Introduction
We don't memorize these graphs. We build them on top of the ones we already know.
Past Knowledge
You can graph sine and cosine waves. You know that and .
Today's Goal
We're learning the "Reciprocal Method": graph or first, then build or on top.
Future Success
This "guide function" technique is a powerful mental model. In Calculus, you'll analyze complex functions by comparing them to simpler ones (Limit Comparison Tests, Squeeze Theorem).
The "Kissing Parabolas" Method
To graph a reciprocal function, follow these steps:
- Ghost the Guide: Lightly sketch the corresponding Sine or Cosine wave.
- Draw Walls: Wherever the guide hits ZERO (the x-axis), draw a vertical asymptote.
- Kiss the Peaks: At every maximum and minimum of the guide, draw a "parabola" shape going away from the axis. The graphs should touch at the peaks.
Worked Examples
Example 1: Graphing Secant
Sketch .
Graph the Guide
First, graph . It has amplitude 2.
Draw Reciprocal
At the zeros (), draw asymptotes. At the peaks (height 2), draw "U" shapes going up. At the valleys (height -2), draw "U" shapes going down.
Example 2: Cosecant with Shift
Find the vertical asymptotes of .
Find Sine Zeros
Cosecant explodes when Sine is zero. So solve .
Sine is zero at .
Solve for x
Example 3: Graph Construction (Advanced)
Graph one period of .
Graph the Modified Cosine Guide
The guide is . Reflect it and shift Up 1.
Range of guide: .
Construct Secant
Asymptotes are where the original unshifted secant would be undefined? NO. We must look at where the guide crosses its OWN midline ($y=1$), or simply where $\cos(x)=0$.
The guide midline is $y=1$. It crosses at $\pi/2$. Asymptotes are still at $\pi/2$.
Practice Quiz
Practice Quiz
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