Calculator Inputs
Example Data Table
| Example | Function | Input x | Integer Part | Output y |
|---|---|---|---|---|
| Basic floor | y = floor(x) | 2.75 | 2 | 2 |
| Vertical stretch | y = 3 floor(x) | 1.90 | 1 | 3 |
| Shifted inside | y = floor(x + 2) | 1.25 | 3 | 3 |
| Ceiling option | y = ceil(x) | -1.40 | -1 | -1 |
Formula Used
This calculator uses the transformed integer function:
y = a × I(bx + c) + d
Here, I means the selected integer rule.
It can be floor, ceiling, rounding, or truncation.
The value a changes vertical scale.
The value b changes horizontal compression.
The value c shifts the input before rounding.
The value d shifts the graph up or down.
For the greatest integer function, I(x) = floor(x).
This means the output is the largest integer less than or equal to x.
For example, floor(3.8) = 3.
Also, floor(-2.3) = -3.
How to Use This Calculator
- Select the integer function type.
- Enter values for a, b, c, and d.
- Set the minimum and maximum x values.
- Choose a step size for the table and graph.
- Press the graph button.
- Review the result summary above the form.
- Use the graph to inspect step behavior.
- Download the CSV or PDF report if needed.
Integer Function Graphing Guide
What This Tool Shows
An integer function graph has flat steps. Each step covers a range of x values. The output changes only when the selected integer rule changes. This makes the graph different from a smooth line. The calculator helps you see each jump clearly. It also creates a table for every sampled x value.
Why Integer Graphs Matter
Integer functions are used in many math lessons. They explain rounding, floors, ceilings, and step changes. They also appear in pricing, schedules, inventory counts, and computer logic. A graph makes these ideas easier to compare. You can test how each coefficient changes the shape. You can also see how vertical and horizontal shifts work.
Understanding Transformations
The value a controls the height of each step. A negative value reflects the graph across the x-axis. The value b changes how fast the steps occur. A larger absolute value creates tighter steps. The value c moves the graph left or right. The value d moves every output up or down. These changes follow normal transformation rules.
Reading the Table
The table shows x, the inside expression, the integer value, and y. The inside expression is bx plus c. The integer value is found using the selected rule. The final y value applies the multiplier and vertical shift. When y changes between nearby rows, a step jump occurred. Use a smaller step size for a more detailed graph.
Best Practice
Start with simple values first. Use a = 1, b = 1, c = 0, and d = 0. Then change one input at a time. This helps you understand each transformation. Export the data when you need a homework record. Use the PDF option for quick reports.
FAQs
1. What is an integer function graph?
An integer function graph shows outputs that change by whole-number rules. It often looks like a staircase because many nearby x values share the same y value.
2. What is the greatest integer function?
The greatest integer function returns the largest integer less than or equal to the input. It is also called the floor function.
3. What does the ceiling option do?
The ceiling option returns the smallest integer greater than or equal to the input. It moves decimal values up to the next whole number.
4. Why does the graph look like steps?
The output remains constant across intervals. It changes only when the integer rule creates a new whole-number value.
5. What does the step size control?
The step size controls how many x values are sampled. Smaller steps give more detail, but they create a larger data table.
6. Can I graph shifted integer functions?
Yes. Use c for inside shifts and d for vertical shifts. You can also use a and b for stretching or reflection.
7. What does truncation mean?
Truncation removes the decimal part and keeps the integer part toward zero. It behaves differently from floor for negative numbers.
8. Can I export my results?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a simple report with summary and table values.