Enter vertex-form parameters
Use y = a(x - h)² + k. Defaults match the requested target graph.
Vertex-form transformation rule
Start with the parent function y = x². Write the new equation as y = a(x - h)² + k.
| Parameter | Meaning | Target value |
|---|---|---|
| a | Vertical scale. A negative value also reflects the graph. | -2 |
| h | Horizontal movement. Positive h moves right. | 2 |
| k | Vertical movement. Positive k moves up. | 2 |
For y = -2(x - 2)² + 2, move right 2, stretch vertically by 2, reflect across the x-axis, then move up 2.
Get a complete transformation report
- Enter a, h, and k from your vertex-form equation.
- Choose a parent x-range and point interval.
- Select the number of decimal places to display.
- Press Show transformation steps.
- Read the result above the form.
- Review the graph and coordinate mapping table.
- Download CSV or PDF when you need saved results.
Sample points for y = -2(x - 2)² + 2
| Parent point | Mapped point | Check using target equation |
|---|---|---|
| (0, 0) | (2, 2) | -2(2 - 2)² + 2 = 2 |
| (1, 1) | (3, 0) | -2(3 - 2)² + 2 = 0 |
| (-1, 1) | (1, 0) | -2(1 - 2)² + 2 = 0 |
| (2, 4) | (4, -6) | -2(4 - 2)² + 2 = -6 |
| (-2, 4) | (0, -6) | -2(0 - 2)² + 2 = -6 |
Understanding the Transformation
The parent parabola is y = x². It has a vertex at the origin. It opens upward. Its axis of symmetry is x = 0. The target equation is y = -2(x - 2)² + 2. Vertex form makes every change visible. The number inside brackets controls horizontal movement. The outside multiplier controls vertical size and reflection. The final constant controls vertical movement. Reading these parts separately avoids common graphing mistakes.
Read the Target Equation
In y = a(x - h)² + k, a equals -2. The value of h equals 2. The value of k equals 2. Therefore, move the parent graph right two units. Next, stretch it vertically by factor two. Then reflect it across the x-axis. Finally, move it up two units. The vertex becomes (2, 2). The axis becomes x = 2. Because a is negative, the parabola opens downward.
Track Points Accurately
Useful parent points include (0, 0), (1, 1), and (-1, 1). A horizontal shift moves every x-coordinate. A vertical stretch changes each y-distance from the axis. Reflection reverses the vertical direction. The final vertical shift changes every y-coordinate equally. For this equation, the parent vertex moves to (2, 2). The points one unit from the vertex become (1, 0) and (3, 0). These are also the x-intercepts.
Use the Calculator Results
Enter values for a, h, and k. The calculator writes the equation and lists transformations. It also finds the vertex, axis, intercepts, opening direction, and range. Adjust the parent-point limits to inspect more coordinates. The graph compares y = x² with the transformed curve. Export the coordinate table when you need a record. Check the displayed order before copying steps into homework or notes.
Check Your Work
Use the vertex first. Substitute x = 2 into the target equation. The result should be 2. Then test x = 1 and x = 3. Both outputs should be zero. These symmetric points confirm the axis x = 2. A downward opening confirms the negative multiplier. A taller curve confirms the factor two stretch. This method works for any equation written in vertex form.
Do not confuse a right shift with the sign inside parentheses. The expression x minus 2 means right two units. The expression x plus 2 means left two units. Always match equation to vertex form; interpret h.
Transformation questions
1. What is the parent function?
The parent function is y = x². It has vertex (0, 0), opens upward, and has axis x = 0.
2. What does the negative sign do?
A negative a-value reflects the parabola across the x-axis. The graph then opens downward.
3. Why does x - 2 move the graph right?
Vertex form uses the opposite direction inside parentheses. Therefore, x - 2 produces a right shift of two units.
4. What does the factor 2 change?
The absolute value 2 vertically stretches the graph. Each y-distance from the vertex doubles before the final vertical shift.
5. What is the vertex of the target graph?
The vertex is (2, 2). In y = a(x - h)² + k, the vertex is always (h, k).
6. What is the axis of symmetry?
The axis of symmetry is x = 2. It passes through the vertex and divides the parabola into matching halves.
7. Where are the x-intercepts?
The x-intercepts are (1, 0) and (3, 0). Substituting either x-value into the equation gives zero.
8. Does the graph open upward or downward?
It opens downward because a equals -2. Any negative leading multiplier produces a downward opening parabola.
9. Can stretch and reflection be reversed?
Yes. A vertical stretch and reflection both operate on y-values. Their order does not change the final transformed graph.
10. What range does the target graph have?
The range is y ≤ 2. The vertex has the maximum y-value because the parabola opens downward.
11. Can I use different values?
Yes. Change a, h, and k to analyze another vertex-form parabola. The graph, table, and exports update after submission.