Problem 48 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graph of the function by hand. Then use a graphing utility to verify the graph. $$f(x)=[[x]]-3$$

Step-by-Step Solution

Answer

The graph of the function is a floor function graph shifted down 3 units. For each integer value of x from -∞ to +∞, the y-value is that integer minus three. This continues along the x-axis in the form of step-like graph.

Step 1: Understand the function

We will first understand the function \(f(x)=\[[x]]-3\). This is a floor function, which means it rounds down any real number to the nearest integer less than the number. The '-3' simply shifts the graph 3 units down.

Step 2: Determine major points

For every real number x, if x is not an integer, the value of the floor function is the next smaller integer. The critical points occur at each integer from -3 onwards (since we have shifted 3 units down). At each of these critical points, the step function moves to the number that is 3 units less than the integer.

Step 3: Draw the line segments

Draw the line segments for each integer, starting from -3, by placing dots at the integers for the y-value of each integer minus 3. After plotting the points, each horizontal line segment (from one integer to the next but not including the next integer) should be at the value of the previous integer subtract 3.

Step 4: Verify the graph using a graphing utility

Use a graphing utility to verify the sketched graph. Enter \(f(x)=\[[x]]-3\) into the utility, and the resulting graph should match your sketched graph.

Key Concepts

Graphing Piecewise Functions

When you are dealing with piecewise functions, you are essentially working with different "pieces" of a function graphed on the same set of axes. In the case of the floor function expressed as \[ f(x)=\lfloor x \rfloor - 3 \]you will notice that it exhibits a characteristic step-like appearance. Here's why:

• Each segment of the graph represents a portion of the total domain of the function.
• For each integer value within the segment, the function outputs that integer, adjusted by any additional operations, such as the subtraction of 3 in our example.
• The graph remains constant between two consecutive integer values and then "jumps" down to the next value at every integer step.

This step-like pattern is typical of how floor functions behave, and graphing them manually involves marking out where these constant segments start and end. To successfully graph, follow these steps:

• Identify the integer values on the x-axis.
• Subtract 3 from these integer values to determine the corresponding y-values.
• Connect these y-values with horizontal line segments, skipping each integer on the x-axis to create the step function look.

Piecewise functions like these require careful attention to the behavior at each transition point, so keep an eye on where the function jumps to the next integer.

Integer Rounding

Understanding integer rounding is crucial for mastering floor functions. The floor function, denoted as \( \lfloor x \rfloor \), rounds down any real number to the largest integer less than or equal to the number. For example, for a value like 4.5, the floor function will round this down to 4.Here are some key points about integer rounding involving floor functions:

• Integer Values: If the number is already an integer, the floor function will remain unchanged, as it is already at the nearest lower integer.
• Non-integer Values: Any number with a fractional part will be reduced to the nearest lower integer.
• Handling Negative Numbers: Negative numbers work similarly, where the floor function rounds down beyond zero—meaning \( \lfloor -2.3 \rfloor \) will give -3, not -2.

Rounding, as conducted by the floor function, helps facilitate the creation of these piecewise graphs by representing real-world scenarios that require discrete steps or integer values.

Shift Transformations

Shift transformations are simple yet powerful tools in graphing that modify the position of a function on the coordinate plane without altering its shape. In the function \( f(x)=\lfloor x \rfloor - 3 \), you can observe a vertical shift transformation. Here's how it works:

• The original floor function \( \lfloor x \rfloor \) generates an output based solely on rounding down the input to the nearest whole number.
• Subtracting 3 from this output results in a vertical shift downward by 3 units. Thus, every y-value generated by the floor function is reduced by 3.

Shift transformations can be either vertical or horizontal:

• Vertical Shifts: Implemented by adding or subtracting a constant from the function, shifting the graph up or down.
• Horizontal Shifts: Achieved by adding or subtracting a constant from the input of the function, shifting the graph left or right.

By understanding these shifts, students can predict the behavior of functions when combined with other operations, making graphing less daunting and more intuitive.