Problem 20 - Algebra 2

Question

Graph each function on the interval \(0^{\circ} < x < 470^{\circ}\) and \(-300 < y < 300 .\) Evaluate the function at \(x=45^{\circ}, 90^{\circ},\) and \(135^{\circ} .\) $$ y=-100 \tan x $$

Step-by-Step Solution

Answer

The graph of \(y = -100 \times tan(x)\) over the given interval shows a vertically stretched tangent function with asymptotes at \(x = (90 + 180 \times n)^{\circ}\) for \(0 < n < 3\). The function's values at \(x = 45^{\circ}\), \(90^{\circ}\), and \(135^{\circ}\) are respectively -100, undefined, and 100.

Step 1: Understanding the Tangent Function

The basic tangent function, \(y = tan(x)\), has a period of \(\pi\) radians (or 180 degrees) and its range is all real numbers, with asymptotes (undefined values) at \(\pi/2 + n\pi\) for integers n. Here we are working with \(y = -100tan(x)\), which is just a vertically stretched version of the basic tangent function, this time with the range being all real numbers multiplied by -100.

Step 2: Graphing the Tangent Function

Now we can graph \(y = -100tan(x)\) over the interval \(0^{\circ} < x < 470^{\circ}\) and \(-300 < y < 300\). We first draw the basic tangent curves at \(x = 180^{\circ} \times n\) for \(0 < n < 3\), each curve approaching asymptotes at \(x = (90 + 180 \times n)^{\circ}\). We then scale these curves vertically by a factor of -100.

Step 3: Evaluating the Function at Specific Points

We also have to evaluate the function at \(x=45^{\circ}, 90^{\circ}, 135^{\circ}\). At \(x = 45^{\circ}\), \(y = -100 \times tan(45^{\circ}) = -100\). At \(x = 90^{\circ}\), \(y = -100 \times tan(90^{\circ})\) is undefined because tan(90 degrees) is undefined. At \(x = 135^{\circ}\), \(y = -100 \times tan(135^{\circ}) = 100\) (because tan(135) = -1).

Key Concepts

Tangent Function

The tangent function, represented as \(y = \tan(x)\), is a fundamental concept in trigonometry. It is one of the six primary trigonometric functions and is unique due to its specific behavior and properties. Unlike sine and cosine, which have a limited range of values between -1 and 1, the tangent function's range is all real numbers. This means it can take any value from negative infinity to positive infinity.

• **Characteristics**: The tangent function has an interesting behavior - it repeats every 180 degrees (or \(\pi\) radians in the unit circle).
• **Basic Shape**: The graph of the tangent function has a series of "pipes" or vertical patterns, with each section increasing from negative to positive infinity within each period.

In our particular exercise, the function is transformed to \(y = -100\tan(x)\), meaning the amplitude is scaled by -100. This flips the graph vertically and stretches it, but the periodic nature of the tangent function remains.

Asymptotes

Asymptotes are integral in understanding the behavior of the tangent function's graph. For the basic tangent function \(y = \tan(x)\), asymptotes occur where the function is undefined. As you approach these undefined points, the function's value increases or decreases without bound.

• **Locations**: The asymptotes for \(\tan(x)\) occur at \(x = \frac{(2n+1)\pi}{2}\), where \(n\) is an integer. In degrees, that's at every \(90^{\circ} + 180^{\circ}n\).
• **Behavior**: At these points, the graph of the tangent function seems to shoot up or down infinitely.

In the exercise with \(y = -100\tan(x)\), the positions of these asymptotes remain the same, but the way the graph approaches them can appear more extreme due to the vertical stretching.

Function Evaluation

Evaluating a function at specific points allows us to pinpoint certain values on its graph, providing a clearer understanding of how it behaves. For the function \(y = -100\tan(x)\), we evaluate at three important angles - \(x = 45^{\circ}\), \(90^{\circ}\), and \(135^{\circ}\).

• **At \(45^{\circ}\)**: Since \(\tan(45^{\circ}) = 1\), the value for \(y = -100\times 1 = -100\).
• **At \(90^{\circ}\)**: The tangent function is undefined here because you would be dividing by zero, hence \(y\) does not exist.
• **At \(135^{\circ}\)**: With \(\tan(135^{\circ}) = -1\), the value for \(y = -100 \times (-1) = 100\).

These specific evaluations reveal both the vertical stretching and flipping of the basic tangent graph.

Periodicity of Trigonometric Functions

The concept of periodicity is essential to understanding trigonometric functions, particularly the tangent function. A function is periodic if it repeats its values in regular intervals, or periods. This characteristic is very apparent in trigonometric functions.

• **Tangent's Period**: For the function \(y = \tan(x)\), the period is \(180^{\circ}\) or \(\pi\) radians. This means the function's graph repeats every 180 degrees.
• **Importance**: Recognizing the periodic nature helps in predicting values over large domains without graphing every piece. It simplifies calculating function values and understanding graphs.

In our example, the function \(y = -100\tan(x)\) retains the same period of 180 degrees despite the transformation. The frequent repetition of the tangent's vertical patterns across the interval from \(0^{\circ} < x < 470^{\circ}\) showcases this periodic nature, making it predictable and manageable.