Sine and Cosine Function Graphs - AP PreCalculus Study Guide

The Core Idea: Sine and Cosine Function Graphs

The graphs of sine and cosine functions, known as sinusoids or sinusoidal waves, are fundamental in modeling periodic phenomena that exhibit a smooth, repetitive oscillation. These functions are characterized by their wavelike shape that repeats at regular intervals. The core idea of this topic is to understand how the parameters in the general sinusoidal equations, $y=a\sin (b(x-c))+d$ and $y=a\cos (b(x-c))+d$, transform the basic graphs of $y=\sin (x)$ and $y=\cos (x)$.

By analyzing these parameters, we can precisely describe any sinusoidal wave's key features: its height (amplitude), its vertical center (midline), the length of one full cycle (period), and its horizontal starting position (phase shift). This framework allows us to both sketch an accurate graph from a given equation and, conversely, to derive a specific equation that models a given sinusoidal graph. The relationship between sine and cosine is simply a horizontal shift, meaning any sinusoidal graph can be described by either function with the appropriate transformation.

Key Formulas & Parameters

The general form for a sinusoidal function is given by $f(x)=a\sin (b(x-c))+d$ or $f(x)=a\cos (b(x-c))+d$. Each parameter—$a$, $b$, $c$, and $d$—controls a specific graphical transformation.

1. Midline: $y=d$

• The midline is the horizontal line that passes exactly halfway between the function's maximum and minimum values.

• The parameter $d$ represents a vertical shift. It moves the entire graph up (if $d>0$) or down (if $d<0$) from its default midline of $y=0$.

• Formula: $$d=\frac{\text{Maximum Value}+\text{Minimum Value}}{2}$$

2. Amplitude: $|a|$

• The amplitude is the distance from the midline to either the maximum or minimum value of the function. It represents the "height" of the waves.

• The parameter $a$ controls the vertical stretch or compression. The absolute value, $|a|$, is the amplitude.

• If $a<0$, the function is reflected across its midline. For example, while $y=\cos (x)$ starts at a maximum, $y=-\cos (x)$ starts at a minimum.

• Formula: $$|a|=\frac{\text{Maximum Value}-\text{Minimum Value}}{2}$$

3. Period: $\frac{2\pi }{|b|}$

• The period is the length of the smallest horizontal interval over which the function completes one full cycle. The parent functions $y=\sin (x)$ and $y=\cos (x)$ have a period of $2\pi$.

• The parameter $b$ controls the horizontal stretch or compression. It is related to the period by the formula below. A larger $|b|$ value results in a shorter period (more cycles in a given interval).

• Formula: $$\text{Period}=\frac{2\pi }{|b|}$$

• Conversely, to find $b$ from a given period: $$|b|=\frac{2\pi }{\text{Period}}$$

4. Frequency: $\frac{|b|}{2\pi }$

• Frequency is the reciprocal of the period. It represents the number of cycles the function completes over a horizontal interval of length $2\pi$.

• Formula: $$\text{Frequency}=\frac{1}{\text{Period}}=\frac{|b|}{2\pi }$$

5. Phase Shift: $c$

• The phase shift is the horizontal translation of the sinusoidal function from its parent function's standard starting position.

• The parameter $c$ in the form $f(x)=a\sin (b(x-c))+d$ represents the horizontal shift.

• If $c>0$, the graph shifts to the right by $c$ units.

• If $c<0$, the graph shifts to the left by $|c|$ units. This is often seen as $(x+c)=(x-(-c))$.

• Crucial Note: To identify the phase shift correctly, the function must be in the factored form $b(x-c)$. If given $a\sin (bx-k)+d$, you must first factor out $b$ to get $a\sin \left(b\left(x-\frac{k}{b}\right)\right)+d$, making the phase shift $c=\frac{k}{b}$.

Understanding The Sine vs. Cosine Relationship

A critical concept is that the sine and cosine functions are fundamentally the same wave, just shifted horizontally relative to each other. The graph of $y=\sin (x)$ is identical to the graph of $y=\cos (x)$ shifted to the right by $\frac{\pi }{2}$ units.

