| Date | May 2022 | Marks available | 2 | Reference code | 22M.1.AHL.TZ2.3 |
| Level | Additional Higher Level | Paper | Paper 1 (without calculator) | Time zone | Time zone 2 |
| Command term | Solve | Question number | 3 | Adapted from | N/A |
Question
A function ff is defined by f(x)=2x-1x+1f(x)=2x−1x+1, where x∈ℝ, x≠-1.
The graph of y=f(x) has a vertical asymptote and a horizontal asymptote.
Write down the equation of the vertical asymptote.
Write down the equation of the horizontal asymptote.
On the set of axes below, sketch the graph of y=f(x).
On your sketch, clearly indicate the asymptotes and the position of any points of intersection with the axes.
Hence, solve the inequality 0<2x-1x+1<2.
Solve the inequality 0<2|x|-1|x|+1<2.
Markscheme
x=-1 A1
[1 mark]
y=2 A1
[1 mark]
rational function shape with two branches in opposite quadrants, with two correctly positioned asymptotes and asymptotic behaviour shown A1
axes intercepts clearly shown at x=12 and y=-1 A1A1
[3 marks]
x>12 A1
Note: Accept correct alternative correct notation, such as (12, ∞) and ]12,∞[.
[1 mark]
EITHER
attempts to sketch y=2|x|-1|x|+1 (M1)
OR
attempts to solve 2|x|-1=0 (M1)
Note: Award the (M1) if x=12 and x=-12 are identified.
THEN
x<-12 or x>12 A1
Note: Accept the use of a comma. Condone the use of ‘and’. Accept correct alternative notation.
[2 marks]
Examiners report
Syllabus sections
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19M.1.AHL.TZ2.H_11b:
State the range of g.
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19M.1.AHL.TZ2.H_11c:
Sketch the graphs of y=f(x) and y=g(x) on the same axes, clearly stating the points of intersection with any axes.
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17M.1.AHL.TZ2.H_9a.i:
Showing any x and y intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.
y=f(x);
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SPM.1.AHL.TZ0.4:
The following diagram shows the graph of y=f(x). The graph has a horizontal asymptote at y=−1. The graph crosses the x-axis at x=−1 and x=1, and the y-axis at y=2.
On the following set of axes, sketch the graph of y=[f(x)]2+1, clearly showing any asymptotes with their equations and the coordinates of any local maxima or minima.
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19M.1.AHL.TZ2.H_11d:
Find the coordinates of P.
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19M.1.AHL.TZ2.H_11a:
Describe the transformation by which f(x) is transformed to g(x).
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22M.1.AHL.TZ2.11a:
Sketch the curve y=f(x), clearly indicating any asymptotes with their equations. State the coordinates of any local maximum or minimum points and any points of intersection with the coordinate axes.
