2020 HSC Mathematics Extension 1
Question 11(c):
Graph transformations
The diagram shows the graph of . (Accessible description of the given sketch: a continuous curve with a local maximum on the negative -axis side between vertical features near and ; it crosses/meets key heights, has zeros at and , and a local minimum value between those zeros. Use the official paper for the precise sketch.)
Sketch the graph of .
How to recognise this question
Question type: graph transformations
- The command and notation point to graph transformations.
- The requested response is a sketch.
How to handle it: track how each transformation moves key points, asymptotes and intercepts of the base graph.
Watch out: Do not select a formula until its domain, interval, sign and units match the question.
Step-by-step answer
- Zeros of become vertical asymptotes of . From the sketch, expect asymptotes at and .
- Points where are fixed points of the reciprocal map (same coordinates).
- Signs are preserved: where , also ; where , also .
- Where is large, is small (near the -axis). Where has a local minimum magnitude away from zero, has a local extreme.
- In particular, a local minimum maps to a local maximum (NESA sample features asymptotes at and local maximum at ).
- NESA: 3 marks for a correct sketch; partial credit for some correct features; 1 mark for asymptotes at or the local max at .
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 3 total marks for this group. Where the official marking guide provides useful partial-credit criteria, those criteria are stated in the steps above.
Try your hand at it again by answering these questions
Question 1
The graph of has zeros at and , is positive on , and has a maximum value at . Describe the key features of .
Question 2
Suppose , , , and for near . List three features of .
Question 3
A function satisfies , , , a local min. Sketch reasoning for near these values.