# Sine and cosine function graphs

> AP Precalculus · Trigonometric and Polar Functions
> Source: https://www.owlsprep.com/study/ap-precalculus-u3-sine-and-cosine-function-graphs/

This module covers core characteristics of parent sine and cosine graphs, transformations of sinusoidal functions, key graph features, and constructing sinusoidal functions from graphs or parameters for AP Precalculus.

**Prerequisites:** Unit circle definitions of sine and cosine; Basic function transformation rules; Linear equation solving

## Learning objectives

- Identify key features of parent sine and cosine graphs
- Calculate amplitude, period, phase shift and midline for transformed sinusoids
- Construct a sinusoidal function from a graph or given features
- Avoid common exam pitfalls when working with sinusoidal graphs

## Key Features of Parent Sine and Cosine Graphs

The untransformed (parent) sine and cosine functions are $y = \sin x$ and $y = \cos x$, where $x$ is measured in radians, the standard for AP Precalculus unless explicitly stated otherwise. Both functions share core properties: a domain of all real numbers $(-\infty, \infty)$, a range of $[-1, 1]$, and a fundamental period of $2\pi$, meaning one full cycle of oscillation completes over an interval of length $2\pi$.

For the parent sine function $y = \sin x$: the y-intercept is at $(0,0)$, x-intercepts occur at every $x = k\pi$ for any integer $k$, the maximum value of 1 occurs at $x = \frac{\pi}{2} + 2k\pi$, and the minimum value of -1 occurs at $x = \frac{3\pi}{2} + 2k\pi$. It is an odd function, symmetric about the origin. For the parent cosine function $y = \cos x$: the y-intercept is at $(0,1)$, x-intercepts occur at $x = \frac{\pi}{2} + k\pi$ for any integer $k$, the maximum value of 1 occurs at $x = 2k\pi$, and the minimum value of -1 occurs at $x = \pi + 2k\pi$. It is an even function, symmetric about the y-axis.

The concavity of these graphs also follows a fixed pattern. Using the Unit 1 definition, each curve is concave down on the arcs surrounding its maximum points — where the rate of change is decreasing — and concave up on the arcs surrounding its minimum points — where the rate of change is increasing. The concavity switches at each midline crossing (the x-intercepts of the parent graphs), so every full cycle contains one concave-down arc over the crest and one concave-up arc over the trough.

**Worked example:** Identify all maximum points of $y = \cos x$ on the interval $[-3\pi, 3\pi]$.

1. Recall that for parent cosine, maxima occur at $x = 2k\pi$ for all integers $k$, where the function equals its maximum value of 1.
2. Find all integers $k$ such that $-3\pi \leq 2k\pi \leq 3\pi$, which simplifies to:
3. $$-1.5 \leq k \leq 1.5$$
4. The valid integer values of $k$ are $k = -1, 0, 1$.
5. Substitute back to get the maximum points:
6. $$(-2\pi, 1), (0, 1), (2\pi, 1)$$

> **Exam tip:** Always confirm that your solutions lie within the interval specified in the question—AP exam questions regularly test your ability to restrict solutions to a given domain, and full credit is only given for solutions inside the interval.

## Transformations of Sinusoidal Functions

Any translated, stretched, or reflected sine/cosine graph can be written in the standard general form:

$$f(x) = A\sin\left(B(x - C)\right) + D \quad \text{or} \quad f(x) = A\cos\left(B(x - C)\right) + D$$

- $|A|$ = **Amplitude**: the vertical distance from the midline (center line of the graph) to any maximum or minimum. If $A < 0$, the graph is reflected over the midline.
- Period = $\frac{2\pi}{|B|}$: the length of one full cycle of the graph. Larger $|B|$ compresses the graph horizontally, resulting in a shorter period (faster oscillation). The sign of $B$ only reflects the graph horizontally, it does not change the period.
- $C$ = **Phase Shift**: the horizontal shift of the graph. If $C > 0$, the graph shifts $C$ units right; if $C < 0$, it shifts $C$ units left.
- $D$ = **Vertical Shift**: the midline of the graph is the horizontal line $y = D$.

> **warning**
>
> If the function is given in the form $\sin(Bx - C)$ (not factored), you must factor out $B$ to get the correct phase shift: $\sin(Bx - C) = \sin\left(B\left(x - \frac{C}{B}\right)\right)$, so the phase shift is $\frac{C}{B}$, not $C$.

