Problem 5 - 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. $$ y=-2+4 \cos x $$

Step-by-Step Solution

Answer

The graph is a vertically stretched and shifted cosine wave oscillating between -6 and 2.

Step 1: Identifying the Base Function

The base function here is \( \cos x \). It's important to recognize this as our starting point, which is a basic cosine wave with an amplitude of 1, a period of \( 2\pi \), and it oscillates between -1 and 1.

Step 2: Applying Vertical Stretching

The coefficient 4 in front of \( \cos x \) indicates a vertical stretching. This changes the amplitude of the cosine wave from 1 to 4. The function now oscillates between -4 and 4.

Step 3: Applying Vertical Shift

The term \(-2\) causes a vertical shift of the graph by 2 units downward. This means the entire cosine graph is shifted down, and it will now oscillate between -6 and 2.

Step 4: Sketching the Transformed Graph

Graph one cycle of the transformed cosine function. Start from the point \((0, 2)\), reach the peak at \((\frac{\pi}{2}, -2)\), go back to the midpoint at \((\pi, -6)\), reach the trough at \((\frac{3\pi}{2}, -2)\), and finally complete the cycle back to the starting point at \((2\pi, 2)\).

Key Concepts

Cosine Function

The cosine function is a fundamental trigonometric function, often denoted as \( \cos x \). It is defined for all real numbers and oscillates in a smooth, wave-like pattern. The key characteristics of the basic cosine function include:

• An amplitude of 1, meaning it reaches a maximum height of 1 and a minimum of -1.
• A period of \( 2\pi \), indicating that the function repeats itself every \( 2\pi \) units along the x-axis.
• It starts at its maximum at \( x = 0 \), descends to 0 by \( \frac{\pi}{2} \), reaches a minimum at \( x = \pi \), ascends back to 0 by \( \frac{3\pi}{2} \), and completes the cycle at \( x = 2\pi \).

The cosine function is useful in modeling periodic behaviors and serves as a building block for more complex trigonometric transformations.

Vertical Stretching

Vertical stretching transforms the amplitude of a function. For the function \( y = a \cos x \), the amplitude is magnified by the factor \( |a| \). When \( a > 1 \), the graph is vertically stretched, making it taller; if \( 0 < a < 1 \), it is vertically compressed.

In the given exercise, \( y = 4 \cos x \) means the amplitude changes from 1 to 4. This results in the wave oscillating between 4 and -4 instead of 1 and -1. Every point on the original cosine curve is essentially pulled away from the x-axis, increasing the height of peaks and the depth of troughs.

Vertical Shift

Vertical shifts move the entire graph up or down along the y-axis. A constant is added or subtracted from the function to achieve this. For a function \( y = \cos x \pm d \), adding \( d \) shifts the graph up by \( d \) units, while subtracting shifts it down.

In this problem, \( y = -2 + 4 \cos x \) represents a shift of 2 units downward due to the \( -2 \). The function's oscillation center changes from 0 to -2, resulting in a new range of -6 to 2. Essentially, each point on the cosine wave is moved 2 units lower on the graph, modifying where the troughs, midpoints, and peaks appear.

Graph Transformations

Graph transformations involve shifting, stretching, compressing, and reflecting the basic graph functions. The aim is to modify the graph’s shape or position without altering its overall wave pattern fundamentally.

The exercise requires combining multiple transformations to graph the function \( y = -2 + 4 \cos x \). Here's how each transformation plays its role:

• **Vertical Stretching**: Changes the height of the wave, creating large amplitudes.
• **Vertical Shifting**: Alters the vertical position, moving the center line up or down.
• **Graphing Steps for One Cycle**: Begin at \((0, 2)\) as the new maximum, drop to \((\frac{\pi}{2}, -2)\) at the original zero point, reach the lowest point \((\pi, -6)\), and climb back up to complete the cycle at \((2\pi, 2)\).

These transformations shape the final graph according to prescribed mathematical operations, ensuring you not only see a complete visual representation but understand the effect of each mathematical change applied.