What is the difference between the effect of $n$ in $y = n \sin x$ and $y = \sin nx$?

In $y = n \sin x$, $n$ is a vertical stretch factor that multiplies the $y$-coordinates, changing the amplitude. In $y = \sin nx$, $n$ is a horizontal stretch factor that divides the $x$-coordinates, changing the period of the function.

How does the direction of a horizontal translation compare to the sign of the constant $c$ in $f(x + c)$?

The translation is in the opposite direction of the sign. A positive $c$ ($x + c$) shifts the graph to the left (negative direction), while a negative $c$ ($x - c$) shifts the graph to the right (positive direction).

How does a vertical stretch differ from a vertical translation in terms of the graph's midline?

A vertical stretch ($a \cdot f(x)$) expands the graph away from the midline but keeps the midline in the same position. A vertical translation ($f(x) + d$) moves the entire graph, including the midline, up or down by $d$ units.

What error occurs if you assume $y = \cos(2x)$ has a period of $720^\circ$?

This is a misconception that the multiplier $2$ doubles the period. In reality, the multiplier $2$ doubles the frequency, meaning the period is halved to $180^\circ$ ($360/2$).

Why is it a mistake to say the amplitude of $y = 3 \tan x$ is 3?

The tangent function has no amplitude because its range is $(-\infty, \infty)$ and it does not have a maximum or minimum value. The 3 is a vertical stretch factor that makes the curve steeper, but 'amplitude' is a term reserved for bounded waves like sine and cosine.

What happens to the x-intercepts of $y = \sin x$ if you apply a vertical stretch $y = 5 \sin x$?

The x-intercepts remain unchanged. Because the $y$-coordinate at an intercept is 0, multiplying it by the stretch factor ($5 \times 0$) still results in 0.

Define the 'Period' of a transformed sine function $y = \sin(bx)$.

The period is the horizontal distance required for the function to complete one full cycle. It is calculated as $\frac{360^\circ}{|b|}$ or $\frac{2\pi}{|b|}$.

What is a 'Phase Shift' in the context of trigonometric functions?

A phase shift is a horizontal translation of a periodic function. It represents how much the start of the wave cycle has been moved from the origin along the x-axis.

What is the formula for the new amplitude of $y = a \cos(x) + d$?

The amplitude is given by $|a|$. The vertical shift $d$ does not affect the amplitude, as amplitude is the distance from the midline to the peak, not the distance from the x-axis.

How do you calculate the new x-coordinates for the function $y = f(bx + c)$?

You first account for the shift and then the scale, or solve the inequality $0 \le bx + c \le 360$. Effectively, you subtract $c$ from the original x-value and then divide by $b$.

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Trigonometry

Transformations of Trigonometric Functions

Summary

Transforming trigonometric functions involves modifying the base graphs of sine, cosine, and tangent through translations, stretches, and reflections. These operations alter the physical properties of the waves, such as their amplitude, period, and phase, by applying mathematical operations directly to the input variable or the function's output.

1. Definition & Core Concepts

A diagram comparing a standard sine wave (dashed) with a vertically stretched sine wave (solid red), illustrating how the amplitude increases while the x-intercepts remain constant.

2. Underlying Principles

3. Methods & Techniques

4. Key Distinctions

5. Exam Strategy & Tips

6. Common Pitfalls & Misconceptions

Confusing Stretch and Squash: Students often think $y = \sin(2x)$ makes the graph twice as wide, but it actually makes it twice as thin (squashed) because the function completes its cycle in half the time.
Amplitude of Tangent: Note that while $y = 2 \tan(x)$ stretches the graph vertically, the tangent function does not have a defined 'amplitude' because its range is infinite. The stretch just makes the curve appear steeper.
Order of Operations: When multiple transformations are present, applying them in the wrong order (e.g., shifting before stretching) can lead to incorrect intercepts. Generally, follow the order: Horizontal Shift $\rightarrow$ Horizontal Stretch $\rightarrow$ Vertical Stretch $\rightarrow$ Vertical Shift.

Transformations of Trigonometric Functions

Summary

1. Definition & Core Concepts

Trigonometric transformations are mathematical operations applied to the parent functions $y = \sin(x)$ , $y = \cos(x)$ , and $y = \tan(x)$ to change their position or shape.

A vertical transformation affects the $y$ -coordinates of the graph, typically altering the amplitude or vertical position of the wave.

A horizontal transformation affects the $x$ -coordinates, which changes the period (how often the wave repeats) or the phase shift (horizontal position).

The general transformed form can be expressed as $y = a \cdot f(b(x + c)) + d$ , where each constant represents a specific geometric change.

A diagram comparing a standard sine wave (dashed) with a vertically stretched sine wave (solid red), illustrating how the amplitude increases while the x-intercepts remain constant.

2. Underlying Principles

Vertical Scaling: In the function $y = a \cdot f(x)$ , the constant $a$ acts as a multiplier for every output value. If $|a| > 1$ , the graph stretches away from the x-axis; if $0 < |a| < 1$ , it compresses toward the x-axis.

Horizontal Scaling: In the function $y = f(bx)$ , the constant $b$ affects the input speed. Because the function reaches its cycle values $b$ times faster, the graph is horizontally compressed by a factor of $1/b$ .

Reciprocal Relationship: Horizontal transformations are often counter-intuitive because they perform the inverse operation of the constant shown (e.g., multiplying $x$ by 2 results in a graph half as wide).

Reflections: A negative sign outside the function ( $-f(x)$ ) reflects the graph over the x-axis, while a negative sign inside ( $f(-x)$ ) reflects it over the y-axis.