 3sin! Polar Graphs Sketch the graphs: y = 3cos(2x) r = 3cos(2!)
- 22. Polar Graphs Sketch the graphs: y = 3cos(3x) r = 3cos(3!) Polar Graphs Sketch the graphs: y = 3cos(4x) r = 3cos(4!)
- 23. Polar Graphs In the graph of r = a cos(b!): What is the role of a? What is the role of b? Polar Graphs Let's verify our conjectures about the graphs of y = acos(bx) and r = acos(b!).
- 24. Polar Graphs Sketch the graphs: y2 = 4sin(2x) r2 = 4sin(2!) Polar Graphs Sketch the graphs: y2 = 9cos(3x) r2 = 9cos(3!)
- 25. Polar Graphs In the graph of r2 = a cos(b!): What is the role of a? What is the role of b? Polar Graphs Let's verify our conjectures about these graphs. y2 = acos(bx) r2 = acos(b!)