Plot curves and inspect shifted coordinates instantly. Save tables, verify formulas, and study graph movement. Track each vertical translation with simple precise mathematical outputs.
The graph compares the original function and the vertically shifted function across the selected x-range.
| x | Original y = f(x) | Shifted y = f(x) + k | Difference |
|---|---|---|---|
| -2 | 4 | 7 | 3 |
| -1 | 1 | 4 | 3 |
| 0 | 0 | 3 | 3 |
| 1 | 1 | 4 | 3 |
| 2 | 4 | 7 | 3 |
g(x) = f(x) + k
In a vertical shift, the output of the original function changes by a constant amount k. If k > 0, the graph moves upward. If k < 0, the graph moves downward. Every y-value increases or decreases by the same amount, while all x-values stay unchanged.
For any chosen x-value, the calculator first evaluates the original function f(x). It then adds the shift constant k to obtain g(x). The difference column confirms that the shift remains constant across the full table.
Vertical shifting changes only output values. When a constant k is added, every point (x, y) becomes (x, y + k). If a parabola has vertex at (2, -1) and the shift is +4, the new vertex becomes (2, 3). This makes translation analysis efficient because the x-coordinate pattern remains fixed across all comparable points.
For quadratic functions, a vertical shift moves the entire curve without changing opening direction or axis of symmetry. For example, f(x) = x² - 4x + 1 and g(x) = x² - 4x + 6 differ by +5. The minimum value rises by 5 units, while curvature remains identical. This is useful in optimization tasks where baseline output levels change but structural behavior stays constant.
In sine and cosine models, vertical shifting changes the midline. If f(x) = 3sin(x) and g(x) = 3sin(x) + 2, the amplitude stays 3, but the midline moves from 0 to 2. Maximum values rise from 3 to 5, and minimum values rise from -3 to -1. This is common in seasonal, wave, and oscillation modeling.
Analysts often apply vertical shifts when observed data follows the same pattern as a base model but starts from a different reference level. For instance, if a production curve matches a known template yet all measured outputs are 12 units higher, a vertical shift of +12 aligns the model quickly. This preserves trend shape while correcting the baseline.
The data table confirms translation consistency. The difference column should remain constant for every valid x-value. If rows show differences of 3, 3, 3, 3, and 3, the graph has shifted upward by exactly 3 units. On the plot, the original and shifted curves remain parallel in vertical separation, even when the curve itself is nonlinear.
This tool supports instruction, verification, and quick scenario testing. Students can check transformed equations, teachers can demonstrate graph movement, and analysts can compare baseline adjustments in mathematical models. With direct evaluation, plotted comparison, CSV output, and PDF export, the calculator turns a simple transformation rule into a structured workflow for numerical and visual review.
It changes every y-value by the same constant amount. The graph moves up or down, while x-values, shape, slope pattern, and spread stay unchanged.
If the constant k is positive, the shift is upward. Each output increases by that value, so the entire graph appears higher on the coordinate plane.
Yes, it can. Moving a graph up or down may create, remove, or move x-intercepts because the curve meets the x-axis at different locations.
Yes. A vertical shift does not stretch, compress, or reflect the graph. It only translates the function without changing its underlying shape.
It verifies that every output changed by the same constant. A consistent difference confirms that the transformation is a pure vertical shift.
Yes. The calculator supports multiple function families, so you can compare original and shifted outputs for linear, polynomial, trigonometric, radical, and exponential expressions.
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.