Problem 14 - Algebra 2 and Trigonometry

Question

In \(3-14,\) sketch one cycle of the graph. $$ y=-\cos x $$

Step-by-Step Solution

Answer

Plot points \((0, -1), (\pi/2, 0), (\pi, 1), (3\pi/2, 0), (2\pi, -1)\), then draw a curve through them.

Step 1: Understand the Basic Shape

The function given is a cosine function: \(y = -\cos x\). The basic shape of \(\cos x\) is a wave that starts at 1, descends to -1, and returns to 1 over an interval of \(2\pi\). For \(-\cos x\), this shape is flipped vertically. This means the function starts at -1, rises to 1, and returns to -1 over the same interval.

Step 2: Identify Key Points

For one cycle of \(y = -\cos x\) over the interval \([0, 2\pi]\), the key points are determined by the cosine function: \((0, -1), (\pi/2, 0), (\pi, 1), (3\pi/2, 0), (2\pi, -1)\).

Step 3: Plot Key Points

Start by marking these points on the graph. At \(x = 0\), \(y = -1\); at \(x = \pi/2\), \(y = 0\); at \(x = \pi\), \(y = 1\); at \(x = 3\pi/2\), \(y = 0\); and at \(x = 2\pi\), \(y = -1\). These points define the shape of one cycle of the graph.

Step 4: Draw the Cosine Curve

Connect the plotted points smoothly to form a continuous wave-like curve. This curve should resemble an inverted cosine wave, creating a single period of the \(-\cos x\) function over the interval \([0, 2\pi]\).

Key Concepts

Cosine Function

The cosine function, represented as \( y = \cos x \), is one of the fundamental trigonometric functions in mathematics. It creates a wave-like graph that oscillates between +1 and -1. This wave is periodic, meaning it repeats its shape over a consistent interval.

Here are some key features of the cosine function:

• **Amplitude:** It measures the height of the wave from the center line to the peak. For \( y = \cos x \), the amplitude is 1.
• **Period:** This is the length of one complete cycle of the wave. For the cosine function, the standard period is \( 2\pi \).
• **Vertical Shift:** This is an up or down shift along the y-axis. For \( y = \cos x \), the center line before any shifts is the x-axis.

When the cosine function is negated, as in \( y = -\cos x \), the graph is flipped over the x-axis, reversing the peaks and troughs. Hence, instead of starting at \( y = 1 \), the function \( y = -\cos x \) starts at \( y = -1 \), making its path an inverted wave.

Graph Sketching

Graph sketching involves drawing the graph of a function accurately based on its equation. For the cosine function, the graph sketching process includes understanding the key points along its cycle.

To sketch the graph of \( y = -\cos x \):

• **Identify Key Points:** As shown in the exercise, crucial points for one cycle are \((0, -1), (\frac{\pi}{2}, 0), (\pi, 1), (\frac{3\pi}{2}, 0), (2\pi, -1)\).
• **Plot Key Points:** It's essential to mark these coordinates on a graph. This anchors the curve precisely.
• **Draw the Curve:** Connect these points with a smooth, wave-like line that depicts one full cycle. Be sure to create a curved, not straight, connection to represent the natural shape of the cosine wave.

Each segment of the curve reflects gradual changes in the value of \( y \) with respect to \( x \), often rounded to highlight the natural curvature inherent in trigonometric functions like cosine.

Phase Shift

Phase shift refers to a horizontal shift along the x-axis in trigonometric functions. It alters the starting position of the wave without changing its shape or orientation.

Though the exercise does not incorporate phase shifts, understanding them provides valuable insight into graph transformations:

• **Calculation:** Phase shifts are typically represented in the form \( y = \cos(x - c) \), where \( c \) is the phase shift.
• **Impact:** A positive \( c \) shifts the graph to the right, while a negative \( c \) shifts it to the left.
• **Example:** For \( y = \cos(x - \frac{\pi}{4}) \), the graph of \( \cos x \) shifts to the right by \( \frac{\pi}{4} \).

Phase shifts are a powerful tool in trigonometry, allowing precise control over the timing and alignment of periodic signals. They play a critical role in scenarios ranging from sound waves to AC circuits.