A Level H2 Mathematics Graphs Coordinate Geometry Quiz

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A-Level Maths H2 Quiz - Graphs Coordinate Geometry

Name: ____________________
Class: ____________________
Date: ____________________
Score: ________ / 55

Duration: 90 Minutes
Total Marks: 55

Instructions:

• Answer all questions.
• Use of a non-CAS Graphing Calculator (GC) is permitted.
• Show all necessary working.
• Sketch graphs clearly, labeling all key features (intercepts, asymptotes, stationary points).

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Section A: Function Sketching and Analysis (Questions 1–8)

1. Sketch the graph of $y = \frac{2x}{x-3}$ for $x \neq 3$. Label the asymptotes and intercepts.
   
   
   [3 marks]

2. Sketch the graph of $y = e^{2x} - 4$. State the coordinates of the $y$-intercept and the equation of the horizontal asymptote.
   
   
   [3 marks]

3. Sketch the graph of $y = \ln(x+2)$ for $x > -2$. Label the $x$-intercept and the vertical asymptote.
   
   
   [3 marks]

4. Given $f(x) = x^2 - 4x + 3$, sketch the graph of $y = |f(x)|$.
   
   
   [3 marks]

5. Sketch the graph of $y = \frac{1}{x^2-1}$. Clearly mark the vertical asymptotes and the $y$-intercept.
   
   
   [3 marks]

6. Sketch the graph of $y = 2\sin(2x)$ for $0 \le x \le \pi$. Label all intercepts.
   
   
   [3 marks]

7. Sketch the graph of $y = \frac{x+1}{x^2+1}$. Identify any stationary points using your GC.
   
   
   [4 marks]

8. Sketch the graph of $y = 3^{x-1} + 2$. State the equation of the horizontal asymptote.
   
   
   [3 marks]

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Section B: Data Interpretation and Scatter Diagrams (Questions 9–14)

9. A set of data consists of $(x, y)$ pairs: $(1, 2), (2, 5), (3, 7), (4, 11), (5, 14)$. Sketch a scatter diagram of this data on a small coordinate plane.
   
   
   [2 marks]

10. Based on the data in Question 9, comment on the suitability of a linear model for this data.
    
    
    [2 marks]

11. Sketch a scatter diagram for the data: $(10, 100), (20, 80), (30, 60), (40, 40), (50, 20)$.
    
    
    [2 marks]

12. For the data in Question 11, describe the correlation (positive/negative) and its strength.
    
    
    [2 marks]

13. Given a scatter diagram showing a strong curved relationship, suggest a transformation for $y$ that might linearize the data.
    
    
    [2 marks]

14. If a scatter diagram shows points $(2, 4), (4, 16), (6, 36), (8, 64)$, what is the most likely relationship between $x$ and $y$?
    
    
    [2 marks]

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Section C: Argand Diagram Loci (Questions 15–20)

15. On an Argand diagram, sketch the locus of $z$ such that $|z - (2 + i)| = 3$.
    
    
    [3 marks]

16. On an Argand diagram, sketch the locus of $z$ such that $|z - 1| = |z - 3i|$.
    
    
    [3 marks]

17. On an Argand diagram, sketch the locus of $z$ such that $\arg(z - 2) = \frac{\pi}{4}$.
    
    
    [3 marks]

18. Sketch the region on an Argand diagram satisfying $|z + i| \le 2$.
    
    
    [3 marks]

19. Sketch the locus of $z$ such that $\text{Re}(z) = 2$.
    
    
    [2 marks]

20. On a single Argand diagram, sketch the loci $|z - 2| = 2$ and $\arg(z) = \frac{\pi}{3}$. Label the point of intersection.
    
