Sketching curves

Understanding Sketching Curves for Polar Equations

• Polar equations describe mathematical curves as points designated by a radius and an angle from the origin.
• Sketching these curves requires a systematic substitution of different angle values and subsequent calculation of their corresponding radii.
• Representative values of θ and the resulting magnitudes of r constitute each plotted point. Forming a line through these points reveals the sketch of the curve.

Useful Curve Sketching Methods

• For symmetrical curves, test for symmetry around the x-axis, y-axis, and the origin. These can reduce the sketching process to half or even a quarter of the theta space.
• Asymptotic behaviour can be detected through setting r to zero or infinity and solving for θ to identify directions of approach.
• Check critical points (maxima, minima, and zeros of the resulting r-value) by evaluating the derivative of the function.
• Plot ease points (simple values of θ like 0, π/2, π, 3π/2) as these can often be calculated mentally, helping you get a sense of the curve.

Equation-Specific Sketching Considerations

• For the polar equation r = a, the sketch will be a circle with radius a, centred at the origin.
• The polar equation r = aθ will sketch a spiral radiating from the pole.
• Polar equations r = a / cos(θ - α) yield a straight line at angle α from the x-axis.

Using Technology

• Software tools like graphing calculators or graphing software can assist in polar curve sketching.
• Always cross-verify hand-drawn sketches with software solutions to check accuracy.

Analysis Post-Sketching

• Once a curve has been sketched, identify and label the key features of the graph such as asymptotes, zeros, poles, maxima, minima, and identify the symmetries.
• Compare these features with the original polar equation, ensuring a comprehensive understanding of the graph.