Problem 15 - Algebra 1: Concepts and Skills

Question

The distance \(d\) (in miles) that sound travels in air in time \(t\) (in seconds) is represented by the function \(d=0.2 t .\) Make a table of the input \(t\) and the output \(d .\) Use \(t\) values of \(0,5,10,15,20,25,\) and \(30 .\) Use your table to help you draw the graph of the function.

Step-by-Step Solution

Answer

The table of values from the function \(d = 0.2t\) is \((0, 0), (5, 1), (10, 2), (15, 3), (20, 4), (25, 5), (30, 6)\). The graph of this function is a straight line passing through these points.

Step 1: Define the Function

Let's first illustrate the function. The function is \(d = 0.2t\), where \(d\) is the distance sound travels in air in miles and \(t\) is the time in seconds.

Step 2: Create the Table

Given the time values \(0, 5, 10, 15, 20, 25,\) and \(30\), we will substitute each value into the function to find the corresponding distance. This will yield the following pairs: \((0, 0), (5, 1), (10, 2), (15, 3), (20, 4), (25, 5), (30, 6)\).

Step 3: Draw the Graph

Now we will plot these pairs of points on a graph. The time (t) will be plotted along the x axis, and the distance (d) will be plotted along the y axis. Connect the points to display the linear function.

Key Concepts

Function Representation

Understanding how to represent a function is crucial in mathematics, especially when it comes to graphing. A function establishes a relationship between two variables, typically an input and an output. In the given exercise, the function provided is d = 0.2t, which represents the distance d that sound travels in air, in miles, as a function of time t, measured in seconds. This function is linear, indicated by the constant rate of change, or slope, which is 0.2. This means for every second of time, the distance increases by 0.2 miles.

In mathematical terms, t is the independent variable, and d is the dependent variable. This relationship shows that the value of d depends on the value assigned to t. When representing this function graphically, the independent variable is placed on the x-axis, while the dependent variable is plotted on the y-axis. By using a range of t values, we can calculate their corresponding d values to create a visual representation of the function on a graph.

Creating a Table for a Function

Creating a table is a fundamental step in the process of graphing a function. It sets up a clear visualization of how the input affects the output. For our function d = 0.2t, a table organizes the relationship between time t and distance d. Begin by selecting a series of values for the time t, which serves as the independent variable. In this scenario, we use t values of 0, 5, 10, 15, 20, 25, and 30 seconds.

Next, we apply each of these t values to the function to calculate the corresponding distance. The function creates ordered pairs (t, d), which can easily be depicted in a two-column table. On the left, list the time values, and on the right, write down the calculated distance for each time value. This table not only aids in understanding the function's behavior but also forms the basis for plotting these points on the graph.

Plotting Points on a Graph

With the table of values ready from our function d = 0.2t, the next step is plotting these values on a coordinate graph. Each ordered pair from the table, such as (0, 0) or (10, 2), corresponds to a point in the graph. To plot a point, first locate the time value on the x-axis (horizontal), then find the associated distance on the y-axis (vertical), and mark where these two values intersect.

After marking all the points, we can observe a pattern emerging; since our function is linear, the points will lie in a straight line. Draw a line through these points, extending it in both directions to indicate that the function continues beyond the plotted points. Using a ruler will help ensure the line is straight. This graph offers a visual interpretation of how the distance that sound travels increases linearly with time, reflecting the real-world phenomena the function represents. Graphing makes the understanding of functional relationships more intuitive and can be applied to various disciplines and real-life situations.