Problem 61 - Precalculus Mathematics for Calculus

Question

Graph the three functions on a common screen. How are the graphs related? $$y=x^{2}, \quad y=-x^{2}, \quad y=x^{2} \sin x$$

Step-by-Step Solution

Answer

Each graph is parabolic, modified by sine in the oscillatory graph.

Step 1: Plot the graph of y = x^2

To graph the function \(y = x^2\), start by plotting several points, such as \((-2, 4)\), \((-1, 1)\), \((0, 0)\), \((1, 1)\), and \((2, 4)\). Connect these points with a smooth, upward-opening parabola centered at the origin.

Step 2: Plot the graph of y = -x^2

Next, graph the function \(y = -x^2\) by using points like \((-2, -4)\), \((-1, -1)\), \((0, 0)\), \((1, -1)\), and \((2, -4)\). Connect these points with a smooth, downward-opening parabola also centered at the origin.

Step 3: Plot the graph of y = x^2 sin x

Finally, graph the function \(y = x^2 \sin x\). This function combines a parabolic shape with the oscillatory behavior of the sine function. Use points like \((0,0)\), when \(x = \pi \approx 3.14\) it becomes close to zero due to sine, and check other oscillation points. The graph will look like a wavy pattern superimposed on a parabolic shape.

Step 4: Analyze the relationship among the graphs

All three graphs are parabolic but differ in their orientation and modification by the sine function. \(y = x^2\) and \(y = -x^2\) are standard parabolas, one opening upwards and the other downwards. \(y = x^2 \sin x\) oscillates between parabolas due to the sine factor.

Key Concepts

Parabola

A parabola is one of the most common types of curve found in mathematics. It is a symmetrical open plane curve which is formed by the graph of a quadratic function. In its general form, a parabola can be represented by the equation \(y = ax^2 + bx + c\).

The key features of parabolas to remember include:

• Vertex: This is the highest or lowest point on the graph, depending on the parabola's orientation.
• Axis of Symmetry: A vertical line that passes through the vertex, given by the equation \(x = -\frac{b}{2a}\).
• Direction: If \(a > 0\), the parabola opens upwards. If \(a < 0\), it opens downwards.
• Focus and Directrix: These describe the parabola in terms of its geometrical properties rather than its equation.

In the original problem, the functions \(y = x^2\) and \(y = -x^2\) are examples of parabolas. The first one opens upwards, like a "U"; the second one opens downwards, like an upside-down "U". Both are centered at the origin where the vertex is located.

Sine Function

The sine function is a fundamental periodic function that is widely used in trigonometry. It describes the oscillation phenomena, which are repetitive over a fixed period. Mathematically, it is expressed as \(y = \sin x\).

The properties to note about the sine function include:

• Period: The sine function repeats every \(2\pi\) radians, meaning every \(360^\circ\).
• Amplitude: The maximum value of \(\sin x\) is 1, and the minimum is -1.
• Zeroes: The function crosses the x-axis at multiples of \(\pi\), such as \(0, \pi, 2\pi,\) etc.
• Symmetry: It is an odd function with rotational symmetry about the origin.

In the context of the given exercise, the presence of the sine function in \(y = x^2 \sin x\) causes the graph to exhibit oscillatory behavior, but its range and amplitude differ due to multiplying by \(x^2\). This added multiplication modifies its standard wave pattern, creating peaks and troughs that vary in size as \(x\) changes.

Oscillation

Oscillation refers to any motion that repeats itself over certain intervals, such as the back and forth swing of a pendulum or the fluctuations seen in a sine wave. In mathematics, oscillation is typically encountered in the context of trigonometric functions like sine and cosine.

For example, the sine function itself illustrates a simple harmonic oscillation with its regularly repeating up and down pattern. The main characteristics of oscillatory motion to understand are:

• Frequency: The number of oscillations per unit time.
• Amplitude: The maximum extent of the oscillation measured from the equilibrium position.
• Phase Shift: A horizontal shift of the wave; if applicable, it indicates where the cycle begins.
• Period: The time taken for one complete cycle of oscillation to occur, often \(2\pi\) for the sine wave.

In the function \(y = x^2 \sin x\), the oscillation effect becomes stretched, combined with the quadratic term \(x^2\). The peaks of the graph can extend higher as \(x^2\) increases with both negative and positive x-values, adding a parabolic growth factor to the oscillation. This results in a sine wave that appears enveloped by a parabolic shape, creating a unique graphical behavior.