Transformations of Trigonometric Functions

• Parent Functions: The foundational trigonometric graphs $y = \sin(x)$, $y = \cos(x)$, and $y = \tan(x)$ serve as the starting points for all transformations. These functions are periodic, meaning they repeat their values in regular intervals known as periods.

• General Transformation Form: A fully transformed trigonometric function is often expressed as $y = a f(b(x - c)) + d$. In this model, $f$ represents the trigonometric ratio, while the constants $a, b, c,$ and $d$ determine the specific spatial modifications.

• Vertical vs. Horizontal: Transformations occurring 'outside' the function (like $a$ and $d$) affect the vertical dimension and output values. Conversely, transformations 'inside' the function (like $b$ and $c$) affect the horizontal dimension and the input values.

• Vertical Stretch and Amplitude: The coefficient $a$ in $y = a \sin(x)$ scales the vertical distance from the midline. The Amplitude is defined as the absolute value $|a|$, representing half the distance between the maximum and minimum peaks.

• Horizontal Scale and Period: The coefficient $b$ inside the argument affects the 'speed' of the cycle. The new period $P$ of a sine or cosine function is calculated by $P = \frac{2\pi}{b}$, while for a tangent function it is $P = \frac{\pi}{b}$.

• The Midline Principle: The vertical shift $d$ moves the entire graph up or down. This constant value defines the new horizontal axis of oscillation, often called the Midline or principal axis.

• Step 1: Identify Parameters: Begin by rewriting the equation into the standard form $y = a \sin(b(x - c)) + d$. It is critical to ensure that $b$ is factored out from the $x$ term to correctly identify the phase shift.

• Step 2: Establish the Frame: Determine the midline ($y = d$) and the maximum/minimum boundaries ($y = d \pm |a|$). Sketching these horizontal lines provides a 'corridor' that contains the entire wave.

• Step 3: Map the Cycle: Calculate the period and the phase shift. Determine the starting point $(c, d)$ and the end of one full cycle $(c + P, d)$, then divide this interval into four equal segments to find the locations of peaks, troughs, and intercepts.

Transformation Comparison Table

• Inside vs. Outside Counter-Intuition: Observe that 'inside' transformations (horizontal) often work in the opposite way of the sign. For instance, subtracting $c$ shifts the graph to the right (positive direction), and multiplying by $b > 1$ compresses the graph (smaller period).

• Sine vs. Cosine Origin: Recall that the sine graph typically starts at $(0,0)$ on the midline and goes up, while the cosine graph starts at $(0,1)$ at the maximum peak. Transformations simply relocate these starting points.

• The Factoring Error: A common mistake is reading the phase shift directly from $y = \sin(2x - \pi)$ as $\pi$. In reality, the function must be factored to $y = \sin(2(x - \pi/2))$, revealing the true phase shift is $\pi/2$.

• Negative Amplitude Confusion: If $a$ is negative, the amplitude remains the absolute value $|a|$. The negative sign represents a reflection in the x-axis, meaning the sine wave starts by going down instead of up.

• Period vs. Frequency: Do not confuse the coefficient $b$ with the period. $b$ represents the frequency (how many cycles occur in $2\pi$), whereas the period is the physical length of one cycle.