Problem 44 - A Graphical Approach to Precalculus with Limits

Question

Describe how the graph of the given function can be obtained from the graph of \(y=[x]\). $$y=[x+2]$$

Step-by-Step Solution

Answer

Shift the graph of \(y = [x]\) two units to the left to obtain the graph of \(y = [x+2]\).

Step 1: Understand the Base Function

The base function given is \(y = [x]\). This represents the floor function, also known as the greatest integer function, which takes an input \(x\) and returns the greatest integer less than or equal to \(x\). For example, \([3.7] = 3\) and \([-2.3] = -3\).

Step 2: Analyze the Transformation

The function to graph is \(y = [x + 2]\). Notice that \(x + 2\) shifts the input of the function by 2 units to the left. This means the graph of \(y = [x]\) will be horizontally shifted.

Step 3: Apply the Transformation

To apply the horizontal shift to the graph of \(y = [x]\), move each step (discontinuity) 2 units to the left. A point originally at \((x, [x])\) will move to \((x-2, [x])\). This results in the graph starting its integer steps at \(-2\) instead of \(0\).

Step 4: Verify the Transformed Graph

Identify a few key points: originally, \((0, 0)\) is on the graph of \(y = [x]\). For \(y = [x + 2]\), this point shifts to \((-2, 0)\). Verify by substitution: \((0) + 2 = 2\) and \([2] = 2\), confirming \((0, 2)\) transforms to \((-2, 0)\) relative to its original position.

Key Concepts

Floor Function

In the realm of functions, the **floor function** plays a fundamental role in mathematical transformations. Commonly referred to as the greatest integer function, it maps a real number to the largest integer less than or equal to that number. For instance, when you apply the floor function to 3.7, which is denoted as \([3.7]\), the result is 3. This function effectively "rounds down" the real number to the nearest whole number. It handles negative numbers in a similar manner. For example, \([-2.3]\) becomes -3 because -3 is the greatest integer less than or equal to -2.3.

• Positive numbers: The floor function \

Horizontal Shift

Introducing a horizontal shift to a graph means you are moving the entire function either left or right along the horizontal axis. This occurs through modifying the input variable of the function, often by addition or subtraction within the argument of the function.

Consider the function \(y = [x]\). If you adjust this to \(y = [x + 2]\), you're adding 2 to every input before the flooring operation, which shifts each corresponding output to the left by 2 units.

• Addition inside: Moves function to the left.
• Subtraction inside: Moves function to the right.

This idea of shifting is valuable as it allows the manipulation of function graphs without altering their form, thereby adjusting how they interact or appear across the coordinate plane.

In our specific example, replacing \(x\) with \(x + 2\) in the floor function \(y = [x]\) results in the graph starting at -2 instead of at 0. Each step or discontinuity, which is characteristic of the floor function, shifts leftward by 2 units, adjusting the entire graphical representation appropriately.

Greatest Integer Function

The greatest integer function is another name for the floor function. This concept is critical in understanding the visualization of certain types of functions, particularly those forming step-like graphs. It serves to simplify complex real numbers into integers by essentially "cutting off" the decimal part, thus returning the largest whole number not exceeding the original input.

One way to visualize this is by imagining numbers on a number line, where every point between two integers falls back to the lower of the two. For example:

• \( [1.9] = 1 \)
• \( [-0.1] = -1 \)

This characteristic makes the greatest integer function a series of horizontal segments with jumps as you move between integers. It's instrumental in applications requiring rounding down in operations, such as programming loops or discretizing continuous data inputs.

In the transformation described above, when shifting to \(y = [x + 2]\), this function maintains its greatest integer role, simply with its steps starting earlier on the number line.