Shift algebraic equations across the coordinate plane. Inspect parent functions, translated rules, points, and graphs. Create accurate transformations, save outputs, and understand every step.
Positive horizontal shift moves right. Negative moves left. Positive vertical shift moves up. Negative moves down.
| Equation Type | Original Equation | Shift | Translated Equation |
|---|---|---|---|
| Linear | y = 2x + 3 | Right 4, Up 1 | y = 2x - 4 |
| Quadratic | y = x2 - 4x + 1 | Right 3, Up 2 | y = x2 - 10x + 22 |
| Absolute Value | y = |x| | Left 2, Up 5 | y = |x + 2| + 5 |
| Cubic | y = (x - 1)3 - 2 | Right 2, Down 3 | y = (x - 3)3 - 5 |
| Circle | (x - 1)2 + (y + 2)2 = 16 | Left 3, Up 4 | (x + 2)2 + (y - 2)2 = 16 |
Translation moves every point by the same horizontal and vertical amount.
General function rule: If y = f(x), then the translated graph is y = f(x - h) + k.
Meaning: h shifts the graph horizontally. k shifts the graph vertically.
Linear: If y = mx + b, then the translated line becomes y = m(x - h) + b + k.
Quadratic: If y = ax2 + bx + c, then the translated curve becomes y = a(x - h)2 + b(x - h) + c + k.
Absolute value: If y = a|x - p| + q, translation changes the vertex to (p + h, q + k).
Cubic: If y = a(x - p)3 + q, translation moves the inflection point to (p + h, q + k).
Circle: If (x - p)2 + (y - q)2 = r2, translation changes the center to (p + h, q + k).
It means shifting the graph without changing its basic shape. Every point moves by the same horizontal and vertical amount.
A positive horizontal shift moves the graph to the right. In the translated rule, x is replaced by x minus that amount.
A negative horizontal shift moves the graph to the left. The inside expression becomes x plus the shift magnitude.
A positive vertical shift moves the graph upward. It is added outside the function, not inside the x expression.
No. Translation changes location only. A line keeps its slope, a parabola keeps its width, and a circle keeps its radius.
The leading coefficient stays the same during translation. That preserves the opening direction and the overall shape of the parabola.
Yes. The circle’s center moves by the chosen shifts, while the radius remains exactly the same.
They include the main result details and the translated point table. The PDF also includes the graph image.
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.