Question

Graph the function \(f(x)=\sin 50 x\) using the window given by a \(y\) range of \(-1.5 \leq y \leq 1.5\) and the \(x\) range given by (a) \([-15,15]\) (b) \([-10,10]\) (c) \([-8,8]\) (d) \([-1,1]\) (e) \([-0.25,0.25]\) Indicate briefly which \(x\) -window shows the true behavior of the function, and discuss reasons why the other \(x\) -windows give results that look different.

Short answer

The x-range \([-0.25, 0.25]\) shows the true behavior of \(f(x) = \sin(50x)\) due to its high frequency. Larger ranges blur the oscillations.

Step-by-step solution

Understand the Sine Function

The function to be graphed is \(f(x) = \sin(50x)\). The sine function oscillates between -1 and 1, and the factor 50 compresses the oscillations horizontally, meaning it has a very high frequency.

Graph Configuration for y-range

Configure the graph to show the y-range from -1.5 to 1.5. This y-range is slightly larger than the range of the sine wave (-1 to 1), providing space to visualize the oscillations clearly.

Graphing with Given x-range (a)

For \(x\) range \([-15, 15]\), graph \(f(x) = \sin(50x)\). This results in very tight oscillations due to the compression factor, making individual cycles indistinguishable.

Graphing with Given x-range (b)

For \(x\) range \([-10, 10]\), graph \(f(x) = \sin(50x)\). Similar to the previous case, the high frequency remains, and individual oscillations remain indistinct.

Graphing with Given x-range (c)

For \(x\) range \([-8, 8]\), plot \(f(x) = \sin(50x)\). There will still be rapid oscillations, but this range slightly improves visibility compared to wider ranges.

Graphing with x-range (d) for Better Resolution

For \(x\) range \([-1, 1]\), graph \(f(x) = \sin(50x)\). The oscillations cover more of the window, somewhat showing the nature of the rapid oscillations.

Graphing with x-range (e) for True Behavior

For \(x\) range \([-0.25, 0.25]\), graph \(f(x) = \sin(50x)\). This range helps in showing a clearer oscillatory pattern as the cycles are more visible, reflecting the true behavior of the sine function with high frequency.

Evaluate the Graph Results

The \(x\) range \([-0.25, 0.25]\) best shows the true behavior of \(f(x) = \sin(50x)\) as it provides the scale necessary to view the rapid oscillations clearly. Larger \(x\) ranges make the oscillations appear as a blur due to the high frequency.

Key concepts

Sine Function

The sine function is a fundamental concept in trigonometry, often denoted as \( \sin(x) \). This function maps input values, which are angles in radians, to outputs between -1 and 1. The basic form of the sine function is periodic, meaning it repeats its pattern in regular intervals.
One of its most notable characteristics is its wave-like oscillation within the interval of \(-1\) to \(+1\). This is represented mathematically as: \( f(x) = \sin(x) \)

• The sine function naturally repeats every \(2\pi\), known as its period.
• This periodic behavior makes it an excellent model for various wave phenomena, such as sound or light waves.
• The function transitions smoothly from one value to another with a curve, unlike linear functions.

Applications of the sine function extend throughout physics and engineering, especially in systems involving harmonic motion and oscillations.

Oscillations

Oscillations describe any form of repetitive variation over time, typically in a pattern that can be regular or irregular.
In our function \( f(x) = \sin(50x) \), oscillations occur due to the sine function’s nature to cycle through its values continuously and smoothly within a confined range.
\(\frac{\pi}{2}, 3\frac{\pi}{2} \)The modifying factor, in this case, is the '50', which affects the oscillation frequency, resulting in compressed and dense cycles compared to the standard sine curve.

• Regular oscillations: Sine function terms often appear as steady, predictable oscillations, useful for modeling repetitive events.
• The mathematical expression involves adjusting parameters to achieve different kinds of oscillation patterns.
• Increased frequencies lead to tighter, more rapid oscillations as experienced in \( f(x) = \sin(50x) \).

Recognizing the type and frequency of oscillations are crucial in diagnosing and predicting system behaviors in natural and engineered systems.

Frequency

Frequency refers to the number of cycles a wave completes within a unit interval, indicating how often the function repeats its pattern.
In trigonometric functions like \(\sin(x)\), frequency is typically modified by the coefficient in front of the variable, such as in \(\sin(50x)\).

• In \(f(x) = \sin(50x)\), the 50 denotes that the sine wave oscillates 50 times more frequently than \(\sin(x)\). The period of the function, therefore, becomes \(\frac{2\pi}{50}\).
• Higher frequency leads to a compressed wave, showcasing many oscillations over a short \(x\) interval.
• Frequency is inversely proportional to the period; hence higher frequency means a shorter period.

In practical contexts, frequency is fundamental for understanding wave properties, whether it's sound waves, light frequencies, or oscillation patterns in electrical circuits.

Graphing Techniques

Graphing techniques involve selecting appropriate scales and ranges to accurately depict functions and their characteristics.
For functions with high frequencies such as \( f(x) = \sin(50x) \), graphing requires special attention to axis scales in order to capture the essence of the function.

• Choosing an appropriate \(x\) range is vital. For \( f(x) = \sin(50x)\), a smaller \(x\) window like \([-0.25, 0.25]\) unveils the detail of oscillations.
• The \(y\) range should encompass the full amplitude of oscillations, here set from \(-1.5\) to \(1.5\), slightly more than necessary to ensure clarity.
• Graphing large \(x\) ranges, such as \([-15, 15]\), causes high-density oscillations to appear as a blur, masking the wave-like properties of the function.

Mastering graphing techniques allows clearer visual analysis, making it easier to interpret various kinds of data facets or functions.