トピック / GCSE / Mathematics / Algebra Cubic, reciprocal, exponential and circle graphs 遊んで学ぼう
問題に答えてエネルギーを集めたら、釣りや探検を楽しもう。アカウント不要。
教育者の方へ： Cubic, reciprocal, exponential and circle graphs（Mathematics、Algebra）向けのすぐ使えるレッスンスライド, 復習ノート — レッスンで使うか、学習者がライブゲームとして遊ぶインタラクティブなクラス活動としてトピックを実施できます。
レッスンノート Cubic graphs A cubic has equation y = ax³ + bx² + cx + d with a ≠ 0. The simplest, y = x³, passes through the origin and has rotational symmetry. A cubic with positive x3 rises from bottom-left to top-right; with negative x3 it falls. It can cross the x-axis up to three times. Reciprocal graphs For k ≠ 0, y = k/x is a hyperbola with two separate branches. It is undefined at x = 0. It never touches the axes: the axes are asymptotes . x = 0 is undefined. The graph is symmetrical about the lines y = x and y = -x. Cubic and reciprocal curves Cubic and reciprocal curves O −6 −4 −2 2 4 6 −3 −2 −1 1 2 3 x y y = 1/x y = x³ Exponential graphs y = ax (with a > 0, a \neq 1) grows or decays rapidly. y = 2x passes through (0, 1) because 20 = 1. For a > 1 the graph rises ever more steeply; for 0 < a < 1 (e.g. y = (\frac{1}{2})x ) it decays towards the x-axis, which is an asymptote. Never negative: y > 0 for all x. Exponential growth Exponential growth O 1 2 3 4 5 6 7 8 9 −3 −2 −1 1 2 3 x y y = 2ˣ (0, 1) Circle graphs A circle with centre the origin and radius r has equation x2 + y2 = r2 . For x2 + y2 = 25, the radius is 5. To check if a point lies on the circle, substitute it: (3, 4) gives 9 + 16 = 25, so it does. The tangent to a circle is perpendicular to the radius at the point of contact. x² + y² = 25 x² + y² = 25 r = 5 units
Recognising graphs Link the shape to the equation: line, parabola, cubic S-shape, hyperbola, exponential curve, circle. スライド Sign up free to view the lesson slides
Step through every slide for this topic — plus flashcards and revision notes — with a free account.
練習問題 無料プレビュー — 35問中8問。すべて見るには登録を。 1. What is the shape of the graph of y = \frac{1}{x} called?
Easy A parabolaB hyperbolaC ellipseD cubic2. What is y when x = 0 on y = 2x ?
Easy 3. What is y when x = 3 on y = 2x ?
Easy 4. What is y when x = 2 on y = x3 ?
Easy 5. What is y when x = -2 on y = x3 ?
Easy 6. What is y when x = 4 on y = \frac{1}{x}?
Easy 7. What is y when x = -2 on y = \frac{6}{x}?
Easy 8. Which value of x can't be used in y = \frac{1}{x}?
Easy もっと問題 9. What does the graph of y = 2x approach as x becomes very negative?
Easy A y = 1B y = 0C y = -1D y becomes large10. What is the radius of x2 + y2 = 49?
Easy 11. What is the radius of x2 + y2 = 10?
Easy 12. Which point lies on x2 + y2 = 25?
Medium A (3, 4)B (4, 4)C (5, 5)D (2, 3)13. Which point lies on x2 + y2 = 13?
Medium A (2, 3)B (3, 3)C (1, 4)D (3, 4)14. Which equation gives a circle of radius 6 centred at the origin?
Medium A x2 + y2 = 6B x2 + y2 = 12C x2 + y2 = 36D x + y = 615. Which equation matches a graph that passes through (0, 1) and rises steeply?
Medium A y = 3x B y = x3 C y = \frac{1}{x}D y = x2 16. Which graph is symmetrical about the origin (rotational symmetry) and S-shaped?
Medium A y = x2 B y = x3 C y = 2x D y = |x|17. Which graph has the x-axis as an asymptote?
Medium A y = x3 B y = 2x C y = x2 D y = 2x18. What is y when x = -1 on y = 2x ?
Medium 19. The curve y = 10 \times 2x passes through (0, k). What is k?
Medium 20. A tangent touches the circle x2 + y2 = 25 at (3, 4). What is the gradient of the radius to that point?
Medium A \frac{4}{3}B \frac{3}{4}C -\frac{3}{4}D -\frac{4}{3}21. The tangent at (3, 4) to x2 + y2 = 25 has gradient:
Medium A \frac{4}{3}B \frac{3}{4}C -\frac{3}{4}D -\frac{4}{3}22. The graph y = \frac{1}{x} lies in which quadrants?
Easy A 1st and 2ndB 1st and 3rdC 2nd and 4thD 3rd and 4th23. Which table gives y = x3 for x = -1, 0, 1?
Easy A -1, 0, 1B 1, 0, 1C -3, 0, 3D 1, 0, -124. For y = \frac{1}{x}, what happens to y as x increases from 1 to 100?
Medium A y increasesB y decreases towards 0C y stays the sameD y becomes negative25. Which has no x-intercept?
Medium A y = x3 B y = 2x C y = x2 D y = x - 126. What is the centre of the circle x2 + y2 = 9?
Easy A (3, 3)B (0, 0)C (9, 0)D (0, 9)27. y = x3 passes through the origin.
Easy True or false?
28. The graph of y = 2x can be negative.
Easy True or false?
29. x2 + y2 = 25 is the equation of a circle of radius 5 centred at the origin.
Easy True or false?
30. The graph of y = \frac{1}{x} crosses the y-axis.
Easy True or false?
31. The point (4, 4) lies on x2 + y2 = 25.
Easy True or false?
32. The tangent to a circle is perpendicular to the radius at the point of contact.
Medium True or false?
33. Which graphs pass through (0, 1)?
Medium A y = 2x B y = x2 + 1C y = \frac{1}{x}D y = x3 E y = 5x 34. Which points lie on x2 + y2 = 25?
Medium A (5, 0)B (0, -5)C (3, 4)D (4, 4)E (2, 5)35. Describe the graph of y = 2x .
Medium Unlock all 35 questions & more 無料アカウントを作って、このトピックのすべての問題・スライド・フラッシュカード・復習ノートを見よう。
過去問 このトピックの過去問練習は近日公開。 