IB Mathematics AA HL More Rational Functions Study Notes
IB Mathematics AA HL More Rational Functions Study Notes
IB Mathematics AA HL More Rational Functions Study Notes Offer a clear explanation of More Rational Functions , 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 More Rational Functions.
More Rational Functions: SL 2.8
In this section, we explore more complex rational functions, including their behavior, graph characteristics, and transformations.
Guidance & Clarification:
The Reciprocal Function:
- The reciprocal function is defined as \( f(x) = \frac{1}{x} \), \( x \neq 0 \).
- It exhibits a self-inverse nature, meaning \( f(f(x)) = x \).
Asymptotes:
- Vertical asymptote at \( x = 0 \).
- Horizontal asymptote at \( y = 0 \).
The graph is symmetric with respect to the origin, demonstrating odd function properties.
General Rational Function of Form \( \mathbf{\frac{ax + b}{cx + d}} \):
- Standard form: \( f(x) = \frac{ax + b}{cx + d} \), where \( c \neq 0 \).
Asymptotes:
- Vertical asymptote: \( x = -\frac{d}{c} \) (set \( cx + d = 0 \)).
- Horizontal asymptote: \( y = \frac{a}{c} \) (leading coefficient ratio).
Intercepts:
- X-intercept: Set \( ax + b = 0 \); \( x = -\frac{b}{a} \).
- Y-intercept: Evaluate \( f(0) = \frac{b}{d} \).
Graphing Steps and Key Features:
- Step 1: Identify the asymptotes (both vertical and horizontal).
- Step 2: Determine the x-intercept and y-intercept.
- Step 3: Sketch the graph by plotting key points and analyzing end behavior.
- Step 4: Analyze symmetry:
- If \( f(-x) = -f(x) \), the function is odd (symmetric about the origin).
- If \( f(-x) = f(x) \), the function is even (symmetric about the y-axis).
Example: Analyze \( f(x) = \frac{2x + 3}{x – 1} \).
- Vertical Asymptote: \( x = 1 \).
- Horizontal Asymptote: \( y = 2 \).
- X-intercept: \( x = -\frac{3}{2} \).
- Y-intercept: \( y = -3 \).
Transformation of Rational Functions:
- Vertical shifts: \( f(x) + k \), moves the graph up/down by \( k \) units.
- Horizontal shifts: \( f(x – h) \), moves the graph right/left by \( h \) units.
- Reflections:
- Reflection across x-axis: \( -f(x) \).
- Reflection across y-axis: \( f(-x) \).
- Vertical stretch/compression: \( af(x) \); if \( |a| > 1 \), the graph stretches away from the x-axis; if \( 0 < |a| < 1 \), the graph compresses towards the x-axis.
Examples of Rational Functions
Let’s explore some specific examples of rational functions to understand their behavior, graph characteristics, and transformations.
Example 1: Basic Reciprocal Function
Consider the reciprocal function \( f(x) = \frac{1}{x} \), \( x \neq 0 \).
- Vertical Asymptote: \( x = 0 \).
- Horizontal Asymptote: \( y = 0 \).
- Behavior: As \( x \to 0^+ \) or \( x \to 0^- \), \( f(x) \to \infty \) or \( f(x) \to -\infty \).
- Graph Characteristics: The graph is symmetric about the origin (odd function).
Example 2: Transformation of a Rational Function
Consider the transformation \( g(x) = \frac{1}{x – 2} + 1 \).
- Base Function: \( f(x) = \frac{1}{x} \).
- Horizontal Shift: \( x – 2 \) shifts the graph 2 units to the right.
- Vertical Shift: \( +1 \) shifts the graph 1 unit up.
- Vertical Asymptote: \( x = 2 \).
- Horizontal Asymptote: \( y = 1 \).
IB Mathematics AA SL More Rational Functions Exam Style Worked Out Questions
Question
The following diagram shows the graph of a function \(f\), for −4 ≤ x ≤ 2.
On the same axes, sketch the graph of \(f\left( { – x} \right)\).
Another function, \(g\), can be written in the form \(g\left( x \right) = a \times f\left( {x + b} \right)\). The following diagram shows the graph of \(g\).
Write down the value of a and of b.
▶️Answer/Explanation
Markscheme
A2 N2
[2 marks]
recognizing horizontal shift/translation of 1 unit (M1)
eg b = 1, moved 1 right
recognizing vertical stretch/dilation with scale factor 2 (M1)
eg a = 2, y ×(−2)
a = −2, b = −1 A1A1 N2N2
[4 marks]
Question
Let \(f(x) = 3{(x + 1)^2} – 12\) .
Show that \(f(x) = 3{x^2} + 6x – 9\) .[2]
For the graph of f
(i) write down the coordinates of the vertex;
(ii) write down the equation of the axis of symmetry;
(iii) write down the y-intercept;
(iv) find both x-intercepts.[8]
Hence sketch the graph of f .[2]
Let \(g(x) = {x^2}\) . The graph of f may be obtained from the graph of g by the two transformations:
a stretch of scale factor t in the y-direction
followed by a translation of \(\left( {\begin{array}{*{20}{c}}
p\\
q
\end{array}} \right)\) .
Find \(\left( {\begin{array}{*{20}{c}}
p\\
q
\end{array}} \right)\) and the value of t.[3]
▶️Answer/Explanation
Markscheme
\(f(x) = 3({x^2} + 2x + 1) – 12\) A1
\( = 3{x^2} + 6x + 3 – 12\) A1
\( = 3{x^2} + 6x – 9\) AG N0
[2 marks]
(i) vertex is \(( – 1{\text{, }} – 12)\) A1A1 N2
(ii) \(x = – 1\) (must be an equation) A1 N1
(iii) \((0{\text{, }} – 9)\) A1 N1
(iv) evidence of solving \(f(x) = 0\) (M1)
e.g. factorizing, formula,
correct working A1
e.g. \(3(x + 3)(x – 1) = 0\) , \(x = \frac{{ – 6 \pm \sqrt {36 + 108} }}{6}\)
\(( – 3{\text{, }}0)\), \((1{\text{, }}0)\) A1A1 N1N1
[8 marks]
A1A1 N2
Note: Award A1 for a parabola opening upward, A1 for vertex and intercepts in approximately correct positions.
[2 marks]
\(\left( {\begin{array}{*{20}{c}}
p\\
q
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
{ – 1}\\
{ – 12}
\end{array}} \right)\) , \(t = 3\) (accept \(p = – 1\) , \(q = – 12\) , \(t = 3\) ) A1A1A1 N3
[3 marks]