Tuesday, 3-19: Today everyone will have a chance to complete the polar introduction and the assignment is to work in the online text.
Wednesday, 3-20: Today we wrote polar equations in Cartesian form and Cartesian equations in polar form. Things get messy, but remember that when graphing polar equations, standard form means
A) there are no x or y variables and
B) we solve the equation for r unless that's impossible, in which case we write a relationship as simply as possible (minimum number of trig functions, like terms combined, etc.)
Standard form of a Cartesian equation technically depends on what type of equation we have, but in general form we want
A) no r or theta variables,
B) x and y variables in alphabetical order with decreasing powers of the variables. (Ex. Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0).
Thursday, 3-21: Today we answered questions about the assignments and students investigated graphs a properties of 5 special polar equations.
Friday, 3-22: Today we summarized the conclusions from the special polar shapes worksheet Special Polar Graphs:
1. Limaçons: Basic equations: r = a + b×cos (theta) and r = a + b×sin (theta)
The difference between the cosine and the sine curves is that the cosine curves are symmetric wrt the x axis and the sine curves are symmetric wrt the y axis.
When a > b, the graph has no loop because the radius "a" is larger than the amplitude of the trig function that alters it.
When a < b, the graph does have a loop because at some angles, a - b will produce a negative radius, so points will be plotted "behind" the quadrant that the theta values predict.
If b < 0, the graph will be reflected over the x axis for a sine curve or over the y axis for a cosine curve.
When a = 0, the graph is a circle. :-)
If you know the basic shape, exact coordinates can be found by substituting 4 values for theta: {0, pi/2, pi, 3pi/2}
2. Cardioids: Basic equations: r = a + a×cos(theta) and r = a + a×sin(theta) These are supposed to look like hearts...
The difference between the cosine curve and the sine curve is that the cosine curves are symmetric wrt the x axis and the sine curves are symmetric wrt the y axis. Are all these graphs similar??
3. Lemniscates: Basic equations: r^2 = a^2×cos (2theta) and r^2 = a^2×sin(2theta) These are all "infinity shaped" graphs.
The difference between the sine and cosine graphs is that cosine graphs are symmetric wrt the x axis and the sine graphs are symmetric wrt the Cartesian line y = x (which is the polar graph theta = pi/4.)
4. Roses: Basic equations: r = a×cos(b×theta) and r = a×sin(b×theta)
The difference between sine and cosine graphs is cosine curves always have a petal on the positive x axis. Sine curves have a petal where the value of theta is (pi/(2b)).
When b is even there are 2×b petals.
When b is odd there are only b petals because the graph traces over itself.
The variable a affects the length of the petals.
5. Lines: Basic equation for A×x + B×y = C in polar is r = C/(A×cos(theta) + B×sin(theta))
The slope is -A/B. The x-intercept is x = C/A. The y-intercept is y = C/B.
Then we looked at the difference between where polar graphs cross and simultaneous solutions of polar equations. Their assignment is a worksheet that has about 6 intersection problems and many other review problems. Here are answers to that worksheet.
Monday, 3-25: The assignment is to work problems involving the polar form of conic sections, but the particulars of that are not on the quiz. Below is a derivation of the formula for the polar form of a conic if you are interested. The quiz is still tomorrow and the test will be Friday. Quiz topics can include:
Points:
Convert polar to Cartesian
Convert Cartesian to polar
Plot Polar points
"Deal with negatives" both radii and angles
Write other names for polar points
Equations:
Convert polar to Cartesian
Convert Cartesian to polar
Sketch polar equations of limacons, cardioids, lemniskates, roses, circles, lines
Analyze polar equations (including symmetry and writing equations of graphs)
Find simultaneous solutions of polar equations.
Wednesday, 3-20: Today we wrote polar equations in Cartesian form and Cartesian equations in polar form. Things get messy, but remember that when graphing polar equations, standard form means
A) there are no x or y variables and
B) we solve the equation for r unless that's impossible, in which case we write a relationship as simply as possible (minimum number of trig functions, like terms combined, etc.)
Standard form of a Cartesian equation technically depends on what type of equation we have, but in general form we want
A) no r or theta variables,
B) x and y variables in alphabetical order with decreasing powers of the variables. (Ex. Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0).
Thursday, 3-21: Today we answered questions about the assignments and students investigated graphs a properties of 5 special polar equations.
Friday, 3-22: Today we summarized the conclusions from the special polar shapes worksheet Special Polar Graphs:
1. Limaçons: Basic equations: r = a + b×cos (theta) and r = a + b×sin (theta)
The difference between the cosine and the sine curves is that the cosine curves are symmetric wrt the x axis and the sine curves are symmetric wrt the y axis.
When a > b, the graph has no loop because the radius "a" is larger than the amplitude of the trig function that alters it.
When a < b, the graph does have a loop because at some angles, a - b will produce a negative radius, so points will be plotted "behind" the quadrant that the theta values predict.
If b < 0, the graph will be reflected over the x axis for a sine curve or over the y axis for a cosine curve.
When a = 0, the graph is a circle. :-)
If you know the basic shape, exact coordinates can be found by substituting 4 values for theta: {0, pi/2, pi, 3pi/2}
2. Cardioids: Basic equations: r = a + a×cos(theta) and r = a + a×sin(theta) These are supposed to look like hearts...
The difference between the cosine curve and the sine curve is that the cosine curves are symmetric wrt the x axis and the sine curves are symmetric wrt the y axis. Are all these graphs similar??
3. Lemniscates: Basic equations: r^2 = a^2×cos (2theta) and r^2 = a^2×sin(2theta) These are all "infinity shaped" graphs.
The difference between the sine and cosine graphs is that cosine graphs are symmetric wrt the x axis and the sine graphs are symmetric wrt the Cartesian line y = x (which is the polar graph theta = pi/4.)
4. Roses: Basic equations: r = a×cos(b×theta) and r = a×sin(b×theta)
The difference between sine and cosine graphs is cosine curves always have a petal on the positive x axis. Sine curves have a petal where the value of theta is (pi/(2b)).
When b is even there are 2×b petals.
When b is odd there are only b petals because the graph traces over itself.
The variable a affects the length of the petals.
5. Lines: Basic equation for A×x + B×y = C in polar is r = C/(A×cos(theta) + B×sin(theta))
The slope is -A/B. The x-intercept is x = C/A. The y-intercept is y = C/B.
Then we looked at the difference between where polar graphs cross and simultaneous solutions of polar equations. Their assignment is a worksheet that has about 6 intersection problems and many other review problems. Here are answers to that worksheet.
Monday, 3-25: The assignment is to work problems involving the polar form of conic sections, but the particulars of that are not on the quiz. Below is a derivation of the formula for the polar form of a conic if you are interested. The quiz is still tomorrow and the test will be Friday. Quiz topics can include:
Points:
Convert polar to Cartesian
Convert Cartesian to polar
Plot Polar points
"Deal with negatives" both radii and angles
Write other names for polar points
Equations:
Convert polar to Cartesian
Convert Cartesian to polar
Sketch polar equations of limacons, cardioids, lemniskates, roses, circles, lines
Analyze polar equations (including symmetry and writing equations of graphs)
Find simultaneous solutions of polar equations.