C1 Revision

C1 Algebra and Functions

• Quadratic functions: If f(x)=ax^2+bx+c, the discriminant is b^2-4ac
• For f(x)=0, b^2-4ac>0 = two real, distinct roots,
• b^2-4ac=o = two real, equal roots,
• b^2-4ac<0 = no distinct real roots.
• Factorising, completing the square, using the formula.
• Sketching quadratic functions:
• a) find the point of intersection with the y-axis: put x=0 in y=f(x).
• b) find the points of intersection with the x-axis: solve f(x)=0.
• c)find the maximum/minimum point: use completing the square, symmetry or solve f '(x)=0.
• Other curves: reciprocal (y = 1/x), cubics.
• Expanding brackets, collecting like terms, factorising.
• Simultaneous equations (including one linear and one quadratic).
• Linear and quadratic inequalities.
• Transformation Description
• y=f(x)+a a>0 Translation of y=f(x) (0/a)
• y=f(x+a) a>0 Translation of y=f(x) (-a/0)
• y=af(x) a>0 Stretch in x, SF a
• y=f(ax) a>0 Stretch in y, SF 1/a

C1 Coordinate Geometry

• P(x1,y1) and Q(x2,y2)
• Gradient PQ = y2-y1 / x2-x1
• Distance PQ = square root: (x2-x1)^2 + (y2-y1)^2
• Equation of a straight line:
• (i) Given gradient + y intercept: y=mx+c
• (ii) Given point(x1,y1) on line and m: y-y1=m(x-x1)
• (iii) Given P(x1,y1) + Q(x2,y2) on line:
• y-y1 / y2-y1 = x-x1 / x2-x1
• Midpoint of PQ = (x1+x2 / 2 , y1+y2 / 2)
• Gradient of line l1 is m1, gradient of l2 is m2
• If l1 is parallel to l2, then m1=m2
• If l1 is perpendicular to l2, then m1 x m2 = -1

C1 Sequences and Series

• Sigma notation, e.g. E^4-r=1 (2r+5) = 7 + 9 + 11 + 13
• Un+1 = 3Un+5, n>_1, u1=-2 The first 5 terms of this sequence are -2, -1, 2, 11 and 38
• An arithmetic series is a series in which each term is obtained from the previous term by adding a constant called the common difference, d
• nth term = a + (n-1)d
• Sn = n/2 [2a + (n-1) d] or Sn = n/2 (a+l) where last term l = a + (n-1)d
• Sum of the first n natural numbers: 1 + 2 + 3 + 4 + ..... + n: Sn = n/2 (n+1)

C1 Differentiation

• Notation:If y=f(x) then dy/dx = f'(x) and d^2y/dx^2 = f'(x)
• y dy/dx
• ax^n anx^n-1 (a is constant)
• f(x) +/- g(x) f'(x) +/- g'(x)
• Equation of tangents and normals: Use the following facts:
• a) Gradient of a tangent to a curve = dy/dx
• b) The normal to a curve at a particular point is perpendicular to the tangent at that point
• c) If two perpendicular lines have gradients m1 and m2 then m1xm2 = -1
• d) The equation of a line through (x1,y1) with gradient m is y-y1 = m(x-x1)

C1 Integration

• /ax^ndx = ax^n+1/n+1 +c provided n =/ -1
• /(f'(x) + g '(x)) dx = f(x) + g(x) + c