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Graphing Trig Functions

Trigonometric Functions

Overview Sine & Cosine Transformations Graphing Modeling Other Functions Harmonic

Sine & Cosine Transformations

When discussing functions of the form \(y = f(x)\), it is helpful to discuss the transformations that can be applied to the graphs of the basic functions. For example, remember that the graph of \(y = x^2 + 3\) is the graph of \(y = x^2\) that has been shifted up by 3 units, or that the graph of \(y = \sqrt{x + 2}\) is the graph of \(y = \sqrt{x}\) that has been shifted to the left by 2 units, or the graph of \(y = -5\ln{x}\) is the graph of \(y = \ln{x}\) that is reflected over the \(x\)-axis and stretched vertically by a facctor of 5. A quick summary of the relevant transformations is provided below.

Review of Transformations
Form
Effect
\[y = f(x \pm h)\]
The graph of \(y = f(x)\) is shifted horizontally by \(h\) units to the left if \(+ h\) or to the right if \(- h\).
\[y = f(x) \pm k\]
The graph of \(y = f(x)\) is shifted vertically by \(k\) units upward if \(+ k\) or downward if \(- k\).
\[y = a f(x)\]
The graph of \(y = f(x)\) is stretched vertically by a factor of \(a\) if \(|a| \gt 1\) or compressed vertically by a factor of \(a\) if \(|a| \lt 1\). If \(a \lt 0\), then the graph is also reflected across the \(x\)-axis.
\[y = f(bx)\]
The graph of \(y = f(x)\) is compressed horziontally by a factor of \(b\) if \(|b| \gt 1\) or stretched horizontally by a factor of \(b\) if \(|b| \lt 1\). If \(b \lt 0\), then the graph is also reflected across the \(y\)-axis.

The following video will illustrate how these transformations effect the shape of the graph of \(y = \sin{x}\).

Transformations of y = sin(x)
[https://youtu.be/h7xjb0zfczY]

For a function having the form \(y = a \sin{\left(k\left(x - b\right)\right)} + c\) or \(y = a \cos{\left(k\left(x - b\right)\right)} + c\), we can summarize the transformations as follows. Note that we have some new vocabulary too, but the idea is still the same.

  • We call the value of \(|a|\) the amplitude, which decribes the vertical stretch or compression of the graph. For example, the function \(y = -7\cos{x}\) has an amplitude of 7 and is vertically stretched by a factor of 7. It is also reflected across the \(x\)-axis because the \(a\) value is negative.
  • We call the value of \(b\) the phase shift, which describes the horizontal shift of the graph. The direction of the shift is always opposite of the sign. For example, the function \(y = \sin(x - 4)\) has a horizontal shift to the right of 4 units.
  • The value of \(c\) is the vertical shift. For example, the function \(y = \cos(x) + 1\) has been vertically shifted up 1 unit.
  • The value of \(k\) is related to the period of the graph by the equation \(k = \frac{2\pi}{\text{Period}}\), or \(\text{Period} = \frac{2\pi}{k}\). The value of \(k\) is NOT the period. We get this strange relationship because (1) the period length of the standard \(y = \sin{x}\) or \(y = \cos{x}\) graph is \(2\pi\) and (2) the horziontal stretch varies inversely with the coefficient \(k\). For example, the function \(y = \cos\left(\frac{1}{2}x\right)\) should have a horizontal stretch (not compression) by a factor of 2 since \(\frac{1}{2} \lt 1\), and 2 is the reciprocal of \(\frac{1}{2}\). If the standard period is \(2\pi\), then stretching it by a factor of 2, or doubling it, will result in a period of \(4\pi\). We can confirm this using the formula \(\text{Period} = \frac{2\pi}{k} = \frac{2\pi}{1/2} = 2\pi \cdot \frac{2}{1} = 4\pi\).

Self-Check #1: What transformations have been applied to \(y = \cos{x}\) to get \(y = \cos(x-\pi) - 2\)? (Select the most appropriate response.)


You must select an answer above.

(Answer: B) -- The inner expression \(x - \pi\) corresponds to a phase shift to the right of \(\pi\) units, going the opposite direction of the sign, while the outer expression \(- 2\) corresponds to a vertical shift downward of 2 units.

Self-Check #2: Which graph represents the 1-period portion of \(y = 3\sin{\left(x + \frac{\pi}{2}\right)}\)? (Select the most appropriate response.)

You must select an answer above.

(Answer: A) -- We can disregard graphs B and C because they are 1-period graphs of \(\cos{x}\). The amplitude of 3 means the graph has been vertically stretched by a factor of 3, or that it is 3 times taller than the base \(y = \sin{x}\) graph. The \(x + \frac{\pi}{2}\) inner expression means it has also a phase shift to the left by \(\frac{\pi}{2}\). Graph D is close, but it is shifted to the right. Only graph A is shifted to the left by \(\frac{\pi}{2}\) and is 3 times taller than \(y = \sin{x}\).

Self-Check #3: Which graph represents the 1-period portion of \(y = -\frac{1}{2}\cos{x}\)? (Select the most appropriate response.)

You must select an answer above.

(Answer: D) -- We can disregard graph A because it is a 1-period \(\sin{x}\) graph. We can disregard graph B because it is not reflected over the \(x\)-axis. The amplitude of \(\frac{1}{2}\) means the graph has been compressed vertically by a factor of 2, or that it is half as tall as the base \(y = \cos{x}\) graph. Only graph D has been reflected over the \(x\)-axis and is \(\frac{1}{2}\) as tall as \(y = \cos{x}\).

Self-Check #4: Which graph represents the 1-period portion of \(y = \sin(2x)\)? (Select the most appropriate response.)

You must select an answer above.

(Answer: C) -- We can disregard graphs B and D because they are 1-period graphs of \(\cos{x}\). The \(2x\) inner expression means the graph has been compressed horizontally by a factor of 2, or that its 1-period length is half of its original length. Since a 1-period graph of \(\sin{x}\) is \(2\pi\), half of that would be \(\pi\). However, graph A is twice the orignal 1-period length, having a length of \(4\pi\). Only graph C has a 1-period length that is \(\frac{1}{2}\) of the base 1-period length of \(y = \sin{x}\).