This relationship is captured by the identity:

$$\sin (x)=\cos \left(x-\frac{\pi }{2}\right)$$

Implications for Modeling:

• Flexibility: Any sinusoidal graph can be modeled using either a sine function or a cosine function. The choice of function will simply change the required phase shift ($c$).

• Identifying the Function: The primary difference in their "starting" behavior at $x=0$ (or at the beginning of a phase-shifted cycle) helps determine which function to use for a simpler equation.

  • Cosine: The basic cosine function, $y=\cos (x)$, starts a cycle at a maximum value on the y-axis. Therefore, if a sinusoidal graph begins its cycle (after its phase shift) at a maximum or minimum, a cosine function often provides the most straightforward model.

  • Sine: The basic sine function, $y=\sin (x)$, starts a cycle at its midline value and then increases. If a sinusoidal graph begins its cycle (after its phase shift) at the midline, a sine function is typically the easier choice.

For example, a graph that has a maximum at $x=\frac{\pi }{4}$ could be modeled as a cosine function with a phase shift of $c=\frac{\pi }{4}$. Alternatively, it could be modeled as a sine function, but its phase shift would correspond to the x-value where the graph crosses the midline while increasing.

Core Concepts & Rules

• Periodicity: The graphs of $y=\sin (x)$ and $y=\cos (x)$ are periodic, repeating every $2\pi$ radians. All transformations of these functions remain periodic.

• Amplitude ($|a|$): This parameter determines the vertical distance from the midline to the maximum or minimum value. It is always a non-negative value. A negative $a$ value reflects the graph over the midline.

• Period ($\frac{2\pi }{|b|}$): This is the horizontal length of one complete cycle. It is inversely related to the parameter $b$.

• Frequency ($\frac{|b|}{2\pi }$): This is the number of cycles per unit interval, and it is the reciprocal of the period.

• Midline ($y=d$): This is the horizontal line that serves as the vertical center of the graph. The parameter $d$ shifts the graph vertically.

• Phase Shift ($c$): This is the horizontal shift of the graph. It is determined by the value of $c$ in the factored form $b(x-c)$.

• Sine vs. Cosine Starting Points: The parent sine function starts at $(0,0)$ (midline) and increases. The parent cosine function starts at $(0,1)$ (maximum). This difference is key to choosing a model and determining the phase shift.

• Equivalence: Any sinusoid can be expressed as either a sine or a cosine function with an appropriate phase shift, because $\sin (x)=\cos (x-\frac{\pi }{2})$.

Step-by-Step Example 1: Graphing from an Equation

Problem: Identify the amplitude, period, midline, and phase shift of the function $f(x)=-2\cos \left(\frac{\pi }{2}x-\pi \right)+3$. Then, sketch one full cycle of the graph.

Step 1: Factor the argument to identify the parameters.

The function is given as $f(x)=-2\cos \left(\frac{\pi }{2}x-\pi \right)+3$. To correctly identify the phase shift, we must factor out the $b$ value, which is $\frac{\pi }{2}$, from the argument.

$$\frac{\pi }{2}x-\pi =\frac{\pi }{2}(x-2)$$

So, the function in standard form is $f(x)=-2\cos \left(\frac{\pi }{2}(x-2)\right)+3$.

Step 2: Identify the parameters $a,b,c,d$.

From the standard form, we can identify:

• $a=-2$

• $b=\frac{\pi }{2}$

• $c=2$

• $d=3$

Step 3: Determine the key characteristics.

• Midline:$y=d⟹y=3$

• Amplitude:$|a|=|-2|=2$. The negative sign on $a$ indicates a reflection over the midline.

• Period:$\frac{2\pi }{|b|}=\frac{2\pi }{|\pi /2|}=2\pi \cdot \frac{2}{\pi }=4$. One full cycle has a length of 4 units.

• Phase Shift:$c=2$. The graph is shifted 2 units to the right.

Step 4: Sketch one cycle of the graph.

1. Draw the frame: Draw the midline at $y=3$. Since the amplitude is 2, the maximum value will be $3+2=5$ and the minimum value will be $3-2=1$. Draw horizontal lines at $y=5$ and $y=1$ to bound the graph.

