IB Mathematics SL 2.3 The graph of a function AI SL Paper 2 - Exam Style Questions - New Syllabus
▶️ Answer/Explanation
Markscheme (concise working)
(a) Sketch of \(y=e^x-2x-5\) on \([-4,3]\)
Required features for full credit:
- Roots approximately placed: one in \(-3<x<-2\), one in \(2<x<3\).
- y-intercept & local minimum: \(f(0)=1-0-5=-4\); \(f'(x)=e^x-2=0\Rightarrow x=\ln2\approx0.693\) and \(f(\ln2)=2-2\ln2-5\approx-4.386\).
- Endpoints: \(f(-4)=e^{-4}+8-5\approx3.02\) and \(f(3)=e^3-11\approx9.09\); curve decreasing near \(x\approx0.69\) then increasing fast to the right.
A1 A1 A1
Method 1 (analytic features): compute \(f(0)\), stationary point via \(f'(x)\), evaluate endpoints, locate roots by sign change (or technology) in the stated intervals.
Method 2 (technology check): plot a quick GDC table/graph to confirm the intercepts, minimum near \(x=\ln2\), and end values, then sketch with correct relative positions.
Method 2 (technology check): plot a quick GDC table/graph to confirm the intercepts, minimum near \(x=\ln2\), and end values, then sketch with correct relative positions.
(b) Transformation from \(f\) to \(g\)
Method 1 (composition): \[ f(3x)=e^{3x}-6x-5 \;\Rightarrow\; g(x)=f(3x)-2. \] Thus a horizontal stretch with scale factor \(\displaystyle k=\tfrac{1}{3}\) (compression by 3), followed by a vertical translation \(c=-2\) (down 2).
Method 2 (rule matching): For a general transform \(y=f(qx)+b\), the effect is a horizontal stretch of factor \(\frac{1}{q}\) and vertical translation \(b\). Comparing \(g(x)=e^{3x}-6x-7\) with \(f(x)=e^x-2x-5\) gives \(q=3\Rightarrow k=\tfrac{1}{3}\) and \(b=-2\Rightarrow c=-2\).
Hence \(\boxed{k=\tfrac{1}{3}},\ \boxed{c=-2}\). A1 A1
Method 2 (rule matching): For a general transform \(y=f(qx)+b\), the effect is a horizontal stretch of factor \(\frac{1}{q}\) and vertical translation \(b\). Comparing \(g(x)=e^{3x}-6x-7\) with \(f(x)=e^x-2x-5\) gives \(q=3\Rightarrow k=\tfrac{1}{3}\) and \(b=-2\Rightarrow c=-2\).
Hence \(\boxed{k=\tfrac{1}{3}},\ \boxed{c=-2}\). A1 A1
Total Marks: 5