8 — Trigonometric Functions and Graphs

08/10 — Parent Functions and Basic Graph Features

Trigonometric graphs describe repeating behavior. Sine and cosine produce smooth oscillations, while tangent, secant, cosecant, and cotangent include discontinuities or vertical asymptotes.

The six principal Parent function graphs are:

• $y=\sin x$

• $y=\cos x$

• $y=\tan x=\frac{\sin x}{\cos x}$

• $y=\csc x=\frac{1}{\sin x}$

• $y=\sec x=\frac{1}{\cos x}$

• $y=\cot x=\frac{\cos x}{\sin x}$

Sine and cosine are defined for every real input, have range $[-1,1]$, and repeat every $2\pi$. Tangent and cotangent repeat every $\pi$. Secant and cosecant repeat every $2\pi$.

For tangent, the restricted inputs and vertical asymptotes occur at

where $k$ is any integer. For cosecant and cotangent, they occur at

These restrictions arise where a denominator in a quotient or reciprocal definition equals zero. Tangent and cotangent have all real numbers as their range. Secant and cosecant have range

Takeaway: Identify the parent function first, then use its period, domain restrictions, asymptotes, and range as the foundation for graphing.

08/10 — Transforming Sine and Cosine

A transformed sine or cosine function can be written as

or

The parameters have distinct graphical roles:

• $A$ produces a vertical stretch or compression; $|A|$ is the Amplitude. If $A<0$, the graph is reflected across its midline.

• $B$ produces horizontal compression or stretching and controls the Period.

• $h$ gives the horizontal translation, or Phase shift.

• $D$ gives the vertical translation; the Midline is $y=D$.

The key formulas are

The range is

For one cycle of the parent sine function, use the points

For one cycle of the parent cosine function, use the points

To graph a transformed sine or cosine function, locate the phase shift, divide one period into four equal intervals, and assign maximum, minimum, and midline values according to the parent pattern.

For example,

has amplitude $3$, period $8$, phase shift right by $2$, and midline $y=5$. Its maximum is $8$, its minimum is $2$, and its range is $[2,8]$. The graph begins at a maximum when $x=2$ and returns to that maximum after $8$ units.

Takeaway: Read the factored input $B(x-h)$ before interpreting the horizontal transformation.

08/10 — Transforming Tangent

A transformed tangent function has the form

Its period, phase shift, and vertical shift are

Tangent has no amplitude because its range is unbounded. The coefficient $A$ changes the vertical stretch or compression and reflects the graph when $A<0$.

To locate the Vertical asymptote lines, set the tangent input equal to an odd multiple of $\frac{\pi}{2}$:

Solving for $x$ gives

Between two consecutive asymptotes, the curve passes through its center point $(h,D)$. When $A>0$, the branch increases from left to right; when $A<0$, it decreases.

For

the period is $\frac{\pi}{2}$, the phase shift is right by $\frac{\pi}{4}$, the vertical shift is up by $1$, and the center point is $\left(\frac{\pi}{4},1\right)$. The nearest asymptotes are

Because $A=-2$, the branch decreases between these asymptotes.

Takeaway: For tangent, graph the center point and the two surrounding asymptotes instead of looking for a maximum or minimum.

08/10 — Reciprocal Functions and Asymptotes

The Reciprocal trigonometric functions are built from sine, cosine, and tangent:

To construct a reciprocal graph, begin with the related parent graph. Where the original function is zero, the reciprocal is undefined and has a vertical asymptote. Where the original function equals $1$ or $-1$, the reciprocal also has value $1$ or $-1$.

For secant and cosecant,

and

the period is $\frac{2\pi}{|B|}$, the phase shift is $h$, and the vertical shift is $D$. They have no amplitude because they are unbounded. Their range is

For secant, asymptotes occur where

For cosecant, asymptotes occur where

The branches open upward or downward according to the sign of $A$.

For cotangent,

Its period is $\frac{\pi}{|B|}$, its asymptotes occur where

and its range is all real numbers. The graph is typically decreasing when $A>0$ and increasing when $A<0$.

Takeaway: Reciprocal graphs are controlled by the zeros and extreme values of their related sine, cosine, or tangent graphs.

08/10 — Graphing Procedure and Periodic Modeling

Use the following sequence for any transformed trigonometric graph:

1. Identify the parent function. Decide whether the equation is based on sine, cosine, tangent, secant, cosecant, or cotangent.

2. Factor the input. Rewrite the argument so that the form $B(x-h)$ is visible.

3. Find the period. Use $\frac{2\pi}{|B|}$ for sine, cosine, secant, and cosecant; use $\frac{\pi}{|B|}$ for tangent and cotangent.

4. Locate the phase shift. In $B(x-h)$, the graph shifts horizontally by $h$.

5. Apply vertical transformations. Use $A$ for vertical stretch, compression, or reflection, and $D$ for vertical translation.

6. Mark asymptotes. For tangent, secant, cosecant, and cotangent, find the inputs that make the defining denominator or trigonometric factor zero.

7. Plot reference points. Use quarter-period points for sine and cosine, a center point and asymptotes for tangent, and reciprocal points for secant, cosecant, and cotangent.

8. Repeat the pattern. Extend the graph according to its period.

A Sinusoidal model connects these graph features to periodic data. If a quantity has maximum value $M$, minimum value $m$, and period $P$, then

For a temperature varying between $10^\circ\text{C}$ and $26^\circ\text{C}$ over a $24$-hour cycle,

Thus, one possible model is

where $h$ is selected so that the maximum occurs at the appropriate time.

Final takeaway: Trigonometric parameters translate directly into graph features: vertical distance gives amplitude, cycle length gives period, horizontal starting position gives phase shift, and average level gives the midline.