Problem 67 - Precalculus: Functions and Graphs

Question

Practice sketching the graph of the sine function, taking different units of length on the horizontal and vertical axes. Practice sketching graphs of the cosine and tangent functions in the same manner. Continue this practice until you reach the stage at which, if you were awakened from a sound sleep in the middle of the night and asked to sketch one of these graphs, you could do so in less than thirty seconds.

Step-by-Step Solution

Answer

Sine and cosine are periodic with period 2π, tangent with period π, exhibiting unique oscillation patterns.

Step 1: Understanding the Sine Function

The sine function, denoted as \( y = \sin(x) \), is a periodic function with a fundamental period of \( 2\pi \). It oscillates between 1 and -1, with a wave-like pattern that repeats every \( 2\pi \) units along the horizontal axis. Identify the key points: the origin \((0,0)\), the maximum at \((\pi/2, 1)\), the zero crossing at \((\pi, 0)\), the minimum at \((3\pi/2, -1)\), and another zero at \((2\pi, 0)\). Plot these points on your graph.

Step 2: Sketching the Sine Function

On your graph paper with length units marked, start at the origin \((0,0)\). Draw a smooth wave that reaches \(y = 1\) at \(x = \pi/2\), returns to \(y = 0\) at \(x = \pi\), decreases to \(y = -1\) at \(x = 3\pi/2\), and returns to \(y = 0\) at \(x = 2\pi\). Repeat this pattern to show periodicity on the graph.

Step 3: Understanding the Cosine Function

The cosine function, expressed as \( y = \cos(x) \), is similar to the sine function but shifted horizontally. It also has a period of \( 2\pi \), oscillating between 1 and -1. Key points include the start at maximum \((0, 1)\), zero crossing at \((\pi/2, 0)\), minimum at \((\pi, -1)\), zero crossing again at \((3\pi/2, 0)\), and returning to maximum at \((2\pi, 1)\).

Step 4: Sketching the Cosine Function

Using the key points identified, start the graph at \((0, 1)\). Form a smooth curve that first drops to \(y = 0\) by \(x = \pi/2\), reaches \(y = -1\) at \(x = \pi\), returns to zero at \(3\pi/2\), and completes the cycle at \(y = 1\) by \(2\pi\). Mark these on your graph, and continue the pattern to express periodicity.

Step 5: Understanding the Tangent Function

The tangent function, given by \( y = \tan(x) \), has different characteristics. It has a period of \( \pi \) and exhibits vertical asymptotes where the function is undefined. These occur at \( x = (2n+1)\pi/2 \) where \( n \) is an integer. Key points to plot include \((0, 0)\), with the function approaching infinity as it nears the asymptotes.

Step 6: Sketching the Tangent Function

Plot key point at \((0, 0)\). Draw curves that approach vertical asymptotes at \(x = \pi/2\) and \(x = -\pi/2\), showing the function increasing and decreasing sharply. Repeat every \(\pi\) units to illustrate periodicity, ensuring each section of the curve slopes upward from left to right between asymptotes.

Key Concepts

Sine Function

The sine function, denoted as \( y = \sin(x) \), is one of the fundamental trigonometric functions that describes a smooth periodic wave. This function is periodic, repeating its wave pattern every \( 2\pi \) units along the horizontal axis. In simpler terms, after every \( 2\pi \) units, the pattern of the wave starts again. Key points to help you sketch this wave are:

• The origin \((0,0)\)
• The maximum point \((\pi/2, 1)\)
• A zero crossing at \((\pi, 0)\)
• The minimum point \((3\pi/2, -1)\)
• Another zero at \((2\pi, 0)\)

When sketching, plot these points carefully and join them with a smooth, wave-like curve. Remember, the sine function will oscillate between 1 and -1, never going beyond these values.

Cosine Function

The cosine function, written as \( y = \cos(x) \), is quite similar to the sine function but is horizontally shifted compared to it. Much like the sine function, cosine also has a fundamental period of \( 2\pi \) and oscillates between 1 and -1. If you're sketching the graph, it can be helpful to think of the cosine as starting "higher" than the sine graph, beginning at the top of its wave at \( y = 1 \).

• Maximum at \((0, 1)\)
• Zero crossing at \((\pi/2, 0)\)
• Minimum at \((\pi, -1)\)
• Another zero crossing at \((3\pi/2, 0)\)
• Returning to maximum at \((2\pi, 1)\)

Plot these key points and connect them with a smooth, rolling curve. The cosine graph creates a wave that is identical in shape to the sine graph, but it starts at its maximum value.

Tangent Function

The tangent function, \( y = \tan(x) \), is distinct from the sine and cosine functions. It has a different periodicity, with a period of just \( \pi \). This means that the pattern in its graph repeats every \( \pi \) units. Unlike sine and cosine, tangent does not oscillate within a fixed range. Instead, it can take on any real number value, resulting in vertical asymptotes where the function is undefined. These occur at \( x = (2n+1)\pi/2 \), where \( n \) is an integer.When sketching the tangent graph, take note of these key points:

• The origin \((0, 0)\)
• Approaching vertical asymptotes at \( x = \pi/2, -\pi/2, \dots \)

Tangent curves slope upwards from left to right between each asymptote, giving them a distinct set of repeating "S" shaped forms.

Graph Sketching

Sketching trigonometric graphs can seem daunting, but understanding the basic structure of each function simplifies the process. Begin by identifying and marking the key points, which serve as a guide to form the overall shape of the graph. For both sine and cosine functions, these points include maximums, minimums, and zero crossings, while tangent graphs include key crossings and approach points at asymptotes. Consider the major features:

• Periodicity - how often the wave repeats
• Amplitude - the peak values reached
• Key points and asymptotes - crucial intersections or divergent points

By consistently using the same process to plot these characteristics, anyone can quickly and confidently sketch trigonometric graphs. Practice helps to embed these steps instinctively.

Periodicity

Periodicity is one of the hallmark features of trigonometric functions. It describes how often the function's graph repeats its shape as you move along the x-axis.

• The sine and cosine functions both have periods of \( 2\pi \), meaning their graphs repeat every \( 2\pi \) units.
• The tangent function has a smaller period, repeating every \( \pi \) units.

Understanding these periods is crucial when sketching the graphs. It assures you that the functions maintain a predictable and repeating pattern. So, wherever you are on the graph, you can anticipate how it continues, ensuring that you can sketch any portion of it quickly and accurately. This repetition is what allows trigonometric graphs to model cyclical patterns in real-world phenomena effectively.