Home / IB Mathematics SL 2.3 The graph of a function AI SL Paper 1- Exam Style Questions

IB Mathematics SL 2.3 The graph of a function AI SL Paper 1- Exam Style Questions- New Syllabus

IB DP Maths AI SL Paper 1 - All Topics

Question

Consider \( f(x) \). The graph of \(f\) for \(0<x\le 5\) is shown on the following axes.
(a)
(i) Sketch the graph of \(f\), for \(-5 \le x < 0\), on the same axes.
(ii) Write down the \(x\)-coordinate of the local minimum point. [4]
(b) Use your graphic display calculator to find the solutions to the equation \(f(x)=20\). [2]
(c) Write down the equation of the vertical asymptote for the graph of \(f\). [1]
▶️Answer/Explanation
Markscheme

(a)

(i) Sketching criteria (award A1A1A1):

• Correct shape in the second quadrant only and curve drawn smoothly. A1
• Asymptotic behaviour shown appropriately near the vertical asymptote. A1
• Key points placed approximately: left-most point around \(\;(-5,\,76)\;\) and the local minimum marked in a reasonable position (tolerance ≈ half a square). A1

(Axes and original plotted branch for \(0<x\le5\) are provided on the paper.)
(ii) The \(x\)-coordinate of the local minimum: \(\boxed{-0.941}\) \((\text{more precisely } -0.941035\ldots)\). A1

(b)

Using GDC intersection/solve for \(y_1=f(x)\) and \(y_2=20\): the three solutions are \[ \boxed{x\approx -2.44651,\;\; x\approx -0.252412,\;\; x\approx 2.69892} \] Award A2 for all three correct (A1 for any two). At most A1 if additional spurious solutions are included. If given as coordinates, accept \(({-}2.45,20),\;({-}0.252,20),\;(2.70,20)\) (2 s.f. earns A1A0). A2

(c)

The vertical asymptote is \(\boxed{x=0}\). A1
Total Marks: 7
Working / Notes
• (a)(ii) Read the minimum \(x\)-value from the continuation of the curve into \(-5\le x<0\) (as guided by the given branch for \(0<x\le5\)); technology reading gives \(-0.941035\ldots\).
• (b) On a GDC: plot \(y=f(x)\) and \(y=20\), then use Intersect (or Solve) to obtain the three \(x\)-values. Record to 3–5 s.f. as shown.
• (c) From the behaviour of \(f\) and the provided axes, the curve approaches the \(y\)-axis; hence the vertical asymptote is the line \(x=0\).
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