**Worked example:** Given $f(x) = 3\sin\left(4x - \pi\right) - 1$, find the amplitude, period, phase shift, and midline.

1. Factor $B = 4$ out of the argument: $4x - \pi = 4\left(x - \frac{\pi}{4}\right)$, so the function becomes:
2. $$f(x) = 3\sin\left(4\left(x - \frac{\pi}{4}\right)\right) - 1$$
3. Amplitude is $|A| = |3| = 3$.
4. Period is calculated as:
5. $$\frac{2\pi}{|B|} = \frac{2\pi}{4} = \frac{\pi}{2}$$
6. Phase shift is $\frac{\pi}{4}$ units to the right, since $C = \frac{\pi}{4} > 0$.
7. The midline is $y = D = -1$, which is also a vertical shift of 1 unit down from the parent midline.

> **Exam tip:** If you are ever unsure of your phase shift calculation, plug the shifted starting point into the function to check that it matches the expected output for the parent function.

## Constructing a Sinusoidal Function From Features or Graphs

AP Precalculus regularly asks you to write the equation of a sinusoidal function given its graph or key features. Follow this consistent step-by-step method to solve for $A, B, C, D$:

1. Find $D$ (midline/vertical shift): $D = \frac{\text{maximum value} + \text{minimum value}}{2}$
2. Find $|A|$ (amplitude): $|A| = \text{maximum value} - D = D - \text{minimum value}$
3. Find the period: measure the horizontal distance between two consecutive identical points (e.g., two consecutive maxima), then calculate $B = \frac{2\pi}{\text{period}}$
4. Find $C$ (phase shift): choose to use a sine or cosine base to simplify calculation. If a maximum/minimum is at $x=0$, use cosine with $C=0$ to avoid extra calculation. If a midline point with positive slope is at $x=0$, use sine with $C=0$.

**Worked example:** A sinusoidal function has a minimum at $(0, -2)$ and the next maximum at $(2, 4)$. Write a cosine function in standard form that matches this graph.

1. Calculate $D$ (midline):
2. $$D = \frac{-2 + 4}{2} = 1$$
3. Calculate amplitude: the minimum is at $y=-2$, so $|A| = D - (-2) = 3$. Since the minimum (not maximum) is at $x=0$, $A$ is negative: $A = -3$.
4. Calculate period: the horizontal distance from minimum to next maximum is half a period, so half-period = $2 - 0 = 2$, full period = $4$. Then:
5. $$B = \frac{2\pi}{4} = \frac{\pi}{2}$$
6. The minimum of $A\cos(B(x-C)) + D$ is at $x=0 = C$, so $C = 0$.
7. The final function checks out: $f(0) = -2$ (correct minimum), $f(2) = 4$ (correct maximum):
8. $$f(x) = -3\cos\left(\frac{\pi}{2}x\right) + 1$$

> **Exam tip:** Any sinusoidal function can be written as either a shifted sine or a shifted cosine—both are correct as long as they match the given features, but choosing the form that simplifies $C$ to zero reduces your chance of sign errors.

## AP-Style Practice Worked Examples

**Check your understanding**

Test your understanding of key transformation rules:

1. Which of the following gives the amplitude and period of the function $f(x) = -5\sin\left(\frac{\pi}{4}x\right) + 2$?

   - Amplitude 5, Period $\frac{\pi}{4}$
   - Amplitude 5, Period 8
   - Amplitude 5, Period 4
   - Amplitude $-5$, Period 8

   *Answer:* Amplitude 5, Period 8

   *Why:* Amplitude is always the absolute value of $A$, so amplitude = 5. Period = $\frac{2\pi}{|B|} = \frac{2\pi}{\pi/4} = 8$, which matches this option.

**Worked example:** Let $f(x) = A\cos\left(B(x - C)\right) + D$ be a sinusoidal function with maximum 10, minimum 2, and period $6\pi$. (a) Find $A$ (given $A>0$) and midline $D$. (b) If $f(x)$ has a minimum at $x = \frac{3\pi}{2}$, find $C$. (c) Write the final equation.

1. (a) Calculate midline and amplitude:
2. $$D = \frac{10 + 2}{2} = 6, \quad A = 10 - 6 = 4$$
3. (b) Calculate $B = \frac{2\pi}{6\pi} = \frac{1}{3}$. A minimum of cosine occurs when $B(x-C) = \pi$. Substitute values:
4. $$\frac{1}{3}\left(\frac{3\pi}{2} - C\right) = \pi \implies C = -\frac{3\pi}{2}$$
5. (c) Final equation:
6. $$f(x) = 4\cos\left(\frac{1}{3}\left(x + \frac{3\pi}{2}\right)\right) + 6$$

**Worked example:** Average monthly temperature in a city is sinusoidal, with $t=0$ = January (coldest month). Minimum January temp is 40°F, maximum July temp is 84°F. Write $T(t)$ and find April ($t=3$) temp.

