Lesson 10.5

Graphs of Secant and Cosecant

We don't memorize these graphs. We build them on top of the ones we already know.

Introduction

Secant and Cosecant are the reciprocals of Cosine and Sine.
and .

Just like tangent, when the denominator is zero, we get an asymptote. When the denominator is 1, the reciprocal is also 1.

1

Prerequisite Connection

You can graph sine and cosine waves. You know that and .

2

Today's Increment

We're learning the "Reciprocal Method": graph or first, then build or on top.

3

Why This Matters

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:

  1. Ghost the Guide: Lightly sketch the corresponding Sine or Cosine wave.
  2. Draw Walls: Wherever the guide hits ZERO (the x-axis), draw a vertical asymptote.
  3. 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 .

1

Graph the Guide

First, graph . It has amplitude 2.

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 .

1

Find Sine Zeros

Cosecant explodes when Sine is zero. So solve .

Sine is zero at .

2

Solve for x

Example 3: Graph Construction (Advanced)

Graph one period of .

1

Graph the Modified Cosine Guide

The guide is . Reflect it and shift Up 1.

Range of guide: .

2

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

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