Problem 23 - Calculus with Concepts in Calculus

Question

Sketch the graph of the function. Indicate any intercepts and symmetry, and determine whether the function is even, odd, or neither. $$ \sin 2 x $$

Step-by-Step Solution

Answer

The function \( \sin(2x) \) is odd, periodic with period \( \pi \), and has x-intercepts at \( x = \frac{n\pi}{2} \).

Step 1: Identify the Parent Function

The given function to sketch is \( \, f(x) = \, \sin(2x) \, \). The parent function here is \( \, \sin(x) \, \), which is a periodic function with a period of \( \, 2\pi \, \). It has known key properties, such as being symmetric about the origin, with intercepts at integer multiples of \( \, \pi \, \).

Step 2: Determine Period Changes

The period of \( \, \sin(2x) \, \) is determined by the factor of 2 in front of \( x \,\). The formula for the period of \( \sin(kx) \) is \( \frac{2\pi}{k} \, \). For \( \, k = 2 \, \), the period is \( \frac{2\pi}{2} = \pi \, \). This means the function will complete one full cycle of sine every \( \, \pi \, \) instead of \( \, 2\pi \, \).

Step 3: Identify Intercepts

The x-intercepts of \( \sin(2x) \) occur when \( \, \, \sin(2x) = 0 \, \), i.e., when \( \, 2x = n\pi \, \) for integer \( \, n \,\). Thus, \( x = \frac{n\pi}{2} \, \). The origin, \( (0, 0) \, \), is one of these intercepts. This function does not have any y-intercepts beyond the origin.

Step 4: Determine Symmetry and Check for Even or Odd Properties

For symmetry and determining if the function is even, odd, or neither: Calculate \( \, f(-x) = \, \sin(-2x) = -\sin(2x) \, \). Since \( (-x) = -f(x) \, \), the function is odd and symmetric with respect to the origin. This means its graph is rotationally symmetrical about the origin.

Step 5: Graph the Function

To sketch \( \, \sin(2x) \, \), consider the following: Its period is \( \pi \, \), which means the function will repeat every \( \pi \, \). The x-intercepts occur at multiples of \( \frac{\pi}{2} \, \). Sketch a sine curve that crosses the x-axis at these intercepts, peaking at \( \frac{\pi}{4} \, \), and reaching a trough at \( \frac{3\pi}{4} \, \), and then repeat the pattern.

Key Concepts

Sine Function

The sine function, denoted as \( \sin(x) \), is one of the fundamental trigonometric functions used to model periodic phenomena. Its graph is characteristic for its wave-like appearance, known as a sinusoidal wave. The sine function originates from the unit circle, representing the y-coordinate of a point on the circle corresponding to a given angle \( x \).

The basic sine function has several important properties:

• It oscillates between -1 and 1, providing values of 0 at integer multiples of \( \pi \).
• Its period is \( 2\pi \), meaning the function repeats its pattern every \( 2\pi \) units.
• It is an odd function, meaning its graph is symmetric about the origin. We will explore this further in subsequent sections.

Overall, understanding the behavior of the parent function \( \sin(x) \) is crucial when analyzing more complex variations, such as \( \sin(2x) \). This version alters the period, allowing the wave-like pattern to complete a cycle in a shorter interval, specifically \( \pi \) in this case.

Graph Symmetry

Graph symmetry refers to how a graph reflects or rotates around certain points or axes. It helps determine how a trigonometric function behaves visually. In the context of the sine function, understanding symmetry is key in predicting its pattern over different intervals.

For the function \( \sin(x) \):

• The graph is symmetric with respect to the origin. This means if you spin the graph around the origin, it looks the same. This rotational symmetry is indicative of odd functions.
• If you look at a graph over one period from 0 to \( 2\pi \), you will notice a symmetric wave pattern that repeats over subsequent intervals.

In our specific example with \( \sin(2x) \), the symmetry is maintained, but the graph completes its symmetric cycle faster due to a reduced period — emphasizing the symmetry more frequently across the x-axis.

Even and Odd Functions

Even and odd functions have unique symmetry properties that can make them easier to analyze on a graph. An even function is symmetric with respect to the y-axis, whereas an odd function has rotational symmetry about the origin.

To determine if a function is even, check if \( f(-x) = f(x) \). If this is true for all \( x \), then the function is even. For odd functions, \( f(-x) = -f(x) \) must hold true.

In our case, for \( \sin(2x) \), evaluating the function at negative x-values gives \( \sin(-2x) = -\sin(2x) \). This verifies the odd nature of the function as it satisfies \( f(-x) = -f(x) \). This odd property is embodied in the graph by its rotational symmetry through the origin, producing a reflective effect mirrored along both axes.

Periodic Function

A periodic function is one that repeats its values at regular intervals or periods. The sine function is a classic example of this repeating behavior, making it essential in many applications of mathematics and physics related to wave patterns.

The standard sine function \( \sin(x) \) repeats every \( 2\pi \), but changes in its argument, such as \( \sin(2x) \), modify the period. Specifically, the period of \( \sin(kx) \) is given by \( \frac{2\pi}{k} \).

For \( \sin(2x) \), the period becomes \( \pi \). This signifies that the function completes a full cycle in half the interval of the parent function. The x-axis intercepts are also adjusted, now occurring every \( \frac{\pi}{2} \). This periodic property underscores the function's ability to exhibit similar oscillatory behavior, yet confined within a different spatial interval, effectively doubling the number of cycles in a standard range for the parent function.