Function Transformation Game

Function Transformation Quiz/Game
Function transformation describes how simple changes to a function’s equation translate into movements, resizing, or flipping of its graph. Scroll down the page for a more detailed explanation.
 

This game helps you master function transformations (translations, reflections, and stretches). It has three modes:
Visual Mode, Text Mode and Mixed Mode.
 

 

How to Play the Graph Shifter Game

1. Select the mode:
   Visual Mode: You see the original (gray dashed) and the transformed (neon blue) graphs and must identify the rule.
   Text Mode: Asks you to translate between function notation (e.g., f(x-2)) and descriptive language (e.g., “Shift Right 2”).
   Mixed Mode: A combination of both.
   
2. Look at the Problem: Select one of the given answers.
   
3. Check Your Work: If you selected the right answer, it will be highlighted in green. If you are wrong, it will be highlighted in red and the correct answer will be highlighted in green.
   
4. Get a New Problem: Click “Next Scan” for a new problem.
   Your score is tracked, showing how many you’ve gotten right.
   
5. Finish Game When you have completed 10 questions, your final score will be displayed.
   
6. Play Again Click “Rebot System” to restart the game.
    
   

Understanding Function Transformations
Function transformation describes how the graph of a parent function, y = f(x), is shifted, stretched, compressed, or reflected based on specific changes to its equation.
The general form of a transformed function is often written as:
\(g(x) = a \cdot f(b(x - h)) + k\)

Variables	Type of Transformation	Effect on Graph
a	Vertical Scaling/Reflection	Vertical Stretch, Compression, or Reflection over the x-axis.
b	Horizontal Scaling/Reflection	Horizontal Stretch, Compression, or Reflection over the y-axis.
h	Horizontal Shift	Translates the graph left or right.
k	Vertical Shift	Translates the graph up or down.

1. Translations (Shifts)
Translations move the graph without changing its size or shape.
A. Vertical Shifts (Adding k Outside f(x))
A change made outside the function affects the output values (y) in the expected way.
g(x) = f(x) + k
If k > 0: The graph shifts up k units.
Example: g(x) = x² + 5 shifts the parabola y=x² up by 5.
If k < 0: The graph shifts down |k| units.
Example: g(x) = √x - 2 shifts the square root graph down by 2.

B. Horizontal Shifts (Subtracting h Inside f(x))
A change made inside the function affects the input values (x). This is counter-intuitive, meaning the graph moves in the opposite direction of the sign.
g(x) = f(x - h)
If h is positive (e.g., x - 3): The graph shifts right h units.
Example: g(x) = (x - 4)³ shifts the cube function y=x³ right by 4.
If h is negative (e.g., x + 3, which is x - (-3)): The graph shifts left |h| units.
Example: g(x) = |x + 1| shifts the V-shaped graph y=|x| left by 1.

2. Scaling and Reflections (Multipliers)
Scaling changes the size or shape of the graph (stretching or compressing). Reflections flip the graph across an axis.
A. Vertical Scaling and Reflection (Multiplying by a Outside f(x))
The factor a affects the y-values directly.
g(x) = a f(x)
If |a| > 1: Vertical Stretch by a factor of |a|.
Example: g(x) = 3 sin(x) triples the amplitude of the sine wave.
If 0 < |a| < 1: Vertical Compression by a factor of |a|.
Example: \(g(x) = \frac{1}{2} x^2\) makes the parabola wider.
If a < 0: Reflection across the x-axis (flips the graph upside down).
Example: g(x) = -f(x)

B. Horizontal Scaling and Reflection (Multiplying by b Inside f(x))
The factor b affects the x-values and is counter-intuitive. The graph is scaled by a factor of \(\frac{1}{|b|}\).
g(x) = f(bx)
If |b| > 1: Horizontal Compression by a factor of \(\frac{1}{|b|}\).
Example: g(x) = cos(2x) compresses the cosine wave by half (period is π).
If 0 < |b| < 1: Horizontal Stretch by a factor of \(\frac{1}{|b|}\).
Example: \(g(x) = f(\frac{1}{3}x)\) stretches the graph horizontally by 3.
If b < 0: Reflection across the y-axis (flips the graph left-to-right).
Example: g(x) = f(-x)

Order of Operations (The Transformation Sequence)
When multiple transformations are applied, the order matters! A good way to remember the sequence is to apply the transformations in the same order as they appear in the algebraic expression (closest to x first, then outwards), or more simply:
Horizontal Transformations (Shift h and Scale b): Usually done first, applied to the x-input.
Vertical Scaling/Reflection (a): Multiplication/division affects the y-values.
Vertical Translation (k): Addition/subtraction affects the final y-value.

Example Sequence
Consider the transformation from f(x) to g(x) = -2f(x+3) - 1.
Horizontal Shift h=-3): Shift Left 3 units (f(x+3)).
Vertical Stretch and Reflection (a=-2):
Stretch vertically by a factor of 2.
Reflect across the x-axis.
Vertical Shift (k=-1): Shift Down 1 unit.
 
This video gives a clear, step-by-step approach to explain the function transformations.

• Show Video

 

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