Problem 6 - 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. $$ y=1-2 \sin x $$

Step-by-Step Solution

Answer

Graph by reflecting, vertically stretching, and shifting up the sine function. Amplitude is 2, period stays \(2\pi\), and midline is \(y=1\).

Step 1: Identify the Base Function

The base function here is the sine wave, which is represented as \(y = \sin x\). This is your starting point for the graph sketching.

Step 2: Apply Vertical Stretching and Reflection

In the given function, \(y = 1 - 2\sin x\), the \(-2\sin x\) indicates that the sine function is vertically stretched by a factor of 2 and then reflected about the x-axis. This means that the amplitude of the sine wave will be 2 instead of 1, and its peaks and troughs will be inverted.

Step 3: Apply Vertical Shifting

The \(1\) in the equation \(y = 1 - 2\sin x\) shifts the entire graph upward by 1 unit. This means the midline of the sine wave will now be at \(y = 1\).

Step 4: Sketch the Function

Using the transformed characteristics, sketch the graph: \(-2\sin x\) first flips the sine function upside down and stretches it, resulting in peaks of -2 and -2 changing the amplitude from 1 to 2. The midline is shifted to \(y = 1\), making maximum and minimum points at \(y = 3\) and \(y = -1\), respectively. One cycle of the sine function from 0 to \(2\pi\) is now resketched with these new transformations.

Key Concepts

Sine Function

The sine function, represented by \( y = \sin x \), is a fundamental trigonometric function that graphs as a smooth, wave-like curve. This curve oscillates between -1 and 1 and repeats every \( 2\pi \) radians, creating a perfect wave. The sine wave starts at the origin (0,0), reaches its maximum at \( \pi/2 \), returns to zero at \( \pi \), reaches its minimum at \( 3\pi/2 \), and completes a cycle at \( 2\pi \).

The original exercise requires us to transform this basic sine function to match the function \( y = 1 - 2\sin x \). To understand these transformations, we need to learn about how vertical stretching, reflection, and vertical shifting work.

• **Oscillation**: The regular cycle of peaks and troughs.
• **Amplitude**: The distance from the midline to a peak or trough (originally 1 for \( \sin x \)).
• **Period**: The length of one complete wave cycle (\( 2\pi \) for \( \sin x \)).

Vertical Stretching

Vertical stretching refers to enlarging the amplitude of a function. In the equation \( y = -2\sin x \), the \( 2\) in front of the \( \sin x \) stretches the sine wave vertically. This means every point on the graph is moved further from the x-axis compared to the original sine function. The higher amplitude shows larger waves, with new peaks reaching 2 and troughs reaching -2, compared to the normal range of 1 to -1 in \( y = \sin x \).

When analyzing transformations:

• **Amplitude Change**: Multiply the original amplitude by 2 (from 1 to 2).
• **Graph Impact**: The "stretch" makes the waves taller.

Understanding stretching helps predict how these changes make each point on the graph appear more pronounced and spaced vertically.

Reflection

Reflection in this context flips the graph over the x-axis. The negative sign in \( y = -2\sin x \) is responsible for the reflection. Generally, this changes the direction of the sine wave. Where it used to rise, it now falls, and where it descended, it now ascends.

Understanding reflection is crucial because:

• **Inversion Effect**: Peaks become troughs, and troughs become peaks.
• **Symmetry Shift**: This process alters the graph's symmetry about the x-axis.

Combining reflection with stretching from the previous step lets us visualize not just taller waves, but also inverted ones, creating a new shape for the sine wave.

Vertical Shifting

Vertical shifting moves the entire sine wave up or down. This is indicated by the constant term in the equation \( y = 1 - 2\sin x \). In this case, the graph is shifted up by 1 unit.

To understand how vertical shifting affects the graph, consider:

• **Midline Change**: The midline, originally at \( y=0 \), shifts to \( y=1 \).
• **Position Change**: All points shift upward, making new maximum points at \( y=3 \) and new minimum points at \( y=-1 \).

This shift ensures that the entire sine wave elevates vertically, resulting in a graph where the central axis isn't at zero but at 1, completing our transformation of the sine function to fit the equation \( y = 1 - 2\sin x \).