2. The diagram shows the parabola y = (7 x)(l + x). The points A and C are the x-intercepts and the point B is the maximum point. f : x (x p)(x q).

Transcription

1 Quadratics 1. (a) Factorize x 2 3x 10. Solve the equation x 2 3x 10 = 0. (Total 4 marks) 2. The diagram shows the parabola y = (7 x)(l + x). The points A and C are the x-intercepts and the point B is the maximum point. y B A 0 C x Find the coordinates of A, B and C. (Total 4 marks) 3. The diagram represents the graph of the function f : x (x p)(x q). y x C (a) Write down the values of p and q. The function has a minimum value at the point C. Find the x-coordinate of C. (Total 4 marks) IB Questionbank Maths SL 1

2 4. The following diagram shows part of the graph of f, where f (x) = x 2 x 2. (a) Find both x-intercepts. Find the x-coordinate of the vertex. (4) 5. Let f(x) = p(x q)(x r). Part of the graph of f is shown below. The graph passes through the points ( 2, 0), (0, 4) and (4, 0). (a) Write down the value of q and of r. Write down the equation of the axis of symmetry. (1) (c) Find the value of p. IB Questionbank Maths SL 2

3 6. Part of the graph of the function y = d (x m) 2 + p is given in the diagram below. The x-intercepts are (1, 0) and (5, 0). The vertex is V(m, 2). (a) Write down the value of (i) m; (ii) p. Find d. 7. The diagram shows part of the graph with equation y = x 2 + px + q. The graph cuts the x-axis at 2 and 3. y x Find the value of (a) p; q. (Total 4 marks) IB Questionbank Maths SL 3

4 8. Part of the graph of f (x) = (x p) (x q) is shown below. The vertex is at C. The graph crosses the y-axis at B. (a) Write down the value of p and of q. Find the coordinates of C. (c) Write down the y-coordinate of B. 9. The diagram shows part of the graph of y = a (x h) 2 + k. The graph has its vertex at P, and passes through the point A with coordinates (1, 0). y P 2 1 A x (a) Write down the value of (i) h; (ii) k. Calculate the value of a. IB Questionbank Maths SL 4

5 10. Let f(x) = 8x 2x 2. Part of the graph of f is shown below. (a) Find the x-intercepts of the graph. (4) (i) Write down the equation of the axis of symmetry. (ii) Find the y-coordinate of the vertex. (Total 7 marks) IB Questionbank Maths SL 5

6 11. The diagram shows the graph of the function y = ax 2 + bx + c. y x Complete the table below to show whether each expression is positive, negative or zero. Expression positive negative zero a c b 2 4ac b (Total 4 marks) 12. Consider the function f (x) = 2x 2 8x + 5. (a) Express f (x) in the form a (x p) 2 + q, where a, p, q. Find the minimum value of f (x). 13. (a) Express f (x) = x 2 6x + 14 in the form f (x) = (x h) 2 + k, where h and k are to be determined. Hence, or otherwise, write down the coordinates of the vertex of the parabola with equation y x 2 6x (Total 4 marks) IB Questionbank Maths SL 6

7 14. The quadratic function f is defined by f(x) = 3x 2 12x (a) Write f in the form f(x) = 3(x h) 2 k. The graph of f is translated 3 units in the positive x-direction and 5 units in the positive y-direction. Find the function g for the translated graph, giving your answer in the form g(x) = 3(x p) 2 + q. 15. Let f(x) = 3x 2. The graph of f is translated 1 unit to the right and 2 units down. The graph of g is the image of the graph of f after this translation. (a) Write down the coordinates of the vertex of the graph of g. Express g in the form g(x) = 3(x p) 2 + q. The graph of h is the reflection of the graph of g in the x-axis. (c) Write down the coordinates of the vertex of the graph of h. 16. Let f (x) = a (x 4) (a) Write down the coordinates of the vertex of the curve of f. Given that f (7) = 10, find the value of a. (c) Hence find the y-intercept of the curve of f. 17. The function f is given by f (x) = x 2 6x + 13, for x 3. (a) Write f (x) in the form (x a) 2 + b. Find the inverse function f 1. (c) State the domain of f 1. IB Questionbank Maths SL 7

8 18. Let f (x) = 2x 2 12x + 5. (a) Express f(x) in the form f(x) = 2(x h) 2 k. Write down the vertex of the graph of f. (c) Write down the equation of the axis of symmetry of the graph of f. (d) Find the y-intercept of the graph of f. (1) (e) The x-intercepts of f can be written as Find the value of p, of q, and of r. p ± r q, where p, q, r. (7) (Total 15 marks) 19. The equation of a curve may be written in the form y = a(x p)(x q). The curve intersects the x-axis at A( 2, 0) and B(4, 0). The curve of y = f (x) is shown in the diagram below. y 4 2 A B x (a) (i) Write down the value of p and of q. (ii) Given that the point (6, 8) is on the curve, find the value of a. (iii) Write the equation of the curve in the form y = ax 2 + bx + c. (5) (i) Find d y. dx (ii) A tangent is drawn to the curve at a point P. The gradient of this tangent is 7. Find the coordinates of P. (4) IB Questionbank Maths SL 8

