IB Mathematics AI AHL Transformations of graphs MAI Study Notes - New Syllabus
IB Mathematics AI AHL Transformations of graphs MAI Study Notes
LEARNING OBJECTIVE
- Transformations of graphs.
Key Concepts:
- Translations of Graphs
- Reflections of Graphs
- Stretches of Graphs
- Composite Transformations of Graphs
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TRANSFORMATIONS OF FUNCTIONS
◆ BASIC TRANSFORMATIONS
Consider \( f(x) \).
Vertical Transformations:
\( f(x) + a \): Shift \( a \) units up.
\( f(x) – a \): Shift \( a \) units down.
\( bf(x) \): Vertical stretch by factor \( b \).
\( \frac{f(x)}{b} \): Vertical shrink by factor \( \frac{1}{b} \).
\( -f(x) \): Reflection over \( x \)-axis.
Horizontal Transformations:
\( f(x + a) \): Shift \( a \) units left.
\( f(x – a) \): Shift \( a \) units right.
\( f(bx) \): Horizontal shrink by factor \( \frac{1}{b} \).
\( f\left(\frac{x}{b}\right) \): Horizontal stretch by factor \( b \).
\( f(-x) \): Reflection over \( y \)-axis.
Example For \( f(x) = x^2 \): ▶️Answer/ExplanationSolution: For \( f(x) = x^2 \): |
◆ INVERSE FUNCTION TRANSFORMATION
The graph of \( f^{-1} \) is the reflection of \( f \) over \( y = x \).
Example For \( f(x) = x^2 \) (\( x \geq 0 \)) , \( f^{-1}(x) = \sqrt{x} \). ▶️Answer/ExplanationSolution:
The image of the point A(2,4) is A΄(4,2). |
ASYMPTOTES
Vertical Asymptote: \( x = a \) where \( f(x) \) is undefined.
Horizontal Asymptote: \( y = b \) as \( x \rightarrow \pm \infty \).
Example \( f(x) = \frac{1}{x} \): \( g(x) = \frac{1}{x – 1} + 2 \): ▶️Answer/ExplanationSolution: For \( f(x) = \frac{1}{x} \): For \( g(x) = \frac{1}{x – 1} + 2 \): |