    
    [5 marks]

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Answer Key - A-Level Maths H2 Quiz: Graphs Coordinate Geometry

1. Graph of $y = \frac{2x}{x-3}$

   • Vertical asymptote: $x=3$.
   • Horizontal asymptote: $y=2$.
   • $x$-intercept: $(0,0)$.
   • $y$-intercept: $(0,0)$.
   • Shape: Hyperbola in quadrants relative to asymptotes.
   • [3 marks]

2. Graph of $y = e^{2x} - 4$

   • $y$-intercept: $(0, -3)$.
   • Horizontal asymptote: $y = -4$.
   • Shape: Exponential growth.
   • [3 marks]

3. Graph of $y = \ln(x+2)$

   • Vertical asymptote: $x = -2$.
   • $x$-intercept: $(-1, 0)$.
   • Shape: Logarithmic curve.
   • [3 marks]

4. Graph of $y = |x^2 - 4x + 3|$

   • Roots of $x^2 - 4x + 3$ are $x=1, 3$.
   • Graph is $y = x^2 - 4x + 3$ for $x \le 1$ or $x \ge 3$, and reflected for $1 < x < 3$.
   • Vertex of original was $(2, -1)$, now $(2, 1)$.
   • [3 marks]

5. Graph of $y = \frac{1}{x^2-1}$

   • Vertical asymptotes: $x=1, x=-1$.
   • $y$-intercept: $(0, -1)$.
   • Horizontal asymptote: $y=0$.
   • [3 marks]

6. Graph of $y = 2\sin(2x)$

   • Intercepts: $(0,0), (\frac{\pi}{4}, 0), (\frac{\pi}{2}, 0), (\frac{3\pi}{4}, 0), (\pi, 0)$.
   • Amplitude: 2. Period: $\pi$.
   • [3 marks]

7. Graph of $y = \frac{x+1}{x^2+1}$

   • $y$-intercept: $(0, 1)$. $x$-intercept: $(-1, 0)$.
   • Horizontal asymptote: $y=0$.
   • Stationary points (via GC): Max at approx $(0.414, 1.207)$, Min at approx $(-2.414, -0.207)$.
   • [4 marks]

8. Graph of $y = 3^{x-1} + 2$

   • Horizontal asymptote: $y = 2$.
   • $y$-intercept: $(0, 2\frac{1}{3})$.
   • Shape: Exponential growth.
   • [3 marks]

9. Scatter Diagram

   • Points plotted correctly on axes. $x$-axis (1-5), $y$-axis (2-14).
   • [2 marks]

10. Suitability

    • Points follow a very strong linear trend. Linear model is highly suitable.
    • [2 marks]

11. Scatter Diagram

    • Points plotted correctly. $x$-axis (10-50), $y$-axis (20-100).
    • [2 marks]

12. Correlation

    • Strong negative linear correlation.
    • [2 marks]

13. Transformation

    • $\ln y$ or $1/y$ (depending on the specific curve, usually $\ln y$ for exponential growth).
    • [2 marks]

14. Relationship

    • $y = x^2$ (Quadratic relationship).
    • [2 marks]

15. Locus $|z - (2 + i)| = 3$

    • Circle with centre $(2, 1)$ and radius 3.
    • [3 marks]

16. Locus $|z - 1| = |z - 3i|$

    • Perpendicular bisector of the line segment joining $(1, 0)$ and $(0, 3)$.
    • Equation: $y = \frac{1}{3}x + \frac{4}{3}$ (or sketched correctly).
    • [3 marks]

17. Locus $\arg(z - 2) = \frac{\pi}{4}$

    • Half-line (ray) starting at $(2, 0)$ extending at $45^\circ$ to the positive real axis.
    • Point $(2,0)$ is an open circle (excluded).
    • [3 marks]

18. Region $|z + i| \le 2$

    • Interior and boundary of a circle with centre $(0, -1)$ and radius 2.
    • [3 marks]

19. Locus $\text{Re}(z) = 2$

    • Vertical line passing through $(2, 0)$.
    • [2 marks]

20. Combined Loci

    • Circle: Centre $(2, 0)$, radius 2.
    • Ray: From $(0, 0)$ at $60^\circ$.
    • Intersection: Solve $r = 4\cos\theta$ or geometrically. Point is $(1, \sqrt{3})$.
    • [5 marks]