Problem 121 - College Algebra Essentials

Question

Use transformations of the graph of the greatest integer function, \(f(x)=\operatorname{int}(x),\) to graph each function. $$h(x)=\operatorname{int}(-x)+1$$

Step-by-Step Solution

Answer

The graph of the function \(h(x)=\operatorname{int}(-x)+1\) is a reflection in the y-axis of the graph of the greatest integer function, shifted up by one unit.

Step 1: Graph the original function

The original function is the greatest integer function \(f(x)=\operatorname{int}(x)\). Its graph is a series of constant segments, where each segment goes from one integer to the next, exclusive. Each segment lies on and just below the integer line.

Step 2: Apply the reflection

The \(-x\) in the function \(h(x)=\operatorname{int}(-x)+1\) indicates a reflection in the y-axis. So, take the graph of \(f(x)\) and reflect it over the y-axis. This means flipping it horizontally.

Step 3: Apply the upward shift

The +1 indicates a shift upward by one unit. Take the reflected graph from Step 2 and shift it up by one unit to get the graph of \(h(x)=\operatorname{int}(-x)+1\).

Key Concepts

graph transformations

Transformations of graphs involve changing the position, orientation, or size of the graph of a function. In the case of the greatest integer function, these transformations can alter how the steps of the function appear on the graph. Graph transformations are essential to understand because they help us visualize how functions change under different operations.

• Reflection: flipping the graph across a line (like the x-axis or y-axis).
• Vertical Shift: moving the graph up or down without changing its shape.
• Horizontal Shift: shifting the graph left or right.
• Stretching or Compressing: changing the graph's scale vertically or horizontally.

The greatest integer function, written as \(f(x) = \operatorname{int}(x)\), looks like a series of steps on a graph. Understanding the basic shape makes it easier to apply these transformations.

piecewise functions

The greatest integer function is a classic example of a piecewise function. Piecewise functions are defined by different expressions depending on the input value. They can be tricky to visualize, but recognizing their structure is crucial.For the greatest integer function:

• The function \(f(x) = \operatorname{int}(x)\) takes an input \(x\), rounds it down to the nearest integer, then outputs that integer.
• Its graph comprises horizontal steps, where each step represents an interval from one integer to the next, but it doesn't include the right endpoint.
• This structure creates a graph that looks like a staircase, where each step aligns with an integer value on the y-axis.

Understanding piecewise functions helps in graphing them accurately and applying transformations.

reflection transformation

Reflection is a transformation that produces a mirror image of the original graph across a specific line. In the context of our greatest integer function, applying a reflection across the y-axis is important. For the function \(h(x) = \operatorname{int}(-x) + 1\):

• Reflecting \(f(x) = \operatorname{int}(x)\) over the y-axis means every point \(x, y\) on the original graph maps to \(-x, y\), essentially flipping the graph horizontally.
• This means the steps of the function are reversed in direction but maintain their height and positioning relative to the x-axis.

This transformation is crucial for altering the orientation, allowing us to manipulate the graph's layout while preserving the step pattern of the greatest integer function.

vertical shift

Vertical shifts change the position of the graph up or down. They affect every point on the graph by adding or subtracting a specific value from the y-coordinate of each point. For \(h(x) = \operatorname{int}(-x) + 1\), the +1 represents a vertical shift. Here's how it works:

• Add 1 to every y-value of the reflected greatest integer function.
• This moves the entire graph one unit up, positioning each step one level higher on the y-axis compared to the reflected graph.

Vertical shifts are straightforward but essential because they adjust the entire graph's position without changing its shape or direction, allowing for precise control over where the function appears on a graph.