S3-4.A1.c Use a graphing software to investigate how the positions of the graph y = ax² + bx + c vary with the sign of b² - 4ac, and describe the graph when b² - 4ac < 0.
S3-4.A1.d Use a graphing software to investigate the relationship between the number of points of intersection and the nature of solutions of a pair of simultaneous equations, one linear and one quadratic.
S3-4.A1.e Examine the solution of a quadratic equation and that of its related quadratic inequality (e.g. 4x² + x - 5 + 0 and 4x² + x - 5 > 0), and describe both solutions and their relationship.
S3-4.G2.6 Transformation of given relationships, including y = axn and y = kb×, to linear form to determine the unknown constants from a straight line graph.
S3-4.G2.a Relate the gradient of a straight line to the tangent of the angle between the line and the positive direction of the x-axis, and deduce the relationship between the gradients of (i) two parallel lines and (ii) two perpendicular lines.
S3-4.G2.b Discuss how to solve geometry problems involving finding (i) the equation of a line perpendicular or parallel to a given line, (ii) the coordinates of the midpoint of a line segment (horizontal, vertical and oblique), and (iii) equation of the perpendicular bisector of a line segment.
S3-4.C1.8 Stationary points (maximum and minimum turning points and stationary points of inflexion)
S3-4.C1.9 Use of second derivative test to discriminate between maxima and minima
S3-4.C1.10 Apply differentiation to gradients, tangents and normals, connected rates of change and maxima and minima problems
S3-4.C1.11 Integration as the reverse of differentiation
S3-4.C1.12 Integration of xn for any rational n, sin x, cos x, sec² x and e to the x power, together with constant multiples, sums and differences
S3-4.C1.13 Integration of (ax + b)n for any rational n, sin (ax + b), cos(ax + b) and e (ax + b)
S3-4.C1.14 Definite integral as area under a curve
S3-4.C1.15 Evaluation of definite integrals
S3-4.C1.16 Finding the area of a region bounded by a curve and line(s) (excluding area of region between 2 curves)
S3-4.C1.17 Finding areas of regions below the x-axis
S3-4.C1.18 Application of differentiation and integration to problems involving displacement, velocity and acceleration of a particle moving in a straight line
S3-4.C1.a Relate the derivative of a function to the gradient of the tangent to a curve at a given point, including horizontal and vertical tangents.
S3-4.C1.b Distinguish between constant, average and instantaneous rate of change with reference to graphs.
S3-4.C1.c Relate the sign of the first derivative of a function to the behaviour of the function (increasing or decreasing), locate points on the graph where the derivative is zero, and describe the behaviour of the function before, at and after these points.
S3-4.C1.d Discuss cases where the second derivative test to discriminate between maxima and minima fails (e.g. y = x3, y = x4) and instead, use the first derivative test.
S3-4.C1.e Discuss examples of problems in real-world contexts (e.g. business and sciences), involving the use of differentiation.
S3-4.C1.f Explain what d/dx (f(x)), Γf(x)dx and Γba f (x) dx represent and make connections between
S3-4.C1.f.1 derivative and indefinite integral;
S3-4.C1.f.2 definite and indefinite integrals.
S3-4.C1.g Relate the area bounded by a curve and the y-axis to the area under the curve.
S3-4.C1.h Model the motion of a particle in a straight line, using displacement, velocity and acceleration as vectors (e.g. velocity in the positive direction of x-axis is positive), and explain the physical meanings of ds/dt and dv/dt, and their signs in relation to the motion.