91Ó°ÊÓ

The floor function, denoted as \( \lfloor x \rfloor \), is a mathematical function that maps a real number to the greatest integer less than or equal to that number. This function is integral to understanding the first part of our function \( y = \lfloor x \rfloor - \left\lfloor x - \frac{1}{2} \right\rfloor \). For example:

• \( \lfloor 3.7 \rfloor = 3 \)
• \( \lfloor -2.1 \rfloor = -3 \)
• \( \lfloor 4 \rfloor = 4 \) (since 4 is already an integer)

When graphing these functions, observe the behavior of integers and non-integers differently. At each integer value of \( x \), \( \lfloor x \rfloor = x \) and \( \left\lfloor x - 0.5 \right\rfloor = x - 1 \). This results in \( y = 1 \). For non-integer values, both floor functions yield the same integer right below \( x \), leading to \( y = 0 \).
You will see steps in the graph where it climbs vertically to 1 at integer values and stays at 0 between them.

The fractional part function \( \text{FRACPT}(x) \) gives the non-integer part of a real number, defined as \( x - \lfloor x \rfloor \). This presents a value between 0 and just under 1 for any real number \( x \). Understanding this is key to part (b) of the given exercise with the function \( y = \left| \text{FRACPT}(x) - \frac{1}{2} \right| \). For example:

• \( \text{FRACPT}(3.7) = 3.7 - 3 = 0.7 \)
• \( \text{FRACPT}(-2.3) = -2.3 + 3 = 0.7 \)
• \( \text{FRACPT}(4) = 4 - 4 = 0 \)

By subtracting 0.5 and taking the absolute value, the function reflects symmetrically creating a V-shape centered around each half-integer value \( n + 0.5 \) for integers \( n \). This makes the function oscillate, forming a pattern of peaks at these half points.

Graph analysis involves interpreting and visualizing how these functions behave over different domains. For the function \( y = \lfloor x \rfloor - \left\lfloor x - \frac{1}{2} \right\rfloor \), the graph is made of step-like features. This happens because the outcome depends on whether \( x \) is an integer:

• At integers, \( y = 1 \)
• In between integers, \( y = 0 \)

The transitions happen exactly at integer values, resulting in a series of horizontal lines at these points.
In contrast, for \( y = \left| \text{FRACPT}(x) - \frac{1}{2} \right| \), the graph forms a continuous sequence of V-shapes. Each V is symmetric relative to lines \( x = n + 0.5 \). The curve touches 0 at integer values and reaches a maximum \( \frac{1}{2} \) at midpoints, establishing a wave-like appearance that smoothly rises and falls.

The absolute value function, denoted as \( |x| \), represents the distance of a number from zero on the number line. It ensures that the result is always non-negative. This function is important in part (b) for \( y = \left| \text{FRACPT}(x) - \frac{1}{2} \right| \). Here's why it matters:

• The function \( \text{FRACPT}(x) - \frac{1}{2} \) can be either positive or negative depending on \( x \).
• Taking the absolute value guarantees the result is positive, plotting a symmetrical V-shape between integer values.
• This symmetry results in a consistent pattern across the graph, centered on half-integers.

Graphs involving absolute value functions typically showcase sharp changes in direction, like V-shapes or zigzag lines, due to the reflection property inherent in absolute values.