translating and scaling sine and cosine functions ( Gizmo)
The correct answer and explanation is :
Correct Answer (based on common Gizmo-style questions for “Translating and Scaling Sine and Cosine Functions”):
When transforming sine and cosine functions, the general form is:
$y = A \cdot \sin(B(x – C)) + D \quad \text{or} \quad y = A \cdot \cos(B(x – C)) + D$
- A = amplitude (vertical stretch/compression and reflection if negative)
- B = affects the period: Period = $\frac{2\pi}{|B|}$
- C = horizontal shift (phase shift)
- D = vertical shift (up/down)
So, if the function is:
$$
y = -2 \cdot \cos\left(2(x – \frac{\pi}{4})\right) + 1
$$
- Amplitude = 2 (reflected over x-axis due to the negative)
- Period = $\frac{2\pi}{2} = \pi$
- Phase shift = $\frac{\pi}{4}$ to the right
- Vertical shift = 1 unit up
300-Word Explanation:
In trigonometry, sine and cosine functions describe wave-like patterns, and by applying transformations, we can shift or scale these graphs to model various periodic behaviors. These transformations are controlled using the parameters in the function’s general form:
$$
y = A \cdot \sin(B(x – C)) + D \quad \text{or} \quad y = A \cdot \cos(B(x – C)) + D
$$
Amplitude (A) determines the height of the wave from the midline. If A is negative, the wave is reflected over the x-axis. For example, an amplitude of -2 means the wave reaches 2 units below and above the midline but starts downward due to the reflection.
The B value affects the period, which is how long it takes the function to complete one full cycle. A larger B value compresses the wave horizontally. The period is calculated with $\frac{2\pi}{|B|}$, showing how the wave’s frequency changes.
C represents the horizontal shift, also called the phase shift. If C is positive, the graph shifts right; if negative, it shifts left. This does not affect the shape of the wave, only its position along the x-axis.
D is the vertical shift, moving the entire wave up or down the y-axis. This changes the midline of the function.
Using these transformations in a Gizmo simulation helps students visually understand how altering each parameter affects the sine or cosine wave. These concepts are fundamental in both mathematics and real-world applications like sound waves, tides, and alternating current electricity.