2. Determine the start and end of the cycle: The phase shift is 2, so our cycle starts at $x=2$. The period is 4, so the cycle ends at $x=2+4=6$.

3. Plot the key points: A cosine cycle has five key points: start, quarter-point, mid-point, three-quarter-point, and end.

   • Start ($x=2$): This is a reflected cosine function ($a<0$), so it starts at its minimum on the midline. The point is $(2,1)$.

   • Quarter-point ($x=3$): The graph will cross the midline. The point is $(3,3)$.

   • Mid-point ($x=4$): The graph will reach its maximum. The point is $(4,5)$.

   • Three-quarter-point ($x=5$): The graph will cross the midline again. The point is $(5,3)$.

   • End ($x=6$): The graph will return to its minimum. The point is $(6,1)$.

4. Draw the curve: Connect the five points with a smooth, sinusoidal curve.

Step-by-Step Example 2: Finding an Equation from a Graph

Problem: A sinusoidal function $g(x)$ is graphed below. It has a maximum at $(1,5)$ and a subsequent minimum at $(3,-1)$. Determine a possible equation for $g(x)$ using both a cosine and a sine function.

Step 1: Find the midline ($d$).

The midline is halfway between the maximum and minimum values.

$$d=\frac{\text{Max}+\text{Min}}{2}=\frac{5+(-1)}{2}=\frac{4}{2}=2$$

The midline is $y=2$.

Step 2: Find the amplitude ($|a|$).

The amplitude is the distance from the midline to the maximum (or minimum).

$$|a|=\text{Max}-\text{Midline}=5-2=3$$

The amplitude is 3.

Step 3: Find the period and the parameter $b$.

The graph goes from a maximum to a minimum between $x=1$ and $x=3$. This represents half of a full period.

• Half Period = $3-1=2$

• Full Period = $2\times (\text{Half Period})=2\times 2=4$

Now, use the period to find $b$.

$$|b|=\frac{2\pi }{\text{Period}}=\frac{2\pi }{4}=\frac{\pi }{2}$$

We will use $b=\frac{\pi }{2}$.

Step 4: Determine a cosine equation.

A cosine function is often easiest when a maximum or minimum point is clearly identifiable. The graph has a maximum at $x=1$.

• A standard (non-reflected) cosine function starts at a maximum.

• Since our graph has a maximum at $x=1$, we can model this with a non-reflected cosine function ($a=3$) shifted 1 unit to the right.

• This means our phase shift is $c=1$.

• Putting it all together: $a=3,b=\frac{\pi }{2},c=1,d=2$.

  $$g(x)=3\cos \left(\frac{\pi }{2}(x-1)\right)+2$$

Step 5: Determine a sine equation.

A sine function starts at its midline and increases. We need to find a point on the graph where it crosses the midline $y=2$ with a positive slope.

• The maximum is at $x=1$ and the minimum is at $x=3$. The midline crossing between them occurs at $x=2$. At this point, the graph is decreasing.

• The cycle starts before $x=1$. The point where the graph crosses the midline while increasing would be one-quarter period before the maximum at $x=1$.

• Quarter Period = $\frac{1}{4}\times \text{Period}=\frac{1}{4}\times 4=1$.

• The starting point for a sine model is at $x=(\text{x of max})-(\text{Quarter Period})=1-1=0$.

• So, we can use a non-reflected sine function ($a=3$) with a phase shift of $c=0$.

• Putting it all together: $a=3,b=\frac{\pi }{2},c=0,d=2$.

  $$g(x)=3\sin \left(\frac{\pi }{2}(x-0)\right)+2=3\sin \left(\frac{\pi }{2}x\right)+2$$

Both equations are valid representations of the given graph.

Using Your Calculator

A graphing calculator is an excellent tool for verifying the equation you have derived from a graph or for visualizing a given function. It is not used for analytical derivation but for confirmation.

Task: Verify that the equation $g(x)=3\cos \left(\frac{\pi }{2}(x-1)\right)+2$ from Example 2 correctly models a function with a maximum at $(1,5)$ and a minimum at $(3,-1)$.

