1.5Transformation of Functions
Apply vertical and horizontal shifts, reflections across axes, and vertical and horizontal stretches and compressions to transform parent functions.
Transformations let you graph any function by starting from a parent function and applying shifts, flips, and stretches. This skill is used constantly in graphing trig functions, exponentials, and conics.
Essential Question
How do changes to a function's equation — adding constants, multiplying by factors, or negating inputs or outputs — predictably move, flip, or stretch its graph?
Lesson Overview
A transformation changes the position, size, or orientation of a graph without changing its basic shape. All transformations start from a parent function (like f(x) = x², √x, or |x|) and apply one or more changes. There are four types: shifts (translations), reflections, stretches, and compressions. The general form g(x) = a·f(x − h) + k captures all of them: h controls horizontal shift, k controls vertical shift, a controls vertical stretch/compression and reflection. The trickiest rule: horizontal transformations work opposite to what you expect — f(x − h) shifts RIGHT by h, and f(x + h) shifts LEFT by h.
General Form
g(x) = a · f(x − h) + k
- h: horizontal shift (right if h > 0, left if h < 0)
- k: vertical shift (up if k > 0, down if k < 0)
- |a| > 1: vertical stretch (taller)
- 0 < |a| < 1: vertical compression (flatter)
- a < 0: reflection over x-axis
Order of Transformations
- Horizontal shift (replace x with x − h)
- Horizontal stretch/compression (replace x with bx)
- Reflection over y-axis (replace x with −x)
- Vertical stretch/compression (multiply by |a|)
- Reflection over x-axis (if a < 0, negate)
- Vertical shift (add k)
Transformation Summary
| Notation | Type | Effect on Graph |
|---|---|---|
| f(x) + k | Vertical shift up k | Moves graph UP k units |
| f(x) − k | Vertical shift down k | Moves graph DOWN k units |
| f(x + h) | Horizontal shift left h | Moves graph LEFT h units |
| f(x − h) | Horizontal shift right h | Moves graph RIGHT h units |
| −f(x) | Reflect over x-axis | Flips graph vertically |
| f(−x) | Reflect over y-axis | Flips graph horizontally |
| a·f(x), a > 1 | Vertical stretch | Stretches graph away from x-axis |
| a·f(x), 0 < a < 1 | Vertical compression | Compresses graph toward x-axis |
| f(bx), b > 1 | Horizontal compression | Compresses graph toward y-axis |
| f(bx), 0 < b < 1 | Horizontal stretch | Stretches graph away from y-axis |
Key insight: horizontal transformations work OPPOSITE to what you expect — f(x − h) shifts RIGHT, f(x + h) shifts LEFT.
Before & After: f(x) = x² vs g(x) = 2(x − 1)² + 3
Horizontal shift
(x − 1): shifts RIGHT 1. Remember: subtract = right, add = left.
Vertical stretch
Coefficient 2: stretches the parabola vertically (narrower).
Vertical shift
+ 3: shifts the entire graph UP 3 units. Vertex moves to (1, 3).
Reflections: f(x) vs −f(x) vs f(−x)
−f(x): reflect over x-axis
Negate the OUTPUT. Every y-value flips sign. Graph flips vertically.
f(−x): reflect over y-axis
Negate the INPUT. Every x-value flips sign. Graph flips horizontally.
Worked Examples
Describe all transformations applied to f(x) = x² to obtain g(x) = 3(x − 2)² + 1.
Compare to g(x) = a·f(x − h) + k: a = 3, h = 2, k = 1.
h = 2: horizontal shift RIGHT 2 units.
k = 1: vertical shift UP 1 unit.
a = 3 (positive, greater than 1): vertical stretch by factor of 3.
No reflection (a is positive).
Write the equation of f(x) = √x after: shift left 3, reflect over x-axis, shift down 2.
Start: f(x) = √x
Shift left 3: replace x with (x + 3) → √(x + 3)
Reflect over x-axis: negate the output → −√(x + 3)
Shift down 2: subtract 2 → −√(x + 3) − 2
The graph of f(x) passes through (1, 4), (2, 7), (3, 2). Find the corresponding points on g(x) = f(x − 1) + 3.
g(x) = f(x − 1) + 3 shifts every point RIGHT 1 and UP 3.
(1, 4) → (1+1, 4+3) = (2, 7)
(2, 7) → (2+1, 7+3) = (3, 10)
(3, 2) → (3+1, 2+3) = (4, 5)
Describe the transformation from f(x) = |x| to g(x) = −2|x + 4| − 1.
Compare to a·f(x − h) + k: a = −2, h = −4 (since x + 4 = x − (−4)), k = −1.
h = −4: horizontal shift LEFT 4.
a = −2: |a| = 2 → vertical stretch by 2; negative → reflect over x-axis.
k = −1: vertical shift DOWN 1.
Given f(x) = x³, write the equation for a graph that is compressed horizontally by factor 1/2 and shifted up 5.
Horizontal compression by 1/2 means b = 2 (replace x with 2x): f(2x) = (2x)³ = 8x³.
