Sine and cosine function graphs

AP Precalculus· AP Precalculus CED — Trigonometric and Polar Functions· 14 min read

1. Key Features of Parent Sine and Cosine Graphs★★☆☆☆⏱ 4 min

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 $2 π$ , meaning one full cycle of oscillation completes over an interval of length $2 π$ .

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

📐 Worked Example

Identify all maximum points of $y = cos x$ on the interval $[- 3 π, 3 π]$ .

1
Recall that for parent cosine, maxima occur at $x = 2 k π$ for all integers $k$ , where the function equals its maximum value of 1.
2
Find all integers $k$ such that $- 3 π \leq 2 k π \leq 3 π$ , 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.

2. Transformations of Sinusoidal Functions★★★☆☆⏱ 4 min

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 = $\frac{2 π}{∣ B ∣}$ : 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 $y = D$ .

📐 Worked Example

Given $f (x) = 3 sin (4 x - π) - 1$ , find the amplitude, period, phase shift, and midline.

1
Factor $B = 4$ out of the argument: $4 x - π = 4 (x - \frac{π}{4})$ , 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 $\frac{π}{4}$ units to the right, since $C = \frac{π}{4} > 0$ .
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.

3. Constructing a Sinusoidal Function From Features or Graphs★★★★☆⏱ 4 min

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 $A, B, C, D$ :

Find $D$ (midline/vertical shift): $D = \frac{maximum value + minimum value}{2}$
Find $∣ A ∣$ (amplitude): $∣ A ∣ = maximum value - D = D - minimum value$
Find the period: measure the horizontal distance between two consecutive identical points (e.g., two consecutive maxima), then calculate $B = \frac{2 π}{period}$
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.

4. AP-Style Practice Worked Examples★★★☆☆⏱ 2 min

✓ Quick check

Test your understanding of key transformation rules:

Which of the following gives the amplitude and period of the function $f (x) = - 5 sin (\frac{π}{4} x) + 2$ ?
- Amplitude 5, Period $\frac{π}{4}$
- Amplitude 5, Period 8
- Amplitude 5, Period 4
- Amplitude $- 5$ , Period 8
Reveal answer
1 —
Amplitude is always the absolute value of $A$ , so amplitude = 5. Period = $\frac{2 π}{∣ B ∣} = \frac{2 π}{π /4} = 8$ , which matches this option.

📐 Worked Example

Let $f (x) = A cos (B (x - C)) + D$ be a sinusoidal function with maximum 10, minimum 2, and period $6 π$ . (a) Find $A$ (given $A > 0$ ) and midline $D$ . (b) If $f (x)$ has a minimum at $x = \frac{3 π}{2}$ , 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 $B = \frac{2 π}{6 π} = \frac{1}{3}$ . A minimum of cosine occurs when $B (x - C) = π$ . 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
$T(3) = -22\cos\left(\frac{\pi}{6} \cdot 3\right) + 62 = -22\cos\left(\frac{\pi}{2}\right) + 62 = 62^\circ F$

5. Common Pitfalls

Wrong move:

For $f (x) = sin (2 x - π)$ , reading the phase shift as $π$ units right.

Why:

Students forget to factor out the horizontal scale factor $B$ from the argument, confusing the form $(B x - C)$ with $B (x - C)$ .

Correct move:

Always factor $B$ out of the argument first: $2 x - π = 2 (x - \frac{π}{2})$ , so the phase shift is $\frac{π}{2}$ units right.

Wrong move:

Calculating the period as $\frac{∣ B ∣}{2 π}$ instead of $\frac{2 π}{∣ B ∣}$ .

Why:

Students mix up the inverse relationship between $B$ and period—larger $B$ means shorter period, but the reciprocal flips this relationship.

Correct move:

After calculating period, verify: if $∣ B ∣ > 1$ , period should be less than $2 π$ ; if $∣ B ∣ < 1$ , period should be greater than $2 π$ to confirm.

Wrong move:

Claiming amplitude is negative when $A < 0$ .

Why:

Students confuse the sign of $A$ (which indicates reflection) with the amplitude, which is a distance and always non-negative.

Correct move:

Amplitude is always reported as $∣ A ∣$ ; note the reflection separately if the question asks for transformations.

Wrong move:

Measuring the distance between a maximum and the next minimum as the full period when reading from a graph.

Why:

Maximum and minimum are half a cycle apart, not a full cycle.

Correct move:

Always measure between two consecutive identical points (maximum to maximum, minimum to minimum) to get the full period.

Wrong move:

Setting $D$ equal to the maximum value when constructing an equation.

Why:

Students confuse vertical shift with the maximum value for vertically shifted graphs.

Correct move:

Always calculate $D$ as the average of the maximum and minimum values to get the midline.

Wrong move:

Using degrees to calculate period when no units are specified.

Why:

Introductory courses often mix degree and radian graphing, but AP Precalculus assumes radians for all unspecified cases.

Correct move:

Use radians for all period and phase shift calculations unless the question explicitly says to use degrees.

6. Quick Reference Cheatsheet

Category	Formula / Value	Notes
General standard form (sine)	$f (x) = A sin (B (x - C)) + D$	$B$ must be factored out to read $C$ correctly
General standard form (cosine)	$f (x) = A cos (B (x - C)) + D$	Same factoring requirement as sine
Amplitude	$∣ A ∣$	Always non-negative; negative $A$ = reflection over midline
Period (radians, standard)	$\frac{2 π}{∣ B ∣}$	Adjust to $\frac{36 0 ^{\circ}}{∣ B ∣}$ only if degrees are explicitly specified
Phase Shift	$C$ units	Right if $C > 0$ , left if $C < 0$
Midline (vertical shift)	$y = D = \frac{max + min}{2}$	Center line of the sinusoidal graph
Parent sine x-intercepts	$x = k π, k \in Z$	For all real inputs
Parent cosine x-intercepts	$x = \frac{π}{2} + k π, k \in Z$	For all real inputs
Parent sine extrema	Max at $x = \frac{π}{2} + 2 k π$ , min at $x = \frac{3 π}{2} + 2 k π$	$k \in Z$
Parent cosine extrema	Max at $x = 2 k π$ , min at $x = π + 2 k π$	$k \in Z$

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2024 · AP Precalculus
Construct sinusoid from key features
2023 · AP Precalculus
Identify amplitude/period of transformed function

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.