MATH 1613 - Chapter 6 Notes
Paul Regier
Fall 2020
6.1 Sinusoidal Graphs
- Plot \(f(\theta)=\sin(\theta)\).
- What is the domain and range of sine.
- What is the period of sine?
- Plot \(f(\theta)=\sin(\theta)\).
- What is the domain and range of sine?.
- What is the period of sine?
If \(f(x)\) is a function of \(x\), we say:
- \(f\) is even if \(f(-x)=f(x)\).
- \(f\) is odd if \(f(-x)=-f(x)\).
Is \(f(x)=x\) even or odd?
Is \(f(x)=x^2\) even or odd?
Is \(f(x)=x^3\) even or odd?
Is \(f(x)=\cos(x)\) even or odd?
Is \(f(x)=\sin(x)\) even or odd?
Simplify \(\frac{\sin(-x)}{tan(x)}\).
Sinusoidal graphs (aka sinusoids) can be transformed in four main ways:
- Vertical Stretch
- Horizontal Stretch
- Vertical Shift
- Horizontal Shift
Vertical Stretch
A vertical stretch of a sinuoid changes the amplitude, or distance from the midline to the peak (highest value) or trough (lowest value).
- Sketch the graph a point rotating around a circle of radius 3. Find the equation of the y-coordinate as a function of the angle \(\theta\).
- What is the amplitude of the sinusoid \(f(t) = 0.2 \sin(t)\)?
Horizontal Stretch
A horizontal stretch of a sinuoid changes the period, or distance from one peak to the next peak, or one trough to the next trough.
Sketch the graph of a point on a circle of radius 3 completes one revolution every 2 minutes. Find the equation of the y-coordinate as a function of time \(t\).
What is the period of the sinusoid \(f(t) = \sin(2t)\)?
What is the period of the sinusoid \(f(t) = \sin(4t)\)?
What is the period of the sinusoid \(f(t) = \sin(\frac{1}{2}t)\)?
What is the period of the sinusoid \(f(t) = \sin(\frac{1}{4}t)\)?
What is the period of the sinusoid \(f(t) = \sin(Bt)\)?
\[P = \frac{2\pi}{B}\]
- Graph the function \(f(t) = 2 \cos(\frac{\pi}{2}t)\).
Vertical Shift
A vertical stretch of a sinusoid determines the midline, the line halfway between the value of peaks and troughs.
- Graph the function \(f(t) = \cos(t) + 5\).
- If a sinusoidal function starts on the midline at point (0,3), has an amplitude of 2, and a period of 4, write a formula for the function.
Horizontal Shift
A horizontal shift changes where the sinusoid starts.
- Graph \(f(t) = \sin(x-\frac{\pi}{2})\).
- Graph \(f(t) = \sin(x+\frac{\pi}{2})\).
Note the equation \(f(t) = \sin(x-h)\) is shifted \(h\) to the right, so if \(h\) is negative (e.g. \(h=-2\)), \(f(t) = \sin(x-(-2))=\sin(x+2)\) represents a shift to the left`.
- Sketch the graph of \(f(t) = 3 \sin(\frac{\pi}{4}t-\frac{\pi}{4})\)
Summary
Given the equation in the form
\[f(t) = A \sin( B(t-h) ) + k\]
or \[f(t) = A \cos( B(t-h) ) + k\]
- A is the amplitude (vertical stretch);
- B determines the period, \(P=\frac{2\pi}{B}\);
- k is the vertical shift, which determines the midline of the function; and
- h is the horizontal shift.
Sketch a graph of each of the following functions and determine the amplitude, period, horizontal shift, and midline.
- \(y = 3\sin(8(x+4))+5\)
- \(y = 4\sin( \frac{\pi}{2}(x-3))+7\)
- \(y = 2\sin(3x-21)+4\)
- \(y = \sin(\frac{\pi}{6}x + \pi)-3\)
- Find the formula of the following graph:
A Ferris wheel is 25 meters in diameter and boarded from a platform that is 1 meters above the ground. The six o’clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 10 minutes. The function \(h(t)\) gives your height in meters above the ground \(t\) minutes after the wheel begins to turn.
- Find the amplitude, midline, and period of h(t).
- Find a formula for the height function h(t).
- How high are you off the ground after 5 minutes?
6.2 Graphs of Other Trig Functions
- Sketch a graph of \(f(x)=\tan(x)\).
- What is the period of \(\tan(x)\)?
- What is the domain and range of \(\tan(x)\)?
- Find the equation of the following graph.
- Sketch a graph of \(f(X)=\sec(x)\).
- What is the period of \(\sec(x)\)?
- What is the domain and range of \(\sec(x)\)?
- Recall that \(\sin(x)=\cos(\frac{\pi}{2}-x)\). Use this to show: \[\csc(x)=sec(x-\frac{\pi}{2})\]
- Use this graph \(\csc(x)\).
- What is the domain, range, and period of \(\csc(x)\)?
Sketch a graph of \(f(x)=2\csc(\frac{\pi}{2}x)+1\).
Sketch a graph of \(f(x)=\cot(x)\).
Prove the following identities.
Use these identities to prove \(\tan(\theta)=-\cot(\theta-\frac{\pi}{2})\).
Find the period and horizontal shift of each of the following functions. Then use this information to sketch a graph of the function.
- \(f(x) = 2\tan(4x-31)\)
- \(h(x) = 2\sec(\frac{\pi}{4}(x+1))\)
- \(m(x) = 6\csc(\frac{\pi}{3}x+\pi)\)
If \(\csc(x) = 2\), find \(\csc(-x)\).
If \(\tan(x) = -2\), find \(\cot(-x)\).