Explore horizontal and vertical shifts for many functions. Check tables, equations, domains, and transformed graphs. Download neat summaries for homework, teaching, revision, and analysis.
The calculator uses the transformation model g(x) = A · f(B(x - h)) + k.
A controls vertical stretch, compression, and reflection across the x-axis.
B controls horizontal stretch, compression, and reflection across the y-axis.
h shifts the graph right when positive and left when negative.
k shifts the graph up when positive and down when negative.
When B is greater than 1, the graph compresses horizontally. When 0 < |B| < 1, the graph stretches horizontally.
For trigonometric functions, the chosen angle unit changes evaluation and period values.
Start by selecting a base function family. Choose linear, quadratic, cubic, absolute value, square root, exponential, logarithmic, sine, cosine, or reciprocal. This decides the reference shape before any transformation is applied.
Next, enter the vertical factor A. Use positive values for normal orientation. Use negative values to reflect the graph across the x-axis. Larger absolute values stretch the graph vertically. Values between zero and one compress it vertically.
Then enter the horizontal factor B. Positive values keep the original left-right orientation. Negative values reflect the graph across the y-axis. Larger absolute values compress the graph horizontally. Smaller absolute values stretch it horizontally.
Set h for the horizontal shift and k for the vertical shift. Positive h moves the graph right. Negative h moves it left. Positive k moves the graph upward. Negative k moves it downward.
Choose a plotting window with x minimum and x maximum. This lets you inspect the transformed curve within a useful interval. Increase the sample count for smoother graphs and denser tables.
Press the calculate button. The results appear above the form, directly below the header. You will see the transformed equation, graph, domain, range, intercept estimates, mapped key points, and a sample output table.
Use the CSV button to export the displayed values into a spreadsheet-friendly file. Use the PDF button to save a clean report containing the summary, graph snapshot, and result table.
| Base function | A | B | h | k | Result |
|---|---|---|---|---|---|
| Quadratic | 1 | 1 | 2 | 3 | g(x) = (x - 2)^2 + 3 |
| Absolute Value | -2 | 1 | -1 | 4 | g(x) = -2|x + 1| + 4 |
| Sine | 3 | 2 | 0 | -1 | g(x) = 3sin(2x) - 1 |
| Logarithmic | 1 | 1 | 5 | 0 | g(x) = ln(x - 5) |
Function shifts help students and teachers understand how equations control graph position and shape. A small change in h or k moves a graph immediately. Changes in A and B affect orientation and scaling. Seeing these changes together makes graph behavior easier to predict.
This calculator is useful for classroom demonstrations, algebra practice, precalculus review, and graph interpretation. It helps compare a parent function with a transformed function in one place. That comparison is especially useful when learning domains, ranges, asymptotes, intercepts, and key point mapping.
The graph section shows both the reference curve and the shifted curve. The table section turns those visuals into exact sample values. Export options make it easier to save results for worksheets, notes, reports, and revision tasks.
h changes the horizontal position of the graph. Positive h moves the graph right. Negative h moves it left. The internal subtraction form can feel reversed, so the graph view is helpful.
k changes the vertical position. Positive values move the graph upward. Negative values move it downward. It does not change the basic family shape by itself.
A negative A multiplies every output by a negative number. That reflects the graph across the x-axis. Peaks become valleys, and upward openings become downward openings.
A negative B reflects the graph across the y-axis. It also changes horizontal scaling. For asymmetric functions, this can noticeably change the graph orientation.
Some functions have restricted domains. Square root needs nonnegative input. Logarithmic needs positive input. Reciprocal cannot divide by zero. Undefined rows are skipped in the visible sample table.
Yes. Choose degrees in the angle unit field. The calculator then evaluates trigonometric inputs in degrees and updates period reporting accordingly.
No. They are numerical estimates from the current graph window and sampling density. Increase sample points or adjust the x-range for more detailed approximations.
The CSV export saves the sample output table. The PDF export saves a report with the summary, graph image, and table so you can reuse the results later.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.