Sine and cosine function graphs — AP Precalculus
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 $2\pi$, 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.
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 = $\frac{2\pi}{|B|}$: 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 $y = D$.
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): $D = \frac{\text{maximum value} + \text{minimum value}}{2}$
- Find $|A|$ (amplitude): $|A| = \text{maximum value} - D = D - \text{minimum value}$
- Find the period: measure the horizontal distance between two consecutive identical points (e.g., two consecutive maxima), then calculate $B = \frac{2\pi}{\text{period}}$
- 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 $C=0$.
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
Common Pitfalls
Why: Students forget to factor out the horizontal scale factor $B$ from the argument, confusing the form $(Bx - C)$ with $B(x - C)$.
Why: Students mix up the inverse relationship between $B$ and period—larger $B$ means shorter period, but the reciprocal flips this relationship.
Why: Students confuse the sign of $A$ (which indicates reflection) with the amplitude, which is a distance and always non-negative.
Why: Maximum and minimum are half a cycle apart, not a full cycle.
Why: Students confuse vertical shift with the maximum value for vertically shifted graphs.
Why: Introductory courses often mix degree and radian graphing, but AP Precalculus assumes radians for all unspecified cases.