Problem 5 - Algebra and Trigonometry

Question

Sketch the graph of the function by first making a table of values. \(f(x)=2\)

Step-by-Step Solution

Answer

The graph of \(f(x)=2\) is a horizontal line at \(y=2\).

Step 1: Choose a Set of Input Values for x

To begin sketching the graph of the function \(f(x) = 2\), we first choose a set of input values for \(x\). Although any range could work, let's use the integers from -3 to 3 for simplicity.

Step 2: Calculate f(x) for Each x Value

Now, we calculate the corresponding \(f(x)\) value for each chosen \(x\). Recall that \(f(x) = 2\) for all \(x\). Therefore, no matter what \(x\) we choose, \(f(x)\) will always be 2. This gives us the points (-3, 2), (-2, 2), (-1, 2), (0, 2), (1, 2), (2, 2), and (3, 2).

Step 3: Plot the Points on a Coordinate Plane

Next, plot each of the points you calculated in Step 2 on a coordinate plane. Since the function is constant, all points will lie on the horizontal line \(y = 2\).

Step 4: Draw the Graph

Finally, draw a horizontal line through all the points plotted in Step 3 to complete the graph. This line visually represents the function \(f(x)=2\) and confirms that for every \(x\) value, \(f(x)\) equals 2.

Key Concepts

Understanding the Table of Values

When dealing with functions, creating a table of values is a fundamental method to understand how the function behaves. This is especially useful for sketching graphs. A table of values simply consists of two columns: one for the input, or the independent variable, and one for the output, or the dependent variable. In this particular exercise involving the function \(f(x) = 2\), the table of values reflects that no matter the input \(x\), the output \(f(x)\) remains constant at 2.

• Input Side: You choose several numbers for \(x\), such as -3, -2, -1, 0, 1, 2, and 3.
• Output Side: No calculations are really needed here since \(f(x)=2\) for every \(x\), meaning the output column will list 2 for each of the entry.

This process of creating a table of values not only helps in plotting points but also reinforces the understanding that the function is constant. This table acts as the foundation for the function's graph, making it essential for visualization.

Identifying the Horizontal Line

The graph of a constant function, like \(f(x) = 2\), results in a horizontal line. But, what exactly is a horizontal line? Simply put, a horizontal line is a straight line on the graph that is parallel to the x-axis.

• Direction: It never climbs up or dips down. It stays at the same vertical height along the entire span of the graph.
• Equation: For the function \(f(x) = 2\), the equation of the horizontal line is \(y = 2\).
• Appearance: It looks flat and level, confirming that all output values are identical across inputs.

In our case, the horizontal line is at \(y = 2\), meaning the function outputs 2 no matter what value you substitute in for \(x\). Recognizing this helps simplify the process of graphing the function, as you can predict where the line will be without needing additional computations.

Graphing a Constant Function

Function graphing is a valuable skill that provides a visual representation of a mathematical function. For the function \(f(x) = 2\), each point plotted is consistent because the function is constant. Once the table of values is constructed and points are plotted, drawing the actual graph becomes straightforward.

• Plot Points: Use the table of values to determine points on the graph. Here, they are (-3, 2), (-2, 2), (-1, 2), (0, 2), (1, 2), (2, 2), and (3, 2).
• Form the Line: After plotting these points, simply draw a straight line through them which runs parallel to the x-axis.
• Check Consistency: Ensure that this line is strictly horizontal and positioned at \(y = 2\), as all calculated points should align here.

This results in a neat graph of the function, offering a clear depiction that \(f(x) = 2\) is constant across all \(x\)-values. Understanding this graph solidifies comprehension of how constant functions appear when visualized, making further exercises more approachable.