Function Transformations
Explore transformations, quadratic analysis, and wave parameters.
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About this topic
Linear Function Straight line through the origin.
The identity function — the output equals the input. It is the simplest linear relationship and the foundation of all straight-line equations.
Key formulas & rules
- •Domain: all real numbers
- •Range: all real numbers
- •Passes through (0, 0)
- •Constant slope = 1
- •Odd function: f(−x) = −f(x)
Examples
| Input | Output | Note |
|---|---|---|
| f(2) | 2 | Output matches input |
| f(−3) | −3 | Negative in, negative out |
| f(0) | 0 | Crosses at the origin |
| f(10) | 10 | Constant slope |
Real-world uses
Quadratic Function U-shaped parabola with vertex at origin.
Produces a symmetric parabola. Always non-negative, with its lowest point at the origin. Grows rapidly as |x| increases.
Key formulas & rules
- •Domain: all real numbers
- •Range: y ≥ 0
- •Symmetric about the y-axis (even function)
- •Vertex at (0, 0) — a minimum
- •Only one root: x = 0 (double root)
Examples
| Input | Output | Note |
|---|---|---|
| f(2) | 4 | 2 × 2 |
| f(−2) | 4 | Same as f(2) — symmetry |
| f(3) | 9 | Grows fast |
| f(0) | 0 | Vertex at origin |
Real-world uses
Cubic Function S-shaped curve passing through origin.
An odd function that grows quickly in both directions. It has an inflection point at the origin where the curve changes concavity.
Key formulas & rules
- •Domain: all real numbers
- •Range: all real numbers
- •Passes through (0, 0)
- •Odd function: f(−x) = −f(x)
- •No local maxima or minima
Examples
| Input | Output | Note |
|---|---|---|
| f(2) | 8 | 2 × 2 × 2 |
| f(−2) | −8 | Odd symmetry |
| f(3) | 27 | Grows very fast |
| f(1) | 1 | Identity point |
Real-world uses
Square Root Defined only for x ≥ 0, starts at origin.
The inverse of x² (restricted to x ≥ 0). It grows very slowly for large x and is only defined for non-negative inputs.
Key formulas & rules
- •Domain: x ≥ 0 only
- •Range: y ≥ 0
- •Inverse of y = x² (x ≥ 0)
- •Passes through (0, 0) and (1, 1)
- •Slope decreases as x increases
Examples
| Input | Output | Note |
|---|---|---|
| f(0) | 0 | Starts at origin |
| f(4) | 2 | √4 = 2 |
| f(9) | 3 | √9 = 3 |
| f(100) | 10 | Grows slowly |
Real-world uses
Cube Root Defined for all reals, passes through origin.
The inverse of x³. It is defined for every real number and grows very slowly. It has the same shape as √x but extends into negative x.
Key formulas & rules
- •Domain: all real numbers
- •Range: all real numbers
- •Odd function: f(−x) = −f(x)
- •Inverse of y = x³
- •Passes through (0, 0), (1, 1), (−1, −1)
Examples
| Input | Output | Note |
|---|---|---|
| f(8) | 2 | ∛8 = 2 |
| f(−8) | −2 | ∛(−8) = −2 |
| f(27) | 3 | ∛27 = 3 |
| f(0) | 0 | Origin |
Real-world uses
Absolute Value V-shaped graph with vertex at origin.
Returns the distance of x from zero. The output is always non-negative, and the graph has a sharp corner at the origin.
Key formulas & rules
- •Domain: all real numbers
- •Range: y ≥ 0
- •Even function: f(−x) = f(x)
- •Vertex at (0, 0)
- •Slope = 1 for x > 0, slope = −1 for x < 0
Examples
| Input | Output | Note |
|---|---|---|
| f(5) | 5 | Distance from 0 |
| f(−5) | 5 | Always positive |
| f(0) | 0 | Sharp corner here |
| f(-10) | 10 | Symmetry |
Real-world uses
Reciprocal Hyperbola with asymptotes at x = 0 and y = 0.
