IB Mathematics AA SL Transformations of graphs Study Notes
IB Mathematics AA SL Transformations of graphs Study Notes Offer a clear explanation of Transformations of graphs , including various formula, rules, exam style questions as example to explain the topics. Worked Out examples and common problem types provided here will be sufficient to cover for topic Transformations of graphs
Transformations of Graphs: Translations
Transformations of Graphs: Translations
A translation shifts a graph without changing its shape or orientation.
- Vertical translation: Adding or subtracting a constant outside the function moves the graph up or down.
Example: \( y = f(x) + k \) moves the graph up by \( k \) units (if \( k > 0 \)) or down (if \( k < 0 \)). - Horizontal translation: Adding or subtracting a constant inside the function moves the graph left or right.
Example: \( y = f(x – h) \) moves the graph right by \( h \) units (if \( h > 0 \)) or left (if \( h < 0 \)).
Example: Consider the basic function \( y = x^2 \).
Apply the following translations:
- The function is translated 2 units to the right.
- The function is translated 3 units up.
- The function is translated 1 unit left and 4 units down.
▶️Answer/Explanation
- \( y = (x – 2)^2 \): The graph is translated 2 units to the right.
- \( y = x^2 + 3 \): The graph is translated 3 units up.
- \( y = (x + 1)^2 – 4 \): The graph is translated 1 unit left and 4 units down.
Transformations of Graphs: Reflections
Transformations of Graphs: Reflections
A reflection flips the graph over a given axis.
- Reflection in the x-axis: The graph of \( y = -f(x) \) reflects over the x-axis. All y-values change sign.
- Reflection in the y-axis: The graph of \( y = f(-x) \) reflects over the y-axis. All x-values change sign.
Example: Consider the function \( y = x^3 \).
Apply the following reflections:
- Reflection over the x-axis.
- Reflection over the y-axis.
- Reflection over both axes .
▶️Answer/Explanation
- \( y = -x^3 \): Reflection over the x-axis. The graph is flipped upside down.
- \( y = (-x)^3 = -x^3 \): Reflection over the y-axis. For odd powers like cube, reflection over y-axis gives the same as over x-axis.
- \( y = -(-x)^3 = x^3 \): Reflection over both axes returns to the original.
Transformations of Graphs: Vertical Stretch
Transformations of Graphs: Vertical Stretch
A vertical stretch with scale factor \( p \) means multiplying all y-values of the function by \( p \).
New function: \( y = p \cdot f(x) \)
- If \( p > 1 \), the graph is stretched away from the x-axis (becomes taller).
- If \( 0 < p < 1 \), the graph is compressed towards the x-axis (becomes flatter).
Example: Consider the function \( y = x^2 \).
Apply a vertical stretch with scale factor \( p = 3 \):
▶️Answer/Explanation
Compare the graphs:
- \( y = x^2 \): standard parabola
- \( y = 3x^2 \): narrower, stretched vertically by factor of 3
The points (1,1) on the original graph become (1,3) on the stretched graph.
Transformations of Graphs: Horizontal Stretch
Transformations of Graphs: Horizontal Stretch
A horizontal stretch with scale factor \( \frac{1}{q} \) means replacing\( x \) with \( \frac{x}{q} \) in the function.
New function: \( y = f\left(\frac{x}{q}\right) \)
- If \( q > 1 \), the graph stretches horizontally (wider).
- If \( 0 < q < 1 \), the graph compresses horizontally (narrower).
Example: Consider the function \( y = x^2 \).
Apply a horizontal stretch with scale factor \( \frac{1}{2} \)
▶️Answer/Explanation
Compare the graphs:
- \( y = x^2 \): standard parabola
- \( y = \frac{x^2}{4} \): wider parabola (horizontal stretch)
The point (1,1) on the original graph corresponds to (2,1) on the stretched graph.
Composite Transformations
Composite Transformations of Graphs
A composite transformation occurs when two or more transformations are applied to a function, either simultaneously or in sequence.
Common transformations include:
- Vertical stretch/compression: Multiply the function by a constant → \( y = a f(x) \)
- Horizontal stretch/compression: Replace \( x \) by \( \frac{x}{q} \) → \( y = f\left( \frac{x}{q} \right) \)
- Vertical translation: Add/subtract constant → \( y = f(x) + k \)
- Horizontal translation: Replace \( x \) by \( x – h \) → \( y = f(x – h) \)
- Reflection: Reflect over x- or y-axis → \( y = -f(x) \) or \( y = f(-x) \)
Example: Start with \( y = x^2 \). Apply transformations to sketch \( y = 3x^2 + 2 \).
▶️Answer/Explanation
- Original function: \( y = x^2 \)
- Step 1: Vertical stretch → \( y = 3x^2 \) (makes the parabola narrower)
- Step 2: Vertical shift → \( y = 3x^2 + 2 \) (moves the graph up by 2 units)
The vertex of \( y = x^2 \) is at (0,0). After the transformations:
- \( y = 3x^2 + 2 \) has its vertex at (0, 2).
- The graph is narrower and shifted up by 2.