Problem 10 - Algebra and Trigonometry

Question

Sketch the graph of the function by first making a table of values. \(f(x)=\frac{x-3}{2}, \quad 0 \leq x \leq 5\)

Step-by-Step Solution

Answer

The graph is a straight line passing through the points (0,-1.5), (1,-1), (2,-0.5), (3,0), (4,0.5), and (5,1).

Step 1: Choose Values of x

To create a table of values, we first need to select different values for \(x\) within the given range. We could use integer values from 0 to 5, which are easy to compute: \(x = 0, 1, 2, 3, 4, 5\).

Step 2: Compute Corresponding Values of f(x)

Substitute each chosen \(x\) value into the function \(f(x) = \frac{x-3}{2}\), and compute the corresponding \(f(x)\). For example, for \(x = 0\), \(f(0) = \frac{0-3}{2} = -1.5\). Perform similar computations for \(x = 1, 2, 3, 4, 5\).

Step 3: Complete the Table of Values

Record the results of each computation in a table:| \(x\) | \(f(x)\) ||-------|---------|| 0 | -1.5 || 1 | -1.0 || 2 | -0.5 || 3 | 0.0 || 4 | 0.5 || 5 | 1.0 |

Step 4: Plot the Points on a Graph

Using the table of values, plot each \((x, f(x))\) pair on a coordinate plane. You should have 6 points to plot: (0, -1.5), (1, -1.0), (2, -0.5), (3, 0.0), (4, 0.5), and (5, 1.0).

Step 5: Draw the Graph of the Function

Connect the plotted points with a straight line. Since \(f(x)\) is a linear function, the graph will be a straight line that follows the pattern of the plotted points.

Key Concepts

Graphing Functions

Graphing functions is a fundamental concept in mathematics that allows us to visualize mathematical relationships. To graph a function, we plot its values on a coordinate plane. We use the values of the independent variable, often labeled as \(x\), and the corresponding function values, \(f(x)\). This process provides a visual representation of how the function behaves, which can be particularly useful for understanding trends and relationships.

When graphing linear functions like \(f(x) = \frac{x-3}{2}\), the graph will always be a straight line. This is because linear functions have a constant rate of change, reflected as a slope in the graph. In the function \(f(x) = \frac{x-3}{2}\), this constant rate of change, or slope, is \(\frac{1}{2}\).

To draw the graph:

• Start by plotting the specific \(x\) and \(f(x)\) pairs you calculated in your table of values.
• Once all points are plotted, connect them with a straight line, ensuring the line extends across the defined range of \(x\) values.

Remember, graphing helps us "see" how the function works, making it easier to interpret and analyze.

Table of Values

Creating a table of values is an excellent strategy for graphing functions, especially when you're just starting. This approach involves selecting specific \(x\) values, substituting them into the function, and calculating their corresponding \(f(x)\) values.

Here’s how to create your table of values for \(f(x) = \frac{x-3}{2}\):

• Choose a range for \(x\) values. In this case, we've chosen simple integers between 0 and 5 to make calculations straightforward.
• Substitute each \(x\) value into the function to find \(f(x)\). For example, for \(x = 0\), \(f(0) = \frac{0-3}{2} = -1.5\).
• Repeat the calculation for each \(x\) in your selected range, populating your table with the calculated \(f(x)\) values (e.g., for \(x = 0\) to \(x = 5\), you’ll have \(f(x) = -1.5, -1.0, -0.5, 0.0, 0.5, 1.0\) respectively).

This structured approach not only aids in graphing but also helps in understanding how the function processes each input, providing insights into the function’s behavior across different \(x\) values.

Coordinate Plane

The coordinate plane is like a map for plotting mathematical functions. It consists of two perpendicular axes: the horizontal axis (often called the \(x\)-axis) and the vertical axis (often called the \(y\)-axis or \(f(x)\)-axis in the context of functions). Each point on the plane is represented by an \((x, y)\) pair corresponding to its position relative to these axes.

In the task of graphing the function \(f(x) = \frac{x-3}{2}\), the coordinate plane becomes the canvas for our plotted points. Here’s how it works:

• The \(x\)-axis represents the independent variable values you selected, such as 0 to 5 in this case.
• The \(y\)-axis shows the corresponding \(f(x)\) values calculated from the function.
• Each \((x, f(x))\) pair calculated forms a point which you plot on the plane: e.g., (0, -1.5), (1, -1.0), etc.
• Once all points are plotted, connect them, following the pattern of a straight line, as is characteristic of a linear function.

Understanding the coordinate plane and effectively plotting points are crucial skills in mathematics, setting the stage for exploring more advanced graphing techniques in future learning.