Undefined at x = 0. As x approaches 0 from either side, the output goes to ±∞. As x grows large, the output approaches 0.
Key formulas & rules
- •Domain: all reals except x = 0
- •Range: all reals except y = 0
- •Vertical asymptote at x = 0
- •Horizontal asymptote at y = 0
- •Odd function with two branches
Examples
| Input | Output | Note |
|---|---|---|
| f(1) | 1 | On the curve |
| f(2) | 0.5 | Halving input |
| f(0.5) | 2 | Doubling input |
| f(−1) | −1 | Odd symmetry |
Real-world uses
Sine Wave Smooth wave oscillating between −1 and 1.
A periodic function that repeats every 2π radians. It starts at the origin and rises to a peak at π/2 before returning to zero at π.
Key formulas & rules
- •Domain: all real numbers
- •Range: [−1, 1]
- •Period: 2π ≈ 6.283
- •Odd function: sin(−x) = −sin(x)
- •Passes through the origin
Examples
| Input | Output | Note |
|---|---|---|
| sin(0) | 0 | Starts at origin |
| sin(π/2) | 1 | Peak value |
| sin(π) | 0 | Half a period |
| sin(3π/2) | −1 | Trough value |
Real-world uses
Cosine Wave Like sine but shifted left by π/2.
A periodic function that starts at its peak (y = 1). It is the sine wave shifted horizontally by π/2 radians.
Key formulas & rules
- •Domain: all real numbers
- •Range: [−1, 1]
- •Period: 2π ≈ 6.283
- •Even function: cos(−x) = cos(x)
- •Starts at (0, 1) — the peak
Examples
| Input | Output | Note |
|---|---|---|
| cos(0) | 1 | Starts at peak |
| cos(π/2) | 0 | Zero crossing |
| cos(π) | −1 | Trough |
| cos(2π) | 1 | One full period |
Real-world uses
Exponential Rapid growth; approaches 0 for negative x.
Doubles for every unit increase in x. Always positive, with a horizontal asymptote at y = 0. Grows extremely fast for large positive x.
Key formulas & rules
- •Domain: all real numbers
- •Range: y > 0
- •Passes through (0, 1)
- •Horizontal asymptote at y = 0
- •Doubles each unit increase
Examples
| Input | Output | Note |
|---|---|---|
| 2⁰ | 1 | Always 1 at x = 0 |
| 2¹ | 2 | Doubles |
| 2³ | 8 | Grows fast |
| 2⁻¹ | 0.5 | Fraction for negatives |
Real-world uses
Natural Logarithm Inverse of eˣ; defined only for x > 0.
Answers the question: 'To what power must e be raised to get x?' It grows very slowly and has a vertical asymptote at x = 0.
Key formulas & rules
- •Domain: x > 0 only
- •Range: all real numbers
- •Vertical asymptote at x = 0
- •Passes through (1, 0)
- •Inverse of y = eˣ
Examples
| Input | Output | Note |
|---|---|---|
| ln(1) | 0 | ln(1) = 0 always |
| ln(e) | 1 | e is Euler's number |
| ln(e²) | 2 | Exponent comes down |
| ln(0.5) | −0.693 | Negative for < 1 |
Real-world uses
Modes & formulas
Transformations
a controls vertical stretch/reflection, b controls horizontal stretch/reflection, h shifts left/right, and k shifts up/down. |a| > 1 stretches; 0 < |a| < 1 compresses; negative a flips over the x-axis.
Quadratic Analysis
If Δ > 0, there are two real roots. If Δ = 0, there is one double root. If Δ < 0, there are no real roots. The sign of a determines whether the parabola opens up or down.
Wave Parameters
Amplitude A controls the wave's height. Frequency B controls how many cycles fit in a standard interval. Phase C slides the wave left or right. Vertical shift D moves the entire wave up or down.