Function Transformations
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What Are Function Transformations?
A transformation is a change to a function that alters its graph in a predictable way. By understanding transformations, you can quickly graph complex functions by starting with a simple "parent function" and applying shifts, stretches, and reflections.
The Parent Functions
These are the basic forms you'll transform:
- Linear: f(x) = x
- Quadratic: f(x) = x²
- Cubic: f(x) = x³
- Square root: f(x) = √x
- Absolute value: f(x) = |x|
- Exponential: f(x) = bˣ
- Logarithmic: f(x) = log(x)
The General Transformation Form
Transformation Formula
g(x) = a · f(b(x - h)) + k
Where:
- a = vertical stretch/compression and reflection
- b = horizontal stretch/compression and reflection
- h = horizontal shift
- k = vertical shift
Vertical Transformations
Vertical Shift: f(x) + k
- k > 0: Shift up k units
- k < 0: Shift down |k| units
Vertical Stretch/Compression: a · f(x)
- |a| > 1: Vertical stretch (graph gets "taller")
- 0 < |a| < 1: Vertical compression (graph gets "shorter")
- a < 0: Reflection over x-axis
Horizontal Transformations
Important: Horizontal transformations work "opposite" to what you might expect!
Horizontal Shift: f(x - h)
- h > 0: Shift right h units (subtract inside → move right)
- h < 0: Shift left |h| units (add inside → move left)
Horizontal Stretch/Compression: f(bx)
- |b| > 1: Horizontal compression (graph gets "narrower")
- 0 < |b| < 1: Horizontal stretch (graph gets "wider")
- b < 0: Reflection over y-axis
Order of Transformations
Apply Transformations in This Order:
- Horizontal shift (inside the function)
- Horizontal stretch/compression (inside)
- Reflection over y-axis (if b is negative)
- Vertical stretch/compression (outside)
- Reflection over x-axis (if a is negative)
- Vertical shift (outside)
Summary Table
| Transformation | Notation | Effect on Graph |
|---|---|---|
| Vertical shift up | f(x) + k | Moves up k units |
| Vertical shift down | f(x) - k | Moves down k units |
| Horizontal shift right | f(x - h) | Moves right h units |
| Horizontal shift left | f(x + h) | Moves left h units |
| Vertical stretch | a·f(x), |a| > 1 | Stretches away from x-axis |
| Vertical compression | a·f(x), 0 < |a| < 1 | Compresses toward x-axis |
| Horizontal compression | f(bx), |b| > 1 | Compresses toward y-axis |
| Horizontal stretch | f(bx), 0 < |b| < 1 | Stretches away from y-axis |
| Reflection over x-axis | -f(x) | Flips upside down |
| Reflection over y-axis | f(-x) | Flips left-right |
Examples
Example 1: Describing Transformations
Problem: Describe the transformations applied to f(x) = x² to get g(x) = 3(x - 2)² + 5
Solution:
Compare to the form a·f(x - h) + k:
- a = 3: Vertical stretch by factor of 3
- h = 2: Horizontal shift right 2 units
- k = 5: Vertical shift up 5 units
The vertex moves from (0, 0) to (2, 5), and the parabola is narrower.
Example 2: Writing the Transformed Equation
Problem: Start with f(x) = √x. Write the equation after: shift left 3, reflect over x-axis, shift up 4.
Solution:
Step 1: Shift left 3: √(x + 3)
Step 2: Reflect over x-axis: -√(x + 3)
Step 3: Shift up 4: -√(x + 3) + 4
g(x) = -√(x + 3) + 4
Example 3: Horizontal Compression
Problem: How does g(x) = sin(2x) compare to f(x) = sin(x)?
Solution:
Since b = 2, and |b| > 1, this is a horizontal compression by factor of 1/2.
The period of sin(x) is 2π, so the period of sin(2x) is 2π/2 = π.
The graph completes one full cycle in half the distance - it's compressed horizontally.
Example 4: Multiple Transformations
Problem: Describe all transformations: g(x) = -2|x + 1| - 3
Solution:
Parent function: f(x) = |x| (V-shaped with vertex at origin)
Reading the transformations:
- (x + 1) means h = -1: shift left 1
- The coefficient -2: vertical stretch by 2 AND reflection over x-axis
- The -3 at the end: shift down 3
Vertex moves from (0, 0) to (-1, -3), graph opens downward and is steeper.
Example 5: Finding the Equation from a Graph
Problem: A parabola has vertex at (3, -2) and passes through (4, 1). Find its equation.
Solution:
Step 1: Use vertex form: f(x) = a(x - 3)² - 2
Step 2: Substitute the point (4, 1):
1 = a(4 - 3)² - 2
1 = a(1) - 2
a = 3
f(x) = 3(x - 3)² - 2
Practice
1. Describe all transformations: g(x) = (x + 4)² - 7
2. Describe all transformations: g(x) = -√(x - 5)
3. Start with f(x) = x³. Write the equation after: shift right 2, vertical stretch by 4, shift down 1.
4. Start with f(x) = |x|. Write the equation after: reflect over x-axis, shift left 3, shift up 6.
5. How does g(x) = 2ˣ⁺³ compare to f(x) = 2ˣ?
6. How does g(x) = log(x/2) compare to f(x) = log(x)?
7. The function f(x) = √x is transformed to have a starting point at (-2, 5). Write the equation.
8. What transformation turns f(x) into f(-x)? Into -f(x)?
9. A parabola has vertex at (-1, 4) and passes through (0, 2). Find its equation.
10. If f(2) = 7, what is the value of 3f(x - 4) + 1 when x = 6?
Click to reveal answers
- Shift left 4, shift down 7
- Shift right 5, reflect over x-axis
- g(x) = 4(x - 2)³ - 1
- g(x) = -|x + 3| + 6
- Shift left 3 (the +3 is inside the exponent)
- Horizontal stretch by factor of 2 (since log(x/2) = log((1/2)x))
- g(x) = √(x + 2) + 5
- f(-x) is reflection over y-axis; -f(x) is reflection over x-axis
- f(x) = -2(x + 1)² + 4
- When x = 6: 3f(6-4) + 1 = 3f(2) + 1 = 3(7) + 1 = 22
Check Your Understanding
1. Why does f(x - 3) shift the graph RIGHT instead of left?
Show answer
Think about what input gives you the "original" output. In f(x - 3), you need x = 3 to get f(0). The point that used to be at x = 0 is now at x = 3 - it moved right. In general, horizontal transformations work "opposite" because they affect the input before the function acts on it.
2. What's the difference between 2f(x) and f(2x)?
Show answer
2f(x) is a vertical stretch by factor of 2 - every y-value is doubled. f(2x) is a horizontal compression by factor of 1/2 - the graph is squeezed toward the y-axis. They affect different directions!
3. If you apply a vertical stretch followed by a vertical shift, does the order matter? What about horizontal transformations?
Show answer
Yes, order matters! 2f(x) + 3 is different from 2(f(x) + 3) = 2f(x) + 6. The first shifts up 3 after stretching; the second shifts up 3 before stretching (so the shift gets stretched too). Always apply stretches before shifts when writing in standard form.
4. How can you tell from an equation whether a function has been reflected over the x-axis, y-axis, or both?
Show answer
Look at the signs: a negative sign outside the function (like -f(x)) means x-axis reflection. A negative sign on x inside the function (like f(-x)) means y-axis reflection. Both negatives (like -f(-x)) means reflection over both axes, which is equivalent to a 180° rotation about the origin.
🚀 Next Steps
- Review any concepts that felt challenging
- Move on to the next lesson when ready
- Return to practice problems periodically for review