Problem 42 - A Graphical Approach to Precalculus with Limits

Question

Describe how the graph of the given function can be obtained from the graph of \(y=[x]\). $$y=[-x]$$

Step-by-Step Solution

Answer

The graph is a horizontal reflection of \(y = [x]\) over the y-axis.

Step 1: Recognize the Parent Function

First, identify the parent function from which the given function is derived. Here, the parent function is the greatest integer function, also known as the floor function, represented by \(y = [x]\). It returns the greatest integer less than or equal to \(x\).

Step 2: Analyze the Transformation

Examine the given function \(y = [-x]\). Notice the negative sign inside the brackets. This indicates a transformation called horizontal reflection or reflection over the y-axis. For any \(x\), \([-x]\) finds the greatest integer less than or equal to \(-x\), effectively reflecting the parent function over the y-axis.

Step 3: Illustrate Point Changes

To better understand how the transformation affects specific points, consider a few examples: \(x = 1\) on \(y = [x]\) gives \([1] = 1\). For \(y = [-x]\), \([-1] = -1\). At \(x = 2.7\), \([2.7] = 2\) on \(y = [x]\), while \([-2.7] = -3\) on \(y = [-x]\). All positive integer inputs become their negative greatest integer equivalents in the transformed graph.

Step 4: Draw and Compare the Graphs

Graph both functions to visualize the transformation. The graph of \(y = [x]\) consists of horizontal segments with jumps at every integer \(x\). The graph of \(y = [-x]\) is a horizontally flipped version, with horizontal segments mirrored across the y-axis.

Key Concepts

Greatest Integer Function

The greatest integer function, commonly referred to as the floor function, is a special function that maps a real number to the largest integer less than or equal to it. When we write a function as \( y = [x] \), this is what it's doing:

• If \( x = 2.3 \), then \( [x] = 2 \) because 2 is the largest integer ≤ 2.3.
• If \( x = -2.3 \), then \( [x] = -3 \) because -3 is the largest integer ≤ -2.3.

The graph of \( y = [x] \) is a series of horizontal line segments. It steps up at each integer value \( x \), creating a staircase-like appearance. Each segment runs from an integer to just before the next integer, excluding the endpoint. This function is discontinuous at integer points on the x-axis as it jumps to the next integer value.

Horizontal Reflection

Horizontal reflection is a transformation that flips a graph across the y-axis. This transformation involves changing the sign of the input variable \( x \). For a function \( y = f(x) \), a horizontal reflection would give us the new function \( y = f(-x) \).
In our context with \( y = [-x] \), each point on the parent function \( y = [x] \) is mirrored to the opposite side of the y-axis. This means if the original point is \( (1, [1]) = (1, 1) \), the reflected point after horizontal reflection will be \( (-1, [-1]) = (-1, -1) \).

• A point at \( (3.6, [3.6]) = (3.6, 3) \) when reflected becomes \( (-3.6, [-3.6]) = (-3.6, -4) \).
• This transformation effectively reverses the direction of the function across the vertical axis, making each step on the graph appear as if it has switched places with its mirror image on the other side of the y-axis.

Horizontal reflection is a crucial concept in understanding how transformations can manipulate function graphs, providing insight into symmetry and inversion properties.

Function Graphing

Function graphing involves plotting the set of points that represent a function on a coordinate plane. For \( y = [x] \), this process begins by identifying key points through values of \( x \) and their corresponding greatest integer values, as discussed.
To graph \( y = [x] \):

• Plot points where \( x \) is an integer, marking each with the greatest integer less than \( x \).
• Connect these points horizontally until just before the next integer point, creating a step line.

For the reflected function \( y = [-x] \):

• Repeat the process above but use negative values of \( x \).
• The trick is to consider \( x \) values on \( y = [-x] \) as reflections of their counterparts on \( y = [x] \).

This results in a graph where steps that rise on one side of the y-axis fall in reflection on the other side. Graphing is not only about plotting the function but also about understanding these transformations visually. Practicing these steps enhances the ability to predict how changes in function formulas affect their graphs visually, solidifying the concept of function transformations.