IBDP Maths AHL 2.16 The graphs of the functions AA HL Paper 1- Exam Style Questions- New Syllabus
Question
Consider a function \( f \) characterized by the following features:
- a local maximum at the point \( A\left(-1, -\frac{5}{2}\right) \),
- a local minimum at the point \( B\left(0, \frac{15}{2}\right) \),
- a vertical asymptote defined by the line \( x = -\frac{1}{2} \),
- and an oblique asymptote defined by \( y = 5x + 5 \).
Syllabus Topic Codes (IB Mathematics AA HL):
• AHL 2.13: Rational functions; horizontal, vertical and oblique asymptotes — parts (a), (b)
• AHL 2.16: The graphs of the functions \( y=|f(x)| \) and \( y=\frac{1}{f(x)} \) — parts (a), (b)
▶️ Answer/Explanation
(a) Sketch of \( y = |f(x)| \):
• The vertical asymptote remains at \( x = -\frac{1}{2} \).
• The oblique asymptote \( y = 5x + 5 \) becomes \( y = |5x + 5| \), which is V-shaped with vertex at \( x = -1 \) (where \( 5x+5=0 \)).
• Local maximum at \( A(-1, -\frac{5}{2}) \) reflects to become a local minimum at \( (-1, \frac{5}{2}) \) after taking absolute value.
• Local minimum at \( B(0, \frac{15}{2}) \) stays at \( (0, \frac{15}{2}) \) since positive.
• Portions of the original graph below the x-axis (e.g., near \( A \)) are reflected above the x-axis.
correct vertical asymptote, reflected oblique asymptote, correct turning points after reflection.
(b) Sketch of \( y = \frac{1}{f(x)} \):
• Vertical asymptotes of \( \frac{1}{f} \) occur where \( f(x) = 0 \) (x-intercepts of original \( f \)), and where \( f \) has vertical asymptotes (here \( x = -\frac{1}{2} \)) become zeros of \( \frac{1}{f} \).
• Horizontal asymptote: As \( |x| \to \infty \), \( f(x) \approx 5x+5 \), so \( \frac{1}{f(x)} \approx \frac{1}{5x+5} \to 0 \). Thus \( y = 0 \) is a horizontal asymptote.
• Intercepts: \( \frac{1}{f(x)} = 0 \) occurs where \( f(x) \) has vertical asymptotes (i.e., \( x = -\frac{1}{2} \)). • Original local extrema become local extrema of reciprocal with reversed nature (max ↔ min) and reciprocal y-values.
correct vertical/horizontal asymptotes, intercept at \( (-\frac{1}{2}, 0) \), behaviour near original turning points.