What is the Quadratic Function Transformation Formula?
Quadratic transformations modify the parent parabola f(x) = x2 using the standard vertex form equation:
y = a(b(x – h))2 + k
- Vertex Location: The new vertex of the parabola is at (h, k).
- a (Vertical Stretch / Compression & Reflection):
- If |a| > 1: Vertical stretch (parabola becomes narrower).
- If 0 < |a| < 1: Vertical compression (parabola becomes wider).
- If a < 0: Reflection across the x-axis (parabola opens downwards).
- h (Horizontal Shift): Shifts right if h > 0, shifts left if h < 0. (Note: y = (x – 3)2 shifts right 3).
- k (Vertical Shift): Shifts up if k > 0, shifts down if k < 0.
Parent Function Baseline: f(x) = x2
All quadratic transformations originate from the parent graph y = x2 with its vertex at (0, 0) and axis of symmetry at x = 0.
| Parent x | -2 | -1 | 0 | 1 | 2 |
| Parent y = x2 | 4 | 1 | 0 | 1 | 4 |
Step-by-Step Quadratic Transformations
PART 1: Vertical Shifts (Transformations on the y-axis)
Guided Example: Graph y = x2 – 3
- Shift Direction & Amount: Shift down by 3 units (k = -3)
- Vertex Location: (0, -3)
- Method: Subtract 3 from all parent y-values.
| x | -2 | -1 | 0 | 1 | 2 |
| y = x2 – 3 | 1 | -2 | -3 | -2 | 1 |
Problem 1: Graph y = x2 + 2
- a. Shift Direction & Amount: Shift up by 2 units (k = +2)
- b. Vertex Location: (0, 2)
- c. Table of Values:
| x | -2 | -1 | 0 | 1 | 2 |
| y = x2 + 2 | 6 | 3 | 2 | 3 | 6 |
PART 2: Horizontal Shifts (Transformations on the x-axis)
Problem 2: Graph y = (x + 3)2
- a. Shift Direction & Amount: Shift left by 3 units (h = -3)
- b. Vertex Location: (-3, 0)
- c. Table of Values:
| x | -5 | -4 | -3 | -2 | -1 |
| y = (x + 3)2 | 4 | 1 | 0 | 1 | 4 |
PART 3: Combined Shifts (Both Axes)
Problem 3: Graph y = (x – 1)2 – 4
- a. Horizontal Shift: Right by 1 unit (h = 1)
- b. Vertical Shift: Down by 4 units (k = -4)
- c. Vertex (h, k): (1, -4)
- d. Table of Values:
| x | -1 | 0 | 1 | 2 | 3 |
| y = (x – 1)2 – 4 | 0 | -3 | -4 | -3 | 0 |
PART 4: Reflections Across Axes
Problem 4: Graph y = -(x + 2)2 + 1
- a. Parabola Opens: Downwards (reflected over x-axis since a = -1)
- b. Vertex (h, k): (-2, 1)
- c. Describe All Transformations: Reflection across x-axis, horizontal shift left 2 units, vertical shift up 1 unit
- d. Table of Values:
| x | -4 | -3 | -2 | -1 | 0 |
| y = -(x + 2)2 + 1 | -3 | 0 | 1 | 0 | -3 |
PART 5: Vertical Stretches & Compressions
Problem 5: Graph y = 2x2 (Vertical Stretch)
- a. Transformation Type: Vertical stretch by a factor of 2
- b. Shape: Graph is narrower than parent graph y = x2
- c. Table of Values:
| x | -2 | -1 | 0 | 1 | 2 |
| y = 2x2 | 8 | 2 | 0 | 2 | 8 |
Problem 6: Graph y = ½x2 (Vertical Compression)
- a. Transformation Type: Vertical compression by a factor of ½
- b. Scale Factor: ½ applied to y-values (graph is wider)
- c. Table of Values:
| x | -4 | -2 | 0 | 2 | 4 |
| y = ½x2 | 8 | 2 | 0 | 2 | 8 |
PART 6: Master Challenge (All Transformations Combined)
Problem 7: Graph y = -2(x – 3)2 + 8
- Step 1 (Vertex): (h, k) = (3, 8)
- Step 2 (Reflection): Opens downwards (reflected over x-axis because a = -2)
- Step 3 (Stretch / Compress): Vertical stretch by a factor of 2
- Step 4 (Table of Values):
| x | 1 | 2 | 3 | 4 | 5 |
| y = -2(x – 3)2 + 8 | 0 | 6 | 8 | 6 | 0 |
Frequently Asked Questions (AEO Section)
How do you find the vertex of a quadratic function in vertex form?
In standard vertex form y = a(x – h)2 + k, the vertex coordinate is directly given by (h, k). For example, in y = (x – 3)2 + 8, the vertex is (3, 8).
What causes a parabola to open downwards?
A parabola opens downwards when the vertical stretch factor a is negative (a < 0). This represents a reflection across the x-axis.
How do you tell if a quadratic function is stretched or compressed?
If the absolute value of a is greater than 1 (|a| > 1), the parabola undergoes a vertical stretch and appears narrower. If |a| is between 0 and 1 (0 < |a| < 1), the parabola undergoes a vertical compression and appears wider.
Want to master quadratic transformations and test your understanding? Download our complete, printable Quadratic Function Transformations Practice Worksheet (PDF) equipped with graphing grids, mapping tables, and answer keys!
👉 FOR MORE PRACTICE, DOWNLOAD THIS WORKSHEET (PDF)
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