2. 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).

Transcription

1 Quadratics Vertex Form 1. Part of the graph of the function y = d (x m) + p is given in the diagram below. The x-intercepts are (1, 0) and (5, 0). The vertex is V(m, ). (a) Write down the value of (i) m; (ii) p. Find d.. The diagram shows part of the graph of y = a (x h) + k. The graph has its vertex at P, and passes through the point A with coordinates (1, 0). y P 1 A x (a) Write down the value of (i) h; (ii) k. Calculate the value of a.

2 3. Consider the function f (x) = x 8x + 5. (a) Express f (x) in the form a (x p) + q, where a, p, q. Find the minimum value of f (x). 4. (a) Express f (x) = x 6x + 14 in the form f (x) = (x h) + 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 6x The quadratic function f is defined by f(x) = 3x 1x (a) Write f in the form f(x) = 3(x h) 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) + q. 6. Let f(x) = 3x. The graph of f is translated 1 unit to the right and 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) + q. The graph of h is the reflection of the graph of g in the x-axis. Write down the coordinates of the vertex of the graph of h. 7. Let f (x) = a (x 4) + 8. (a) Write down the coordinates of the vertex of the curve of f. Given that f (7) = 10, find the value of a. Hence find the y-intercept of the curve of f. 8. The function f is given by f (x) = x 6x + 13, for x 3. (a) Write f (x) in the form (x a) + b. Find the inverse function f 1. State the domain of f 1.

3 9. Let f (x) = x 1x + 5. (a) Express f(x) in the form f(x) = (x h) k. Write down the vertex of the graph of f. Write down the equation of the axis of symmetry of the graph of f. (d) Find the y-intercept of the graph of f. (e) p ± q The x-intercepts of f can be written as, where p, q, r r. Find the value of p, of q, and of r. 10. The function f (x) is defined as f (x) = (x h) + k. The diagram below shows part of the graph of f (x). The maximum point on the curve is P (3, ). y 4 P(3, ) x (a) Write down the value of (i) h; (ii) k. Show that f (x) can be written as f (x) = x + 6x 7.

4 11. Let f(x) = x + 4x 6. (a) Express f(x) in the form f(x) = (x h) + k. Write down the equation of the axis of symmetry of the graph of f. Express f(x) in the form f(x) = (x p)(x q). 1. (a) Express y = x 1x + 3 in the form y = (x c) + d. The graph of y = x is transformed into the graph of y = x 1x + 3 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. 13. The quadratic function f is defined by f (x) = 3x 1x (a) Write f in the form f (x) = 3(x h) 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) + q. 14. The following diagram shows part of the graph of f (x) = 5 x with vertex V (0, 5). h Its image y = g (x) after a translation with vector has vertex T (3, 6). k

5 (a) Write down the value of (i) h; (ii) k. Write down an expression for g (x). On the same diagram, sketch the graph of y = g ( x). 15. The diagram shows parts of the graphs of y = x and y = 5 3(x 4). y 8 y = x 6 y = 5 3( x 4) x The graph of y = x may be transformed into the graph of y = 5 3(x 4) 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; q.

6 16. Let f (x) = 3(x + 1) 1. (a) Show that f (x) = 3x + 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) Hence sketch the graph of f. () (d) Let g (x) = x. The graph of f may be obtained from the graph of g by the two transformations: a stretch of scale factor t in the y-direction followed by a translation of p. q p Find and the value of t. q

7 17. 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, 40). (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) + k. Show that f(x) can also be written in the form f(x) = x 10x. 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, for 0 t 6. (d) Find the value of t when the speed of the particle is greatest. 18. 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 = + 0t 5t, 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. 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.

8 19. The diagram shows part of the graph of the curve y = a (x h) + 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 =. () (3) Hence show that the equation of the curve can be written as y = x 1x (1)

9 Answers 1. (a) (i) m = 3 (ii) p = 1 d =. (a) (i) h = 1 (ii) k = a = (a) a =, p =, q = 3 Minimum value occurs at (, 3) 4. (a) f (x) = (x 3) + 5 Vertex is (3, 5) 5. (a) f(x) = 3(x ) 1 3(x 5) (a) (1, ) g (x) = 3(x 1) (1, ) 7. (a) Vertex is (4, 8) a = y = 4 8. (a) f (x) = (x 3) + 4 y = x x 4 9. (a) (x 3) 13

10 Vertex is (3, 13) x = 3 (must be an equation) (d) y-intercept is (0, 5) (accept 5) (e) p = 6, q = 6, r = 10. (a) h = 3 k = = x + x (a) f(x) = (x + 1) 8 x = 1 f(x) = (x 1)(x + 3) 1. (a) y = (x 3) + 5 (i) k = (ii) p = 3 (iii) q = (a) 3(x ) 1 3 (x 5) + 4 ( 14. (a) (i) h = 3 (ii) k = 1 g (x) = f (x 3) + 1 or 6 (x 3)

11 y V T x y = x y = 5 3( x 4) q = 5 k = 3, p = (a) 3x + 6x 9 (i) vertex is ( 1, 1) (ii) x = 1 (must be an equation) (iii) (0, 9) (iv) ( 3, 0), (1, 0)

12 y 3 1 x 9 1 p 1 (d) =, t = 3 1 q 17. (a) f (x) = 10(x + 4)(x 6) f (x) = 10(x 1) + 50 f (x) = x 10x (d) t =1 18. (a) h = 17 (i) (not shown) (ii) t = 3 or (a) h = 3 k = 1 a = x 1x + 19