Problem 47 - A Graphical Approach to College Algebra

Question

Use a graphing calculator in dot mode with window \([-5,5]\) by \([-3,3]\) to graph each equation. (Refer to your descriptions in Exercises 41-44.) $$y=-[x]$$

Step-by-Step Solution

Answer

Graph shows step function decreasing at each integer along \([-5, 5]\) on x-axis.

Step 1: Understand the Equation

The equation \( y = -[x] \) represents the negative of the floor function of \( x \), which is sometimes referred to as the greatest integer function. This function takes any real number \( x \) and outputs the greatest integer less than or equal to \( x \).

Step 2: Set Up the Graphing Calculator

Ensure your graphing calculator is in dot mode. Set the graph window to the specified dimensions: \([-5, 5]\) for the x-axis and \([-3, 3]\) for the y-axis.

Step 3: Enter the Equation

Input the equation \( y = -[x] \) into your graphing calculator. Use the floor function button (possibly labeled as 'int') to represent \([x]\).

Step 4: Analyze the Graph

Once the graph is displayed, observe the plot. \( y = -[x] \) represents horizontal lines for each integer value on the x-axis, shifted down by that integer value, creating steps known as a step function. Each segment of the graph will span from \( x = n \) to \( x = n+1 \) and be at the height of \( y = -n \).

Step 5: Sketch or Record the Graph

From the graph, you will notice distinct horizontal line segments at each integer point along the x-axis within the window range. The lines will appear downward, as the function maps positive values of \( x \) to negative integer values.

Key Concepts

Floor Function

The floor function plays a unique role in mathematics. It is defined as the greatest integer less than or equal to a given number. Think of it like a staircase where each step represents an integer. For example, if you have the number 3.7, the floor function will bring it down to 3. This characteristic makes it distinct from just rounding numbers, because it always goes "down" to the nearest whole number.

• For positive numbers, the floor function is straightforward. For instance, for any number x where 0 ≤ x < 1, the floor of x is always 0.
• For negative numbers, it's a bit different. If you have -2.5, it will map to -3, since -3 is the largest integer smaller than -2.5.

Understanding this function is key to grasping how the equation in the exercise translates numbers into steps on a graph. The negative sign in the function \( y = -[x] \) merely flips these steps vertically on the coordinate plane.

Coordinate Plane

The coordinate plane, or Cartesian plane, is a two-dimensional surface where you can plot points, lines, and curves. It's made up of two perpendicular lines called axes. The horizontal axis is typically the x-axis, and the vertical is the y-axis.

• Each point on the plane is identified by a pair of numbers \((x, y)\), known as coordinates.
• In equations involving the coordinate plane, the x-coordinate tells us how far left or right the point is on the plane, while the y-coordinate reveals how far up or down it is.

By setting the window dimensions to \([-5, 5]\) by \([-3, 3]\), we fix a specific viewing area on the graphing calculator. This means we are focusing on x-values between -5 and 5 and y-values between -3 and 3. Taking advantage of a graphing calculator allows for a visual exploration of complex equations like \( y = -[x] \).

Step Function

Step functions are a fascinating type of function typically represented by horizontal line segments. They "step" from one value to another as input values change. This is exactly how \( y = -[x] \) behaves. This function takes each range of x-values between integers and maps them to a constant y-value.

• As \( x \) moves from one integer to the next, say from 2 to 3, the value \( y \) remains constant. It is not until \( x \) reaches the next integer that the step kicks up or down.
• In \( y = -[x] \), the steps are downward because of the negative sign.

These graphs have a distinct, stair-step appearance, which is why step functions are easily distinguishable in graphing. Each "step" on the graph reflects the output of the floor function at different ranges of \( x \). The visual representation on the coordinate plane, especially using a graphing calculator, helps significantly in understanding the abrupt changes characteristic of step functions.