How to shift, reflect, stretch, and compress any function — plus the general form y = a · f(b(x−h)) + k and even/odd symmetry.
Every transformation starts from one of these eight parent functions. Know their basic shapes cold.
| Name | Equation | Shape / Key Feature | Domain |
|---|---|---|---|
| Linear | y = x | Straight line through origin, slope 1 | (−∞, ∞) |
| Quadratic | y = x² | Upward parabola, vertex at origin | (−∞, ∞) |
| Cubic | y = x³ | S-curve through origin, odd symmetry | (−∞, ∞) |
| Square Root | y = √x | Half-parabola starting at origin | [0, ∞) |
| Absolute Value | y = |x| | V-shape, vertex at origin | (−∞, ∞) |
| Reciprocal | y = 1/x | Two hyperbola branches, never touches axes | (−∞,0) ∪ (0,∞) |
| Exponential | y = bˣ (b > 0, b ≠ 1) | Rapid growth or decay, y-intercept at (0,1) | (−∞, ∞) |
| Logarithmic | y = log_b(x) | Slow growth, x-intercept at (1,0) | (0, ∞) |
Shifts the graph UP k units when k > 0, DOWN |k| units when k < 0. Every point moves the same vertical distance.
Shifts the graph RIGHT h units when h > 0, LEFT |h| units when h < 0. Counterintuitive: subtraction inside = rightward shift.
Multiplies every y-value by a. If |a| > 1: vertical stretch (taller). If 0 < |a| < 1: vertical compression (flatter). If a < 0: also reflects over x-axis.
If |b| > 1: horizontal compression (narrower). If 0 < |b| < 1: horizontal stretch (wider). Effect is the reciprocal of what you might expect.
Flips the graph vertically — every y-value changes sign. The x-axis acts as a mirror.
Flips the graph horizontally — every x-value changes sign. The y-axis acts as a mirror.
y = a · f(b(x − h)) + k
Horizontal shift — right by h (left if h < 0)
Vertical shift — up by k (down if k < 0)
Vertical stretch (|a| > 1), compression (0 < |a| < 1), or reflection (a < 0)
Horizontal compression (|b| > 1) or stretch (0 < |b| < 1)
Apply transformations in this order to avoid errors:
Memory aid: work from the inside out. The transformations closest to x (inside) are applied first; the ones farthest from x (outermost constant) are applied last.
Symmetric about the y-axis. Reflecting the graph across the y-axis produces the same graph.
Examples: y = x², y = x⁴, y = cos x, y = |x|
f(x) = x² → f(−x) = (−x)² = x² ✓
Symmetric about the origin. Rotating 180° about the origin produces the same graph.
Examples: y = x, y = x³, y = sin x
f(x) = x³ → f(−x) = (−x)³ = −x³ = −f(x) ✓
Most functions are neither even nor odd. A nonzero constant is even. The only function that is both even AND odd is f(x) = 0.
Example 1 — Describe all transformations of y = −2(x + 3)² − 1
Write in general form: y = a · f(b(x − h)) + k where f(x) = x²
Rewrite to match general form:
y = −2(x − (−3))² + (−1)
so: a = −2, b = 1, h = −3, k = −1
Result: Start with y = x², shift left 3, stretch vertically by 2, reflect over x-axis, shift down 1.
Example 2 — Write the equation for a square root function shifted right 5 and up 2, then reflected over the x-axis
Start from the parent f(x) = √x. Apply each transformation:
Parent: y = √x
Shift right 5: y = √(x − 5)
Shift up 2: y = √(x − 5) + 2
Reflect over x-axis (multiply by −1): y = −√(x − 5) + 2
Answer: y = −√(x − 5) + 2
Example 3 — Determine if f(x) = 3x⁴ − 5x² + 7 is even, odd, or neither
Step 1: Substitute −x:
f(−x) = 3(−x)⁴ − 5(−x)² + 7
Step 2: Simplify (even powers cancel the negative):
f(−x) = 3x⁴ − 5x² + 7
Step 3: Compare to f(x):
f(−x) = f(x) ✓
Answer: EVEN function — symmetric about the y-axis. (All exponents are even, constant term is fine.)
| Notation | Transformation | Direction / Effect | Common Mistake |
|---|---|---|---|
| f(x) + k | Vertical shift | Up if k > 0, down if k < 0 | Confusing with horizontal |
| f(x − h) | Horizontal shift | RIGHT if h > 0, left if h < 0 | Thinking − means left |
| a·f(x), |a| > 1 | Vertical stretch | Taller, y-values multiplied by a | Forgetting reflection if a < 0 |
| a·f(x), 0 < |a| < 1 | Vertical compression | Flatter, y-values shrink | Applying before shift |
| f(bx), |b| > 1 | Horizontal compression | Narrower (not stretch!) | Thinking large b = stretch |
| f(bx), 0 < |b| < 1 | Horizontal stretch | Wider | Forgetting it's reciprocal |
| −f(x) | Reflect over x-axis | Flips up/down | Forgetting to negate all y |
| f(−x) | Reflect over y-axis | Flips left/right | Confusing with x-axis reflection |
A function transformation changes the graph of a parent function by shifting, reflecting, stretching, or compressing it. The general form y = a·f(b(x−h)) + k captures all four types: h shifts the graph horizontally (right by h), k shifts it vertically (up by k), a stretches or reflects vertically, and b stretches or compresses horizontally. Every transformed graph is built from one of the basic parent functions.
This is the most counterintuitive fact in transformations. In f(x − h), you are replacing x with (x − h). For the function to produce the same output it did at x = 0, you now need x = h. So the entire graph moves RIGHT by h units when h > 0. Think of it this way: f(x − 3) reaches its original y-values 3 units later (to the right) on the x-axis. Inside the function, subtraction means rightward shift; addition means leftward shift.
Substitute −x for x and simplify: (1) If f(−x) = f(x), the function is EVEN — its graph is symmetric about the y-axis. Examples: y = x², y = cos x, y = |x|. (2) If f(−x) = −f(x), the function is ODD — its graph is symmetric about the origin. Examples: y = x³, y = sin x, y = x. (3) If neither condition holds, the function is neither even nor odd. A function cannot be both even and odd unless it is identically zero.
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