Calculator
Formula Used
Base function: f(x) = mx + b
Transformation form: g(x) = a f(c(x - h)) + k
Expanded form: g(x) = (amc)x + (ab - amch + k)
Final slope: M = amc
Final intercept: B = ab - amch + k
X-intercept: x = -B / M, when M is not zero.
Inverse form: x = (y - B) / M, when M is not zero.
How to Use This Calculator
- Enter the base slope and intercept for f(x).
- Enter vertical scale, horizontal factor, and shifts.
- Add an x-value for direct evaluation.
- Add a target y-value for inverse checking.
- Set the table start, step, rows, and rounding.
- Press Calculate to show results above the form.
- Use CSV or PDF buttons to save the current result.
Example Data Table
This example uses f(x) = 2x + 1 and g(x) = 3f(0.5(x - 4)) - 2.
| Input | Value | Meaning |
|---|---|---|
| m | 2 | Base line slope |
| b | 1 | Base y-intercept |
| a | 3 | Vertical stretch |
| c | 0.5 | Horizontal input factor |
| h | 4 | Right shift |
| k | -2 | Downward shift |
Linear Function Transformations
A linear function looks simple because its graph is always a straight line. Yet small transformation values can change its position, direction, and steepness. This calculator helps you study those changes without rewriting every step by hand. Start with a base function, f(x)=mx+b. Then apply vertical scale, horizontal factor, horizontal shift, and vertical shift. The tool expands the transformed rule into slope intercept form.
Why Transformations Matter
Transformations show how one line becomes another line. A vertical scale changes rise for every run. A negative vertical scale reflects the line across the x-axis. The horizontal factor changes how input values are read before the base rule works. A negative horizontal factor can reflect the graph across a vertical reference line. Shifts move the line without changing its straight shape.
What The Calculator Shows
The calculator returns the final equation, slope, y-intercept, and x-intercept. It also evaluates a chosen x-value. You can enter a target y-value to find the inverse result when the slope is not zero. The table gives several sample points, so you can verify the graph with ordered pairs. This is useful for homework, teaching notes, and graph checks.
Interpreting The Result
A positive final slope rises from left to right. A negative final slope falls from left to right. A zero final slope makes a horizontal line. The y-intercept shows where the graph crosses the vertical axis. The x-intercept shows where the output becomes zero. If the transformed slope is zero, no unique inverse exists.
Best Practice
Use simple values first. Then test reflections and fractions. Compare the base equation with the final equation. Check the table before graphing. If an answer looks unexpected, review each transformation separately. Pay close attention to signs inside the input expression. Horizontal changes often feel reversed because they affect x before the base function is evaluated. Save your result as CSV or PDF when you need a record.
Common Mistakes
Many mistakes come from mixing outside changes with inside changes. The value k moves outputs. The value h moves inputs. The value a changes outputs after the base line is used. The value c changes inputs before the base line is used. Keep that order for accurate work.
FAQs
What is a linear function transformation?
It is a change made to a base line. The change can move, stretch, compress, or reflect the graph while keeping it linear.
What does the value a do?
The value a changes outputs after the base function is evaluated. It can stretch, compress, or reflect the graph across the x-axis.
What does the value c do?
The value c changes the input before the base rule works. It affects horizontal stretch, compression, and reflection behavior.
Why is the final graph still a line?
A transformed linear function remains linear because the expanded expression still has the form Mx + B, unless it becomes constant.
Can the calculator find intercepts?
Yes. It finds the y-intercept directly from the expanded form. It finds the x-intercept when the final slope is not zero.
When does no unique inverse exist?
No unique inverse exists when the final slope is zero. In that case, many x-values can produce the same output.
Can I download the results?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a simple printable report.
Is this useful for graphing?
Yes. The generated table gives ordered pairs. Plot those points with the intercepts to check the transformed line.