Write an Equation for the Translation
Example Data Table
| Original item | Translation | Rule used | Translated result |
|---|---|---|---|
| Point (2, 5) | Right 3, down 2 | (x, y) → (x + 3, y - 2) | (5, 3) |
| Line y = 2x + 1 | Right 3, down 2 | b′ = b + k - mh | y = 2x - 7 |
| Parabola y = x² | Right 3, down 2 | y = f(x - 3) - 2 | y = (x - 3)² - 2 |
Formula Used
General graph translation:
y = f(x - h) + k
Here, h is the horizontal shift. A positive value moves the graph right. A negative value moves it left.
Here, k is the vertical shift. A positive value moves the graph up. A negative value moves it down.
Point translation:
(x, y) → (x + h, y + k)
Line translation:
For y = mx + b, the translated line is y = mx + b′, where b′ = b + k - mh.
Circle translation:
For (x - a)² + (y - b)² = r², the new center is (a + h, b + k).
Parabola translation:
For y = a(x - p)² + q, the new vertex is (p + h, q + k).
How to Use This Calculator
- Select the type of object you want to translate.
- Enter the horizontal shift value.
- Enter the vertical shift value.
- Fill the fields for the chosen equation type.
- Press the calculate button.
- Read the translated equation above the form.
- Check the graph and calculated table.
- Use CSV or PDF export for saving results.
Article: Understanding Equation Translation
What Translation Means
A translation moves a graph without turning it. It does not stretch the graph. It does not flip the graph. Every point moves the same distance. This makes translation one of the simplest transformations in coordinate geometry.
Horizontal Movement
A horizontal shift changes the input part of a function. This is why the formula uses x minus h. When h is positive, the graph moves right. When h is negative, the graph moves left. This can feel opposite at first. The reason is that the input must be adjusted before the function gives its output.
Vertical Movement
A vertical shift changes the output. This part is direct. Add k to move the graph upward. Subtract from the output to move it downward. For example, y equals x squared plus four moves the basic parabola four units up.
Using Points
Point translation is very direct. Add h to the x-coordinate. Add k to the y-coordinate. If the point is two comma five, and the translation is right three and down two, the image point is five comma three.
Using Equations
Different equations need different forms. A line keeps its slope after translation. Only the intercept changes. A circle keeps its radius. Only the center moves. A parabola keeps its shape factor. Only the vertex moves.
Why This Tool Helps
This calculator shows the rule, equation, table, and graph together. That helps students check each step. It also helps teachers create examples quickly. The export buttons make it easier to save work for homework, lessons, or review sheets.
FAQs
1. What is an equation translation?
It is a movement of an equation or graph left, right, up, or down. The shape stays the same. Only the position changes.
2. What formula translates a function?
The main formula is y = f(x - h) + k. The value h controls horizontal movement. The value k controls vertical movement.
3. Why is the horizontal shift written as x minus h?
The input must be changed before the function is evaluated. That creates the opposite-looking sign inside the function expression.
4. Does a translation change the graph shape?
No. A translation only moves the graph. It does not stretch, shrink, rotate, or reflect the graph.
5. How do I translate a point?
Add the horizontal shift to x. Add the vertical shift to y. The rule is (x, y) becomes (x + h, y + k).
6. How does a translated line change?
The slope stays the same. The intercept changes according to the horizontal and vertical shifts applied to the line.
7. How does a translated circle change?
The radius stays the same. The center moves by the given horizontal and vertical shift values.
8. Can I export the result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a printable summary of the calculation.