Shift up 5: add 5 → 8x³ + 5.
Alternatively using standard form: g(x) = (2x)³ + 5 = 8x³ + 5.
Guided Practice
External Supplemental Resource
Guided Practice Video: Transformation of Functions
Review horizontal and vertical shifts, reflections, stretches, and compressions before completing the guided practice problems.
Video by The Organic Chemistry Tutor on YouTube
Watch on YouTube ↗Describe all transformations applied to f(x) = x² to obtain g(x) = −(x + 1)² + 4.
Hint: Identify a, h, and k. Remember: (x + 1) means h = −1, so shift LEFT 1. Negative a means reflect over x-axis.
Write the equation of f(x) = x² after: shift right 5, vertical compression by 1/3, shift down 2.
Hint: Apply each transformation in order. Vertical compression by 1/3 means multiply by 1/3.
The graph of f passes through (0, 1), (2, 5), (−1, 3). Find the corresponding points on g(x) = 2f(x) − 4.
Hint: g multiplies every y-value by 2 then subtracts 4. Apply this to each y-coordinate.
Describe the transformation from f(x) = √x to g(x) = √(−x + 3).
Hint: Rewrite as √(−(x − 3)). The negative inside means reflect over y-axis; the −3 inside means shift right 3.
A function f has a maximum at (2, 6). After applying g(x) = −f(x − 3) + 1, where is the new maximum or minimum?
Hint: Shift right 3 moves x from 2 to 5. Reflect over x-axis turns max into min and negates y. Then add 1.
Key Vocabulary
Parent Function
The simplest form of a function family: f(x) = x², f(x) = √x, f(x) = |x|, f(x) = x³, etc. All transformations start here.
Vertical Shift
Adding k to f(x): g(x) = f(x) + k. Moves the graph UP k units (k > 0) or DOWN |k| units (k < 0).
Horizontal Shift
Replacing x with (x − h): g(x) = f(x − h). Shifts RIGHT h units (h > 0) or LEFT |h| units (h < 0). Opposite of what you expect!
Reflection over x-axis
Negating the output: g(x) = −f(x). Every y-value changes sign. The graph flips vertically.
Reflection over y-axis
Negating the input: g(x) = f(−x). Every x-value changes sign. The graph flips horizontally.
Vertical Stretch/Compression
Multiplying by a: g(x) = a·f(x). |a| > 1 stretches (taller). 0 < |a| < 1 compresses (flatter).
Interactive Practice — 5 Questions
Which transformation does g(x) = f(x − 3) represent?
What transformation produces g(x) = −f(x)?
The graph of f(x) = x² is shifted left 2 and up 5. What is the new equation?
In g(x) = 4f(x), what happens to the graph?
A point (3, 7) is on f(x). Where does it go on g(x) = f(x + 1) − 2?
Independent Practice
Independent Practice
Describe all transformations applied to f(x) = √x to obtain g(x) = 2√(x + 3) − 1.
Write the equation of f(x) = |x| after: reflect over x-axis, shift right 4, vertical stretch by 3, shift up 2.
The graph of f passes through (−2, 0), (0, 4), (3, −1). Find the corresponding points on g(x) = f(−x) + 2.
Identify the parent function and describe all transformations for h(x) = −(x − 1)³ + 5.
A function f has a vertex at (0, 0). After applying g(x) = 3f(x + 2) − 4, where is the new vertex? What is the vertical stretch factor?
Common Mistakes
Shifting f(x − 3) to the LEFT — thinking 'minus 3 means left'.
f(x − h) shifts RIGHT by h. f(x − 3) shifts RIGHT 3. f(x + 3) shifts LEFT 3. Horizontal shifts are always opposite to the sign.
Confusing vertical and horizontal stretch — thinking f(2x) stretches vertically.
f(2x) is a HORIZONTAL compression (graph gets narrower). 2·f(x) is a VERTICAL stretch (graph gets taller).
Applying transformations in the wrong order — doing vertical shift before reflection.
Apply transformations in order: horizontal shift → horizontal stretch → y-axis reflection → vertical stretch → x-axis reflection → vertical shift.
Forgetting that a < 0 in a·f(x) means BOTH a vertical stretch AND a reflection over the x-axis.
If a = −3, the graph is stretched by 3 AND reflected over the x-axis. Both effects happen simultaneously.
Math Tips
Horizontal transformations are INSIDE the function (affect x). Vertical transformations are OUTSIDE (affect y). Inside = opposite effect; outside = direct effect.
To find where a specific point goes: apply horizontal changes to x first, then vertical changes to y.
Memory trick for horizontal shifts: f(x − 3) = "x must be 3 MORE to get the same output" → graph shifts RIGHT 3.
The general form g(x) = a·f(x − h) + k tells you everything: h = horizontal shift, k = vertical shift, a = stretch/reflection.
Vertical stretch by a > 1 makes the graph TALLER (points move away from x-axis). Compression by 0 < a < 1 makes it FLATTER (points move toward x-axis).