1. Use negative cosine with $C=0$ (minimum at $t=0$). Period = 12 months. Calculate midline and amplitude:
2. $$D = \frac{40 + 84}{2} = 62, \quad A = -22, \quad B = \frac{2\pi}{12} = \frac{\pi}{6}$$
3. Substitute $t=3$ to get April temperature:
4. $$T(3) = -22\cos\left(\frac{\pi}{6} \cdot 3\right) + 62 = -22\cos\left(\frac{\pi}{2}\right) + 62 = 62^\circ F$$

## Common pitfalls

- **Wrong:** For $f(x) = \sin(2x - \pi)$, reading the phase shift as $\pi$ units right.
  - Why it fails: Students forget to factor out the horizontal scale factor $B$ from the argument, confusing the form $(Bx - C)$ with $B(x - C)$.
  - Correct: Always factor $B$ out of the argument first: $2x - \pi = 2\left(x - \frac{\pi}{2}\right)$, so the phase shift is $\frac{\pi}{2}$ units right.
- **Wrong:** Calculating the period as $\frac{|B|}{2\pi}$ instead of $\frac{2\pi}{|B|}$.
  - Why it fails: Students mix up the inverse relationship between $B$ and period—larger $B$ means shorter period, but the reciprocal flips this relationship.
  - Correct: After calculating period, verify: if $|B| > 1$, period should be less than $2\pi$; if $|B| < 1$, period should be greater than $2\pi$ to confirm.
- **Wrong:** Claiming amplitude is negative when $A < 0$.
  - Why it fails: Students confuse the sign of $A$ (which indicates reflection) with the amplitude, which is a distance and always non-negative.
  - Correct: Amplitude is always reported as $|A|$; note the reflection separately if the question asks for transformations.
- **Wrong:** Measuring the distance between a maximum and the next minimum as the full period when reading from a graph.
  - Why it fails: Maximum and minimum are half a cycle apart, not a full cycle.
  - Correct: Always measure between two consecutive identical points (maximum to maximum, minimum to minimum) to get the full period.
- **Wrong:** Setting $D$ equal to the maximum value when constructing an equation.
  - Why it fails: Students confuse vertical shift with the maximum value for vertically shifted graphs.
  - Correct: Always calculate $D$ as the average of the maximum and minimum values to get the midline.
- **Wrong:** Using degrees to calculate period when no units are specified.
  - Why it fails: Introductory courses often mix degree and radian graphing, but AP Precalculus assumes radians for all unspecified cases.
  - Correct: Use radians for all period and phase shift calculations unless the question explicitly says to use degrees.

## Cheatsheet

| Category | Formula / Value | Notes |
| --- | --- | --- |
| General standard form (sine) | $f(x) = A\sin\left(B(x - C)\right) + D$ | $B$ must be factored out to read $C$ correctly |
| General standard form (cosine) | $f(x) = A\cos\left(B(x - C)\right) + D$ | Same factoring requirement as sine |
| Amplitude | $\|A\|$ | Always non-negative; negative $A$ = reflection over midline |
| Period (radians, standard) | $\frac{2\pi}{\|B\|}$ | Adjust to $\frac{360^\circ}{\|B\|}$ only if degrees are explicitly specified |
| Phase Shift | $C$ units | Right if $C>0$, left if $C<0$ |
| Midline (vertical shift) | $y = D = \frac{\text{max} + \text{min}}{2}$ | Center line of the sinusoidal graph |
| Parent sine x-intercepts | $x = k\pi, \ k \in \mathbb{Z}$ | For all real inputs |
| Parent cosine x-intercepts | $x = \frac{\pi}{2} + k\pi, \ k \in \mathbb{Z}$ | For all real inputs |
| Parent sine extrema | Max at $x = \frac{\pi}{2} + 2k\pi$, min at $x = \frac{3\pi}{2} + 2k\pi$ | $k \in \mathbb{Z}$ |
| Parent cosine extrema | Max at $x = 2k\pi$, min at $x = \pi + 2k\pi$ | $k \in \mathbb{Z}$ |

## What's next

Mastering sine and cosine function graphs is a non-negotiable prerequisite for the next core topics in AP Precalculus Unit 3. Without being able to quickly identify key features, transform graphs, and write sinusoidal equations from context, you will struggle to score full credit on periodic modeling free-response questions, which make up a large portion of Unit 3 exam points. This topic also lays the foundation for polar graphing, where many common polar curves are defined using sinusoidal functions of the angle, and for inverse trigonometric functions, where understanding domain and range restrictions of sine and cosine relies on familiarity with their full graphs. Continue building your understanding of trigonometric functions with the following topics.

- [Inverse trigonometric functions](https://www.owlsprep.com/study/ap-precalculus-u3-inverse-trigonometric-functions/)
- [Sinusoidal Functions](https://www.owlsprep.com/study/ap-precalculus-u3-sinusoidal-functions/)
- [Sinusoidal Function Transformations](https://www.owlsprep.com/study/ap-precalculus-u3-sinusoidal-function-transformations/)

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