9 (c) The line L passes through B(4, 0), and is perpendicular to the tangent to the curve at point B. (i) Find the equation of L. (ii) Find the x-coordinate of the point where L intersects the curve again. (6) (Total 15 marks) 20. The function f (x) is defined as f (x) = (x h) 2 + k. The diagram below shows part of the graph of f (x). The maximum point on the curve is P (3, 2). y 4 2 P(3, 2) x (a) Write down the value of (i) h; (ii) k. Show that f (x) can be written as f (x) = x 2 + 6x 7. (1) (c) Find f ʹ (x). The point Q lies on the curve and has coordinates (4, 1). A straight line L, through Q, is perpendicular to the tangent at Q. (d) (i) Calculate the gradient of L. (ii) Find the equation of L. (iii) The line L intersects the curve again at R. Find the x-coordinate of R. (8) (Total 13 marks) IB Questionbank Maths SL 9

10 21. The quadratic equation 4x 2 + 4kx + 9 = 0, k > 0 has exactly one solution for x. Find the value of k. (Total 4 marks) 22. The quadratic equation kx 2 + (k 3)x + 1 = 0 has two equal real roots. (a) Find the possible values of k. (5) Write down the values of k for which x 2 + (k 3)x + k = 0 has two equal real roots. (Total 7 marks) 23. The equation kx 2 + 3x + 1 = 0 has exactly one solution. Find the value of k. 24. Consider f(x) = 2kx 2 4kx + 1, for k 0. The equation f(x) = 0 has two equal roots. (a) Find the value of k. The line y = p intersects the graph of f. Find all possible values of p. (5) (Total 7 marks) 25. The equation x 2 2kx + 1 = 0 has two distinct real roots. Find the set of all possible values of k. 26. Let f(x) = 2x 2 + 4x 6. (a) Express f(x) in the form f(x) = 2(x h) 2 + k. Write down the equation of the axis of symmetry of the graph of f. (1) (c) Express f(x) in the form f(x) = 2(x p)(x q). IB Questionbank Maths SL 10

11 27. Consider two different quadratic functions of the form f (x) = 4x 2 qx The graph of each function has its vertex on the x-axis. (a) Find both values of q. For the greater value of q, solve f (x) = 0. (c) Find the coordinates of the point of intersection of the two graphs. 28. Let f(x) = ax 2 + bx + c where a, b and c are rational numbers. (a) The point P( 4, 3) lies on the curve of f. Show that 16a 4b + c = 3. The points Q(6, 3) and R( 2, 1) also lie on the curve of f. Write down two other linear equations in a, b and c. (c) These three equations may be written as a matrix equation in the form AX = B, a where X = b. c (i) Write down the matrices A and B. (ii) Write down A 1. (iii) Hence or otherwise, find f(x). (8) (d) Write f(x) in the form f(x) = a(x h) 2 + k, where a, h and k are rational numbers. (Total 15 marks) Let M = 0 0, and O = 3 4. Given that M 2 6M + ki = O, find k. 0 0 IB Questionbank Maths SL 11

12 30. (a) Express y = 2x 2 12x + 23 in the form y = 2(x c) 2 + d. The graph of y = x 2 is transformed into the graph of y = 2x 2 12x + 23 by the transformations a vertical stretch with scale factor k followed by a horizontal translation of p units followed by a vertical translation of q units. Write down the value of (i) k; (ii) p; (iii) q. 31. The quadratic function f is defined by f (x) = 3x 2 12x (a) Write f in the form f (x) = 3(x h) 2 k. The graph of f is translated 3 units in the positive x-direction and 5 units in the positive y-direction. Find the function g for the translated graph, giving your answer in the form g (x) = 3(x p) 2 + q. IB Questionbank Maths SL 12

13 32. The following diagram shows part of the graph of f (x) = 5 x 2 with vertex V (0, 5). Its image y = g (x) after a translation with vector h has vertex T (3, 6). k (a) Write down the value of (i) h; (c) (ii) k. Write down an expression for g (x). On the same diagram, sketch the graph of y = g ( x). IB Questionbank Maths SL 13

14 33. The diagram shows parts of the graphs of y = x 2 and y = 5 3(x 4) 2. y 8 y = x 2 6 y = 5 3( x 4) x The graph of y = x 2 may be transformed into the graph of y = 5 3(x 4) 2 by these transformations. A reflection in the line y = 0 a vertical stretch with scale factor k a horizontal translation of p units a vertical translation of q units. followed by followed by followed by Write down the value of (a) k; p; (c) q. (Total 4 marks) IB Questionbank Maths SL 14

