Problem 19 - 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 \(135^{\circ} .\) $$ y=50 \tan x $$

Step-by-Step Solution

Answer

The function \(y=50 \tan x\) on the interval \(0^{\circ} < x < 470^{\circ}\) and \(-300 < y < 300\) creates a periodic graph similar to the regular tangent function, but stretched vertically by a factor of 50. Evaluating the function at \(x=45^{\circ}\), \(90^{\circ}\), and \(135^{\circ}\), give values of \(y = 50\), undefined, and \(y = -50\) respectively.

Step 1: Sketching the regular function

Start by sketching the function \(y=\tan x\) without the scaling factor for the given interval. The tangent function is periodic with a period of \(180^{\circ}\), and has vertical asymptotes every \(180^{\circ}\).

Step 2: Applying the scaling factor

Now, apply the scaling factor by multiplying the outputs (y-values) of the tangent function by 50. The vertical asymptotes remain the same, but the maximum and minimum y-values within each period change. The function \(y=50 \tan x\) will reach higher peaks and lower valleys compared to the regular tangent function.

Step 3: Evaluating the function at given points

Now, evaluate the function at \(x=45^{\circ}\), \(90^{\circ}\), and \(135^{\circ}\). For \(tan 45^{\circ}\), the function gives a result of 1, so after scaling by 50, we get \(y = 50\). For \(tan 90^{\circ}\), the tangent function is undefined, so \(y\) is undefined for this input. For \(tan 135^{\circ}\), the tangent function gives a result of -1, so after scaling by 50, we get \(y = -50\).

Step 4: Update the graph with evaluations

Finally, update the graph by marking the evaluated points. This will complete the graph for the function \(y=50 \tan x\) over the given interval.

Key Concepts

Tangent Function

The tangent function, denoted as \(y = \tan x\), represents a basic yet intriguing trigonometric function. Unlike sine or cosine, the tangent function starts afresh every \(180^{\circ}\).

It is defined as the ratio of the sine and cosine functions: \(\tan x = \frac{\sin x}{\cos x}\). This definition implies that whenever \(\cos x = 0\), the tangent is undefined, resulting in vertical asymptotes. The graph of \(\tan x\) repeatedly climbs towards infinity as it nears these undefined points.

As you graph \(\tan x\), you'll notice a wave-like shape due to these asymptotes, making it distinctly different from other trigonometric graphs.

Periodic Functions

Trigonometric functions like the tangent function are periodic, meaning they repeat their values in a regular pattern over intervals. For the tangent function, this period is \(180^{\circ}\).

This periodicity means that every \(180^{\circ}\), the function begins a new cycle, producing exactly the same set of values again.

• The basic cycle of \(\tan x\) spans from \(-90^{\circ}\) to \(90^{\circ}\).
• This repetition results from the periodic nature of the sine and cosine components.

For the function \(y = 50 \tan x\), this periodic cycle doesn't change, allowing us to predict the function's behavior over any interval.

Vertical Asymptotes

Vertical asymptotes are invisible lines that the graph of a function approaches but never touches. In the tangent function, these asymptotes occur where \(\cos x = 0\) because \(\tan x\) becomes undefined.

As we graph \(y = 50 \tan x\):

• Vertical asymptotes appear every \(180^{\circ}\) at \(90^{\circ}, 270^{\circ}, \) and so on.
• The function stretches infinitely towards positive or negative infinity near these points.

These lines are key to understanding the graph since they mark the boundaries of each tangent cycle.

Scaling Factor

A scaling factor alters how "tall" or "short" the function's graph appears. It's simply a multiplier for the function's output (y-values). In \(y = 50 \tan x\), the scaling factor is 50.

This means:

• Each y-value from the base function \(y = \tan x\) is multiplied by 50.
• The graph becomes stretched vertically, leading to higher peaks and deeper troughs.

The vertical asymptotes remain unchanged, but the amplitude of the function dramatically increases. Such scaling doesn't affect the function's period or the location of asymptotes, only how intense the graph's fluctuations appear.