Problem 3 - Precalculus with Calculus Previews

Question

Use the techniques of shifting, stretching, compressing, and reflecting to sketch at least one cycle of the graph of the given function. $$ y=2-\sin x $$

Step-by-Step Solution

Answer

The graph is a vertically shifted and reflected sine wave with peaks at 3 and troughs at 1.

Step 1: Identify the Base Function

The given function is \( y = 2 - \sin x \). The base function here is \( \sin x \). Understanding how the base function behaves will help us apply transformations.

Step 2: Recognize Transformations

In \( y = 2 - \sin x \), we identify two transformations: a vertical shift and a reflection. The base function \( \sin x \) is reflected across the x-axis, because of the negative sign before it, giving us \( -\sin x \). It is then shifted upwards by 2 units due to the +2.

Step 3: Graph the Reflected Function

Start by graphing \( -\sin x \). The reflection across the x-axis means for any point \( (x, \sin x) \), we now have \( (x, -\sin x) \). Thus, the maximum amplitude becomes -1, and the minimum becomes 1.

Step 4: Apply the Vertical Shift

Now, take the reflected function \( -\sin x \) and shift the entire graph upward by 2 units. This means the highest value of the sine wave is 2 + 1 = 3, and the lowest value is 2 - 1 = 1.

Step 5: Sketch at Least One Cycle

The sine function \( \sin x \) has a period of \( 2\pi \). Therefore, one full cycle of \( -\sin x \), spanning from 0 to \( 2\pi \), will fit the transformations. Mark the key points: starting at \( (0, 2) \), peak at \( (\frac{\pi}{2}, 3) \), crosses the middle line at \( (\pi, 2) \), minimum at \( (\frac{3\pi}{2}, 1) \), and back to starting level at \( (2\pi, 2) \).

Key Concepts

Vertical Shift

When working with transformations of graphs, a vertical shift is a straightforward adjustment. It involves moving a graph up or down without altering its shape. For the function \( y = 2 - \sin x \), the "+2" in the equation signifies a vertical shift. Specifically, each point on the graph \( -\sin x \) is moved 2 units upwards.
This means, for every point \((x, -\sin x)\), it shifts to \((x, -\sin x + 2)\).
This affects the range of the function:

• The highest value increases by 2 units.
• The lowest value increases by 2 units.

For our function, the maximum of the sine wave becomes \( 3 \), and the minimum becomes \( 1 \). These changes are critical in sketching the correct graph.

Reflection

Reflections are another central aspect of graph transformations. For the function \( y = 2 - \sin x \), the reflection occurs due to the negative sign in front of \( \sin x \).
This reflects the graph over the x-axis. In simpler terms, the peaks and troughs of the wave are inverted. For any point \((x, \sin x)\), it becomes \((x, -\sin x)\).

• The original peaks of \( \sin(x) \) at 1 become troughs at -1 for \( -\sin(x) \).
• Conversely, the original troughs of \( \sin(x) \) at -1 are peaks at 1 for \( -\sin(x) \).

This reflection is an essential step before applying any shifts or further transformations.

Trigonometric Functions

Trigonometric functions, like sine and cosine, are foundational in mathematics, especially in understanding waves and oscillations. The base function in the exercise is \( \sin(x) \), a periodic function that oscillates between -1 and 1. This periodic nature (period \( 2\pi \)) means it repeats its pattern every \( 2\pi \) units.

• Key characteristics include amplitude, period, and frequency.
• \( \sin(x) \) has peaks at \( 1 \), troughs at \( -1 \), and crosses the x-axis periodically.

These are crucial in addressing how transformations will impact the function. For instance, knowing \( \sin(x) \)'s behavior helps us visualize transformations like reflection or vertical shifts.

Sinusoidal Graphs

Sinusoidal graphs are a type of periodic graph that visually represent sine and cosine functions. These graphs are characterized by their wave-like shape, with consistent amplitude and wavelength.
In the problem, we begin with \( y = \sin x \), a classic sine wave.

• The function \( y=2-\sin x \) transforms this wave into a reflected and shifted version.
• Such modifications result in different maximum and minimum points, but the wave nature remains.

Sketching sinusoidal graphs post-transformations involves identifying these new points and drawing the smooth periodic curve that connects them.