15 34. Let f (x) = 3(x + 1) (a) Show that f (x) = 3x 2 + 6x 9. For the graph of f (i) (ii) (iii) (iv) write down the coordinates of the vertex; write down the equation of the axis of symmetry; write down the y-intercept; find both x-intercepts. (8) (c) Hence sketch the graph of f. (d) Let g (x) = x 2. The graph of f may be obtained from the graph of g by the two transformations: Find a stretch of scale factor t in the y-direction followed by p a translation of. q p and the value of t. q (Total 15 marks) IB Questionbank Maths SL 15

16 35. The following diagram shows part of the graph of a quadratic function f. The x-intercepts are at ( 4, 0) and (6, 0) and the y-intercept is at (0, 240). (a) Write down f(x) in the form f(x) = 10(x p)(x q). Find another expression for f(x) in the form f(x) = 10(x h) 2 + k. (4) (c) Show that f(x) can also be written in the form f(x) = x 10x 2. A particle moves along a straight line so that its velocity, v m s 1, at time t seconds is given by v = t 10t 2, for 0 t 6. (d) (i) Find the value of t when the speed of the particle is greatest. (ii) Find the acceleration of the particle when its speed is zero. (7) (Total 15 marks) IB Questionbank Maths SL 16

17 36. A ball is thrown vertically upwards into the air. The height, h metres, of the ball above the ground after t seconds is given by h = t 5t 2, t 0 (a) Find the initial height above the ground of the ball (that is, its height at the instant when it is released). Show that the height of the ball after one second is 17 metres. (c) At a later time the ball is again at a height of 17 metres. (i) (ii) Write down an equation that t must satisfy when the ball is at a height of 17 metres. Solve the equation algebraically. (4) (d) (i) Find d h. dt (ii) (iii) Find the initial velocity of the ball (that is, its velocity at the instant when it is released). Find when the ball reaches its maximum height. (iv) Find the maximum height of the ball. (7) (Total 15 marks) IB Questionbank Maths SL 17

18 37. A family of functions is given by f (x) = x 2 + 3x + k, where k {1, 2, 3, 4, 5, 6, 7}. One of these functions is chosen at random. Calculate the probability that the curve of this function crosses the x-axis. 38. (a) Let y = 16x x 256. Given that y has a maximum value, find (i) the value of x giving the maximum value of y; (ii) this maximum value of y. The triangle XYZ has XZ = 6, YZ = x, XY = z as shown below. The perimeter of triangle XYZ is 16. (4) (i) Express z in terms of x. (ii) Using the cosine rule, express z 2 in terms of x and cos Z. (iii) Hence, show that cos Z = Let the area of triangle XYZ be A. 5x 16. 3x (7) (c) Show that A 2 = 9x 2 sin 2 Z. (d) Hence, show that A 2 = 16x x 256. (4) (e) (i) Hence, write down the maximum area for triangle XYZ. (ii) What type of triangle is the triangle with maximum area? (Total 20 marks) IB Questionbank Maths SL 18

19 39. (a) Consider the equation 4x 2 + kx + 1 = 0. For what values of k does this equation have two equal roots? Let f be the function f (θ ) = 2 cos 2θ + 4 cos θ + 3, for 360 θ 360. Show that this function may be written as f (θ ) = 4 cos 2 θ + 4 cos θ + 1. (1) (c) Consider the equation f (θ ) = 0, for 360 θ 360. (i) (ii) How many distinct values of cos θ satisfy this equation? Find all values of θ which satisfy this equation. (5) (d) Given that f (θ ) = c is satisfied by only three values of θ, find the value of c. (Total 11 marks) 40. The following diagram shows part of the graph of a quadratic function, with equation in the form y = (x p)(x q), where p, q. (a) Write down (i) the value of p and of q; (ii) the equation of the axis of symmetry of the curve. Find the equation of the function in the form y = (x h) 2 + k, where h, k. (c) Find d y. dx (d) Let T be the tangent to the curve at the point (0, 5). Find the equation of T. (Total 10 marks) IB Questionbank Maths SL 19

20 41. The diagram shows part of the graph of the curve y = a (x h) 2 + k, where a, h, k. y P(5, 9) x. (a) The vertex is at the point (3, 1). Write down the value of h and of k. The point P (5, 9) is on the graph. Show that a = 2. (c) Hence show that the equation of the curve can be written as y = 2x 2 12x (1) dy (d) (i) Find. dx A tangent is drawn to the curve at P (5, 9). (ii) (iii) Calculate the gradient of this tangent. Find the equation of this tangent. (4) (Total 10 marks) IB Questionbank Maths SL 20