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Chapter 10: Problem 6

Sketch the curves of the given functions by addition of ordinates. $$y=\frac{1}{4} x^{2}+\cos 3 x$$

Short Answer

Expert verified
Sketch separate curves for \(\frac{1}{4}x^2\) and \(\cos 3x\), add their ordinates point-by-point, and connect these points smoothly to get the final curve.

Step by step solution

01

Understand the Components

The given function is the sum of two separate functions: a quadratic function \(y_1 = \frac{1}{4}x^2\) and a trigonometric function \(y_2 = \cos 3x\). We will analyze these components separately to understand their individual behavior.
02

Analyze the Quadratic Function

The function \(y_1 = \frac{1}{4}x^2\) is a parabola that opens upwards with its vertex at the origin \((0,0)\). The coefficient \(\frac{1}{4}\) makes it wider than the standard parabola \(y = x^2\). We sketch a smooth curve passing through the origin and curving upwards in both directions.
03

Analyze the Trigonometric Function

The function \(y_2 = \cos 3x\) represents a cosine wave. The period of this cosine function is \(\frac{2\pi}{3}\) because the period of \(\cos x\) is \(2\pi\) and the multiplier \(3\) compresses the wave horizontally. This wave oscillates between -1 and 1. Sketch multiple periods of this wave to the same x-axis.
04

Add the Ordinates

To sketch the combined function \(y = \frac{1}{4}x^2 + \cos 3x\), add the ordinates (y-values) of \(y_1\) and \(y_2\) for various x-values. Select some critical points such as peaks, troughs, and zero crossings of \(\cos 3x\), and calculate the corresponding y-values by summing the ordinates of \(y_1\) and \(y_2\). This will help in getting the shape of the combined curve.
05

Sketch the Combined Curve

Using the calculated points from Step 4, sketch the final curve. Begin by plotting a few key points where the ordinance addition was performed, and ensure these points are consistent with the expected behavior of both \(y_1\) and \(y_2\). Connect these points smoothly to reflect the continuous nature of both functions. The resulting curve combines the upward-opening trend of the parabola with the oscillations of the cosine wave.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Functions
Quadratic functions are a type of polynomial function where the highest exponent of the variable is 2. The general form of a quadratic function is: \[ y = ax^2 + bx + c \]Here, \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). These functions graph as parabolas, which are symmetrical, U-shaped curves.
  • The vertex of the parabola is the highest or lowest point. For the function \(y = \frac{1}{4}x^2\), the vertex is at the origin, \((0,0)\).
  • If \(a > 0\), the parabola opens upward. If \(a < 0\), it opens downward. This gives our parabola a gentle upward opening since \(a = \frac{1}{4}\), making it wider than \(y = x^2\).
  • The axis of symmetry is a vertical line that passes through the vertex, given by \(x = -\frac{b}{2a}\), and in standard form, it's just \(x= 0\).
Understanding these characteristics is essential for graphing quadratic functions as it lays the foundation for more complex curve sketching tasks.
Trigonometric Functions
Trigonometric functions like sine and cosine are fundamental in mathematics for modeling periodic phenomena, such as sound waves or the oscillations in the given problem. The function \[ y_2 = \cos(3x) \]is a cosine function with a horizontal compression caused by the multiplier of 3.
  • The basic cosine function, \(\cos x\), has a period of \(2\pi\), which is the distance required to complete one full cycle.
  • By adding a number to \(x\) within the cosine argument, the function's graph is horizontally compressed, changing the period to \(\frac{2\pi}{3}\).
  • The amplitude (height of the wave) remains 1, meaning that the wave oscillates between 1 and -1.
  • The cosine wave has peaks (maximum points) and troughs (minimum points) that occur at regular intervals.
Understanding how trigonometric functions work helps in plotting their graphs accurately, especially when combined with other functions as in this exercise.
Addition of Ordinates
The addition of ordinates is a method used to sum the y-values of multiple functions at specific x-values. This technique is critical when dealing with compounded functions, like in the given exercise.
  • For any x-value, calculate the y-value for each function separately. In our task, these are the quadratic \(y_1\) and the trigonometric \(y_2\) functions.
  • Add these y-values together to get the ordinate for the combined function, \(y = \frac{1}{4}x^2 + \cos(3x)\).
  • Selecting key x-values, such as where the cosine function peaks or crosses the x-axis, helps in effectively applying this method.
  • This approach allows students to visualize how combined functions behave, granting insight into more complex graphing processes.
By patiently adding these ordinates, students can see how the graph's shape emerges from the interactions of different function types.
Curve Sketching
Curve sketching involves drawing a graph of a function based on its important characteristics without the use of plotting software or calculators. This exercise combines both algebraic understanding and visualization.
  • Begin by analyzing and plotting the individual functions separately. Use their key features, such as vertex or amplitude.
  • Through addition of ordinates, calculate key points for those values of x which are visually significant, such as peaks and crossings.
  • Plot these calculated points onto a graph to provide a guideline for sketching the curve.
  • Use these points to draw a smooth curve, ensuring that the transitions between calculated points reflect the physics and math of the function.
By practicing curve sketching, students develop better intuition for how function parameters affect graph shapes, which is useful in more advanced mathematical studies.

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Most popular questions from this chapter

Use the trace feature of a calculator to find the value of a for either \(y=a \sin x\) or \(y=a \cos x,\) whichever is correct such that the given point is on the graph. The amplitude of each function is \(2.50 .\) Thereby determine the function. (All points are located such that the \(x\) value is between \(-\pi\) and \(\pi\).) $$(0.67,-1.55)$$

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