Curved Graphs Quiz: Marking Scheme

This is the marking scheme for quiz: M18 - Curved Graphs Quiz
Topic Guide: M18 - Curved Graphs

Total: 40 marks

Accept equivalent notation and correct methods that produce a correct result.

  1. A — $y=(x-2)^2-3$
  2. B — reciprocal graph with two branches
  3. B — $x=3$
  4. C — tangent has the same gradient as the curve at the point of contact
  1. For $y=x^2-6x+5$:
    • 1 mark: Finds the axis of symmetry as $x=3$, using $x=-b/(2a)$ or an equivalent method.
    • 2 marks: Finds the turning point as $(3,-4)$.
      • 1 mark for substituting $x=3$ correctly.
      • 1 mark for obtaining $y=-4$ and stating the coordinates.
    • 1 mark: Identifies the turning point as a minimum because the coefficient of $x^2$ is positive.

  1. For $y=-3/x$:

    • 1 mark: States vertical asymptote $x=0$.
    • 1 mark: States horizontal asymptote $y=0$.
    • 1 mark: Identifies Quadrants II and IV.
    • 1 mark: Explains that $x=0$ is outside the domain and $y=-3/x$ cannot equal $0$ for any finite $x$.

    Accept “the graph approaches but does not meet the axes” when the domain/range explanation is clear.

  1. For $f(x)=x^2-4x+1$:
    • 2 marks: Completes the square correctly: $$f(x)=(x-2)^2-3.$$
    • 2 marks: Gives turning point $(2,-3)$ and identifies it as a minimum.
      • 1 mark for the coordinates.
      • 1 mark for correct classification with justification.
    • 2 marks: Finds the $x$-intercepts: $$x=2\pm\sqrt{3}.$$
      • 1 mark for applying the quadratic formula or equivalent correct method.
      • 1 mark for the exact solutions or coordinates.
    • 1 mark: Finds the $y$-intercept as $(0,1)$.
  1. For $f(x)=x^3-3x^2-9x+5$:
    • 3 marks: Differentiates and solves for stationary points: $$f’(x)=3x^2-6x-9=3(x-3)(x+1),$$ giving $x=-1$ and $x=3$.
      • 1 mark for the correct derivative.
      • 1 mark for correct factorisation or equivalent solution method.
      • 1 mark for both $x$-values.
    • 2 marks: Calculates $f(-1)=10$ and gives turning point $(-1,10)$.
    • 2 marks: Calculates $f(3)=-22$ and gives turning point $(3,-22)$.
    • 2 marks: Uses the second derivative correctly: $$f’’(x)=6x-6.$$
      • 1 mark for $f’’(-1)=-12<0$ and local maximum.
      • 1 mark for $f’’(3)=12>0$ and local minimum.
    • 2 marks: States correct end behaviour: $$x\to-\infty\Rightarrow f(x)\to-\infty,$$ $$x\to+\infty\Rightarrow f(x)\to+\infty.$$
  1. For $y=3x^4+2x^3-5x+7$:
    • 2 marks: Differentiates correctly: $$\frac{dy}{dx}=12x^3+6x^2-5.$$
    • 1 mark: Substitutes $x=1$ correctly.
    • 1 mark: Gives the gradient as $13$.

Grading Notes

  • Award method marks when a numerical error follows a valid method.
  • Do not award repeated marks for an answer copied without supporting working where working is required.
  • Accept “maximum” and “minimum” for “local maximum” and “local minimum” when context is clear.
  • Accept decimal approximations for Question 7 only when accompanied by, or converted from, the exact form $2\pm\sqrt{3}$.