how does the value of "a" affect Lemniscate curves?
Does it just affect how far the curve is stretched to both sides?
Also when you graph an archimedes spiral for r = theta
theta is greater than or equal to 0 theta >_ 0:
How would the windows on a graphing calc be set ...... would this be x min , xmax or y min, y max
Thanks!
The value of "a" in the equation of a Lemniscate curve does indeed affect how the curve is stretched to both sides. Specifically, the Lemniscate curve is defined by the polar equation r^2 = a^2 * cos(2θ), where r represents the distance from the origin to a point on the curve, and θ represents the angle formed by the polar coordinate.
When "a" is increased, the Lemniscate curve becomes wider, while decreasing "a" makes the curve narrower. In other words, higher values of "a" stretch the curve horizontally, while lower values of "a" compress it.
Regarding the Archimedes spiral, the equation r = θ describes a spiral that starts at the origin (0, 0) and expands outward as the angle θ increases. When graphing this spiral on a graphing calculator, you would typically set the window parameters to accommodate the range of values for both x and y.
Specifically, for the Archimedes spiral with the equation r = θ, you would set the window in the graphing calculator as follows:
- x min: This should be set to a value that includes the smallest x-coordinate necessary to display the spiral. Since the spiral starts at the origin, a good choice for x min would be 0 or a slightly negative value, unless you specifically want to display a portion of the spiral starting at a different angle θ.
- x max: This should be set to a value that includes the largest x-coordinate necessary to display the spiral. Since the spiral expands outward indefinitely, you may need to set a very large positive value for x max to ensure it is displayed properly.
- y min: Since the spiral starts at the origin and does not dip below the x-axis, you can set y min to 0 or a slightly negative value.
- y max: This should be set to a value that allows the entire spiral to be displayed properly. Since the spiral expands outward indefinitely, you may need to set a very large positive value for y max to ensure all parts of the spiral are visible.
By setting the window parameters correctly, you can visualize the Archimedes spiral on a graphing calculator.