Advanced Level Mathematics: Pure Mathematics 1 by Hugh Neill, Douglas Quadling, Julian Gilbey

By Hugh Neill, Douglas Quadling, Julian Gilbey

Written to compare the contents of the Cambridge syllabus. natural arithmetic 1 corresponds to unit P1. It covers quadratics, features, coordinate geometry, round degree, trigonometry, vectors, sequence, differentiation and integration.

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Ax 2 + bx + c? 11 Which of the following could be the equation of y the curve shown in the diagram? f rf 12 Which of the following could be the equation of the curve shown in the diagram? (a) y=-x 2 +3x+4 ! 6 The shapes of graphs of the form y = ~ 2 +bx+ c In Exercise 3C, you should have discovered a number of results, which are summarised in the box below. All the graphs have the same general shape, which is called a parabola. These parabolas have a vertical axis of symmetry. The point where a parabola meets its axis of symmetry is called the vertex.

Y Since the curve cuts the axes at (1,0) and (4,0), as in Fig. 18, the equation has the form y=a(x-l)(x-4). y=2(x-l)(x-4), or y = 2x 2 - •x A If\ Since the point (3,-4) Jies on this curve, -4=a(3-1)(3-4),giving -4=-2a,so a=2. The equation of the curve is therefore Fig. 18 lOx + 8 . Exercise 3E 1 Sketch the following graphs. 2 (a) y=(x-2)(x-4) (b) y=(x+3)(x-l) (c) y=x(x-2) (d) y = (x + 5)(x + 1) (e) y=x(x+3) (f) y = 2(x + l)(x-1) Sketch the following graphs. (x 2 ~ x -12) (g) y = - x 2 (h) y=-(x 2 -7x+12) (i) y = llx - 4x 2 (a) y = x 2 - 2x - 8 - 4x - 4 - 6 2 4 Find the equation, in the form y = x + bx + c , of the parabola which (a) crosses the x~axis at the points (2,0) and (5,0), (b) crosses the x-axis at the points (-7,0) and (-iO,O), (c) passes through the points (-5,0) and (3,0), (d) passes through the points (-3,0) and (1,-16).

2 (a) y=(x-2)(x-4) (b) y=(x+3)(x-l) (c) y=x(x-2) (d) y = (x + 5)(x + 1) (e) y=x(x+3) (f) y = 2(x + l)(x-1) Sketch the following graphs. (x 2 ~ x -12) (g) y = - x 2 (h) y=-(x 2 -7x+12) (i) y = llx - 4x 2 (a) y = x 2 - 2x - 8 - 4x - 4 - 6 2 4 Find the equation, in the form y = x + bx + c , of the parabola which (a) crosses the x~axis at the points (2,0) and (5,0), (b) crosses the x-axis at the points (-7,0) and (-iO,O), (c) passes through the points (-5,0) and (3,0), (d) passes through the points (-3,0) and (1,-16).

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