Steps (TI-84 Style):

1. Enter the Equation:

   • Press the [Y=]` button. * In `Y1`, type `3*cos((π/2)(X-1))+2`. Ensure your calculator is in **Radian mode** by pressing `[MODE]` and checking the setting. 2. **Set the Viewing Window:** * Press the `[WINDOW]` button. * Set the window to match the key features of the problem. * $Xmin: $-1$ (to see left of the maximum)

   • $Xmax$: $5$ (to see the full cycle from $x=1$ to $x=5$)

   • Xscl`: $1 (sets the tick mark spacing on the x-axis)

   • $Ymin‘:$-2$ (to see below the minimum of -1)

   • Ymax`: $6 (to see above the maximum of 5)

   • Yscl`: $1

2. Graph and Verify:

   • Press [GRAPH]. The displayed curve should match the expected shape.

   • Use the calculation tools to find the maximum and minimum.

   • Press [2ND]$then$[TRACE]to open the CALC menu. * Select `4:maximum`. The calculator will ask for a "Left Bound?", "Right Bound?", and "Guess?". Move the cursor to the left of the peak at $x=1, press [ENTER], move to the right, press [ENTER], and press [ENTER]again for the guess. The calculator should display a maximum at or very nearX=1, Y=5`.

   • Repeat the process by selecting 3:minimum to verify the minimum at $(3,-1)$.

AP Exam Quick Hit

Common Question Types

• Given an equation, find a characteristic: "What is the period of the function $f(x)=5\cos (4x+2\pi )-1$?"

  • Solution approach: First, factor the argument: $4x+2\pi =4(x+\frac{\pi }{2})$. Identify $b=4$. Then calculate the period: $Period=$\frac{2\pi}{|b|} = \frac{2\pi}{4} = \frac{\pi}{2}`. - **Given a graph, identify a correct equation:** A graph of a sinusoid is shown. "Which of the following equations could represent the function shown in the graph?" * *Solution approach:* Systematically determine the midline, amplitude, and period from the graph. Use these to eliminate incorrect choices. Then, check the phase shift and any reflection for the remaining options to find the correct match. - **Given characteristics, construct an equation:** "A sinusoidal function has a midline of $y=-3, an amplitude of 7, and a period of $6\pi$. Write a possible equation for this function."

  • Solution approach: Use the characteristics to find the parameters. $d=-3$, $|a|=7$. Find $b$ from the period: $|b|=\frac{2\pi }{6\pi }=\frac{1}{3}$. A possible equation (with $a=7$ and no phase shift) is $f(x)=7\sin (\frac{1}{3}x)-3$.

Common Mistakes

• Confusing Period and $b$: A very common error is to set $b$ equal to the period found from a graph. Remember the relationship is inverse: $Period=$\frac{2\pi}{|b|}$. If the period is $4\pi, then $b=\frac{2\pi }{4\pi }=\frac{1}{2}$, not $4\pi$.

• Incorrect Phase Shift from Unfactored Form: When given an equation like $f(x)=\sin (2x-\pi )$, many students incorrectly state the phase shift is $\pi$. You must factor out $b=2$ first: $f(x)=\sin (2(x-\frac{\pi }{2}))$. The correct phase shift is $\frac{\pi }{2}$ to the right.

• Amplitude is Always Positive: The amplitude is $|a|$, a distance, which cannot be negative. If $a=-5$, the amplitude is 5. The negative sign indicates a reflection over the midline, meaning a cosine model would start at a minimum and a sine model would decrease from the midline initially.

• Sign of Phase Shift: In the form $(x-c)$, the shift is $c$ units to the right. In the form $(x+c)$, which is equivalent to $(x-(-c))$, the shift is $c$ units to the left. Students often reverse this.

• Mixing Up Sine and Cosine Starting Points: Forgetting that a basic cosine function starts at a maximum and a basic sine function starts at the midline (and increases). This mistake leads to choosing the wrong function or an incorrect phase shift when writing an equation from a graph.