This checkpoint assesses your understanding of circles $x^2 + y^2 = r^2$, the hyperbola $y = k/x$, exponential relationships $y = a^x$, comparing non-linear graphs, graphing non-linear relationships and real-world applications. It covers Lessons 12–17.
❓ Multiple Choice (10 questions)
1. What is the radius of the circle $x^2 + y^2 = 49$?
2. What is the centre of the circle $x^2 + y^2 = 16$?
3. The graph of $y = a^x$ (with $a > 1$) crosses the $y$-axis at:
4. Which equation is a hyperbola?
5. For the hyperbola $y = \dfrac{6}{x}$, what is $y$ when $x = 2$?
6. For $y = 2^x$, what is $y$ when $x = 4$?
7. The two branches of $y = \dfrac{5}{x}$ lie in which quadrants?
8. The horizontal asymptote of $y = 3^x$ is:
9. Does the point $(3, 4)$ lie on the circle $x^2 + y^2 = 25$?
10. A bacteria colony doubles every hour. Starting with 50 bacteria, how many are there after 3 hours?
✍ Short Answer (4 questions)
11. Consider the circle $x^2 + y^2 = 36$.
(a) State the centre and radius. (1 mark)
(b) Write the coordinates of the four points where the circle crosses the axes. (1 mark)
(c) Does the point $(6, 1)$ lie on, inside, or outside the circle? Justify. (1 mark)3 MARKS
12. Consider the hyperbola $y = \dfrac{12}{x}$.
(a) Complete the value of $y$ when $x = 3$ and when $x = -4$. (2 marks)
(b) State the equations of the two asymptotes. (1 mark)
(c) State which quadrants the branches lie in and why. (1 mark)4 MARKS
13. Consider the exponential $y = 2^x$.
(a) Complete the table of values for $x = 0, 1, 2, 3$. (2 marks)
(b) State the $y$-intercept and the equation of the horizontal asymptote. (2 marks)4 MARKS
14. Identifying and applying non-linear relationships.
(a) Name the family of graph for each equation: (i) $x^2 + y^2 = 100$, (ii) $y = \dfrac{4}{x}$, (iii) $y = 5^x$. (3 marks)
(b) A car's value depreciates: it is worth \$24000 now and halves in value every 3 years. What will it be worth in 6 years? (2 marks)5 MARKS
1. BThe right side is $r^2 = 49$, so $r = \sqrt{49} = 7$ (not 49 itself).
2. A$x^2 + y^2 = r^2$ is centred at the origin $(0, 0)$.
3. C$a^0 = 1$ for any $a > 0$, so every $y = a^x$ passes through $(0, 1)$.
4. D$y = \dfrac{8}{x}$ is of the form $y = \dfrac{k}{x}$, a hyperbola. (A is a parabola, B a circle, C an exponential.)
5. B$y = \dfrac{6}{2} = 3$.
6. C$2^4 = 2 \times 2 \times 2 \times 2 = 16$.
7. A$k = 5 > 0$, so the branches sit in Quadrants 1 and 3 (both coordinates the same sign).
8. DEvery $y = a^x$ has horizontal asymptote $y = 0$ (the curve approaches but never touches the $x$-axis).
9. B$3^2 + 4^2 = 9 + 16 = 25$, which equals $r^2$, so the point lies on the circle.
10. CDoubling 3 times: $50 \times 2^3 = 50 \times 8 = 400$.
Q11 (3 marks): (a) Centre $(0, 0)$, radius $r = \sqrt{36} = 6$ [1]. (b) $(6, 0)$, $(-6, 0)$, $(0, 6)$, $(0, -6)$ [1]. (c) $6^2 + 1^2 = 36 + 1 = 37 > 36$, so $(6, 1)$ lies outside the circle [1].
Q12 (4 marks): (a) $x = 3$: $y = \dfrac{12}{3} = 4$; $x = -4$: $y = \dfrac{12}{-4} = -3$ [2]. (b) Asymptotes $x = 0$ and $y = 0$ [1]. (c) $k = 12 > 0$, so the branches are in Quadrants 1 and 3 [1].
Q13 (4 marks): (a) $x = 0 \to y = 1$; $x = 1 \to y = 2$; $x = 2 \to y = 4$; $x = 3 \to y = 8$ [2]. (b) $y$-intercept $(0, 1)$; horizontal asymptote $y = 0$ [2].
Q14 (5 marks): (a) (i) circle, (ii) hyperbola, (iii) exponential [3]. (b) 6 years = two halvings: $24000 \to 12000 \to 6000$. Worth \$6000 [2].
Tick when you have finished all questions and checked your answers.