Problem 73 - Precalculus

Question

In Exercises \(67-94,\) add the ordinates of the individual functions to graph each summed function on the indicated interval. $$y=\frac{1}{4} x-\frac{1}{2} \cos [\pi(x-1)], 2 \leq x \leq 6$$

Step-by-Step Solution

Answer

The summed function is graphed by adding the ordinates of both functions over the interval [2, 6].

Step 1: Understand the Functions

The exercise involves two functions: a linear function \(y_1 = \frac{1}{4} x\) and a trigonometric function \(y_2 = -\frac{1}{2} \cos[\pi(x-1)]\). We will add these functions to find their sum over the interval \(2 \leq x \leq 6\).

Step 2: Determine the Sum Function

The sum function \(y = \frac{1}{4} x - \frac{1}{2} \cos[\pi(x-1)]\) is obtained by adding the functions \(y_1 = \frac{1}{4} x\) and \(y_2 = -\frac{1}{2} \cos[\pi(x-1)]\).

Step 3: Calculate Ordinates at Specific Points

Choose specific values of \(x\) within the interval and calculate the ordinates for both functions and their sum. For example, compute \(y\) for \(x = 2\), \(x = 4\), and \(x = 6\).

Step 4: Evaluate at x = 2

For \(x = 2\), calculate: \(y_1 = \frac{1}{4} \times 2 = \frac{1}{2}\) and \(y_2 = -\frac{1}{2} \cos[\pi \times (2-1)] = -\frac{1}{2} \cos[\pi] = \frac{1}{2}\). Sum: \(y = \frac{1}{2} + \frac{1}{2} = 1\).

Step 5: Evaluate at x = 4

For \(x = 4\), compute: \(y_1 = \frac{1}{4} \times 4 = 1\) and \(y_2 = -\frac{1}{2} \cos[\pi \times (4-1)] = -\frac{1}{2} \cos[3\pi] = \frac{1}{2}\). Sum: \(y = 1 + \frac{1}{2} = 1.5\).

Step 6: Evaluate at x = 6

For \(x = 6\), compute: \(y_1 = \frac{1}{4} \times 6 = 1.5\) and \(y_2 = -\frac{1}{2} \cos[\pi \times (6-1)] = -\frac{1}{2} \cos[5\pi] = \frac{1}{2}\). Sum: \(y = 1.5 + \frac{1}{2} = 2\).

Step 7: Graph the Sum Function

Plot the points calculated in the steps above onto the graph: (2, 1), (4, 1.5), and (6, 2). Connect these points to visualize the summed function over the interval.

Key Concepts

Linear Functions

Linear functions are among the simplest types of functions in mathematics.
They are characterized by the equation of the form

• \(y = mx + c\)

where \(m\) represents the slope of the line and \(c\) the y-intercept.
In a linear function, any change in \(x\) results in a constant change in \(y\), which makes the graph of such functions a straight line.

Understanding how linear functions work helps us predict patterns and make projections. The function we have, \(y_1 = \frac{1}{4}x\), represents a line with a gentle slope of \(\frac{1}{4}\).
This means for every unit increase in \(x\), \(y\) increases by \(\frac{1}{4}\). This concept is used often to describe relationships that are directly proportional.
Key properties of linear functions include:

• They have constant rates of change.
• Their graph is always a straight line.
• They are easy to manipulate algebraically.

Understanding these basics is crucial when graphing and interpreting linear relationships.

Trigonometric Functions

Trigonometric functions are based on the relationships found in right triangles and the unit circle.
These functions repeat values in cycles and are fundamental in studying periodic phenomena.
Common trigonometric functions include sine, cosine, and tangent.

In our problem, we encounter a cosine function:

• \(y_2 = -\frac{1}{2} \cos[\pi(x-1)]\).

The cosine function determines the x-coordinate on the unit circle.
It has a periodic nature, meaning it repeats its values at regular intervals.
The "\(-\)" sign in front of our function reflects the cosine curve over the x-axis. Key points to understand about trigonometric functions:

• They have a specific amplitude and period, which determine their height and frequency of cycles.
• The argument of \(\cos\) in our expression is shifted by 1 unit on the x-axis due to \(\pi(x-1)\).
• Trigonometric functions are useful in modeling oscillating systems like sound waves or tides.

By understanding how these functions work, students can interpret and predict various cyclic behaviors in different contexts.

Function Graphing

Graphing functions involves plotting points on a coordinate plane to visualize the behavior of different equations. It provides a visual representation of how a function changes with respect to its variables.When graphing, it's essential to choose specific values of \(x\) to compute corresponding \(y\) values, known as ordinates.
In the given exercise, you graph the composite function

• \(y = \frac{1}{4}x - \frac{1}{2}\cos[\pi(x-1)]\)

by evaluating it at several critical points: \(x = 2, 4, 6\). Steps to graph functions effectively:

• Calculate values at crucial points to understand the function's behavior across relevant intervals.
• Plot these coordinates accurately on a grid.
• Connect plotted points with smooth lines that follow the calculated patterns.

By following these simple steps, students can quickly understand the overall trend of the function they are graphing.Graphing not only helps in visualizing equations but also assists in analyzing and predicting outcomes in practical scenarios.