Problem 67 - Precalculus

Question

In Exercises \(67-94,\) add the ordinates of the individual functions to graph each summed function on the indicated interval. $$y=2 x-\cos (\pi x), 0 \leq x \leq 4$$

Step-by-Step Solution

Answer

Graph the function \(y = 2x - \cos(\pi x)\) over the interval \([0, 4]\) by adding both ordinates.

Step 1: Understanding the Problem

The exercise asks us to graph the sum of ordinates of two functions: \(y = 2x\) and \(y = -\cos(\pi x)\). We need to find the combined function and then graph it over the interval \([0, 4]\).

Step 2: Define the Summed Function

To find the summed function, add the ordinates of the two given functions: \(y = 2x\) and \(y = -\cos(\pi x)\). The resulting function is \(y = 2x - \cos(\pi x)\).

Step 3: Create a Table of Values

Choose values for \(x\) within the interval \([0, 4]\) and calculate corresponding \(y\) values. For example, if \(x = 0\), \(y = 2 \times 0 - \cos(0) = -1\). Repeat this for several values of \(x\) (e.g., \(x = 1, 2, 3, 4\)).

Step 4: Plot the Points

Using the table from Step 3, plot each \(x\) and \(y\) coordinate pair on a graph. For instance, a point would be (0, -1) from our example calculation.

Step 5: Draw the Graph of the Summed Function

Connect the plotted points with a smooth curve to form the graph of the summed function \(y = 2x - \cos(\pi x)\). Ensure the graph covers the entire interval \([0, 4]\).

Key Concepts

Understanding Ordinates

In mathematics, the term "ordinates" refers to the vertical values in a coordinate system. When you plot a point on a graph, it typically consists of an x-coordinate and a y-coordinate, where the y-coordinate is known as the ordinate.
Let's say you have the point (x, y). Here, x is the abscissa, which is the horizontal distance from the origin, and y is the ordinate, the vertical distance from the origin.
In the context of function graphing, ordinates are essential because they help determine the placement of a point in relation to the x-axis. By plotting points using their ordinates, you create a visual representation of a function's behavior.

Exploring the Summed Function

When given several functions, particularly in mathematics problems, you may need to combine them by adding their corresponding ordinates. This process leads us to what is known as a "summed function."
A summed function is simply one function created by adding up the ordinates of multiple individual functions. In our problem, we have two functions: \(y = 2x\) and \(y = -\cos(\pi x)\). The summed function integrates these functions into one, giving us \(y = 2x - \cos(\pi x)\).
This combined function encapsulates the characteristics of both original functions. By graphing the summed function, it becomes easy to see how each original function contributes to the behavior of the whole expression.
Understanding how to build and analyze summed functions enables you to tackle complex mathematical problems where multiple aspects need to be represented at once.

Defining and Using Intervals

Intervals are extremely helpful in defining the domain over which functions—or in this case, summed functions—are considered. An interval specifies a particular range of x-values where you must evaluate and graph your function.
In the problem given, we have a designated interval \([0, 4]\). This means the function needs to be evaluated and depicted graphically from \(x = 0\) to \(x = 4\).
Working within a specified interval assists in focusing the analysis on a relevant section of the function graph, providing a clear window into its behavior between two points. With function graphing, it's crucial to respect these intervals to ensure completeness and accuracy in depiction.