Core 1 revision notes a
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This document provides a summary of core mathematics concepts including: 1) Linear graphs and equations such as y=mx+c and finding the equation of a line. 2) Quadratic equations and graphs including using the quadratic formula, completing the square, and finding the vertex and axis of symmetry. 3) Simultaneous equations and interpreting their solutions geometrically as the intersection of graphs. 4) Other topics covered include surds, polynomials, differentiation, integration, and areas under graphs.
Core 1 revision notes a
- 1. www.mathsbox.org.uk CORE 1 Summary Notes 1 Linear Graphs and Equations y = mx + c gradient = increase in y increase in x y intercept Gradient Facts § Lines that have the same gradient are PARALLEL § If 2 lines are PERPENDICULAR then m1 ´ m2 = – 1 or m2 = 1 m1e.g. 2y = 4x – 8 y = 2x – 4 gradient = 2 gradient of perpendicular line = -½ Finding the equation of a straight line e.g. Find the equation of the line which passes through (2,3) and (4,8) GRADIENT = y1 – y2 x1 – x2 GRADIENT = 3 – 8 = – 5 = 5 2 – 4 – 2 2 Method 1 y – y1 = m(x – x1) y – 3 = 5 (x – 2) 2 y = 5 x – 2 2 2y = 5x – 4 Using the point (2,3) Method 2 y = mx + c 3 = 5 2 ´ 2 + c c = – 2 y = 5 x – 2 2 2y = 5x – 4 Using the point (2,3) Finding the Mid-Point ( x1 y1) and (x2 y2) the midpoint is æ è ç x1 + x2 2 ' y1 + y2 ö 2 ø ÷ Given the points Finding the point of Intersection Treat the equations of the graphs as simultaneous equations and solve Find the point of intersection of the graphs y = 2x – 7 and 5x + 3y = 45 1
- 2. Substituting y = 2x – 7 gives 5x + 3(2x –7) = 45 5x + 6x – 21 = 45 11x = 66 x = 6 y = 2 x 6 – 7 y = 5 Point of intersection = (6, 5) 2 Surds § A root such as √5 that cannot be written exactly as a fraction is IRRATIONAL § An expression that involves irrational roots is in SURD FORM e.g. 3√5 § ab = a ´ b § a b = a b 75 – 12 = 5 ´ 5 ´ 3 – 2 ´ 2 ´ 3 = 5 3 – 2 3 = 3 3 e.g § RATIONALISING THE DENOMINATOR 3 + √2 and 3 √2 is called a pair of CONJUGATES The product of any pair of conjugates is always a rational number e.g. (3 + √2)(3 √2) = 9 3√2+3√22 = 7 2 1 – 5 Rationalise the denominator of 2 1 – 5 ´ 1 + 5 1 + 5 = 2 + 2 5 1 – 5 = 2 + 2 5 – 4 = – 1 – 5 2 3. Quadratic Graphs and Equations Solution of quadratic equations § Factorisation x = x 2 – 3x – 4 = 0 (x + 1)(x – 4) = 0 – 1 or x = 4 2 www.mathsbox.org.uk
- 3. § Completing the square x 2 – 4x – 3 = 0 (x – 2) 2 – (2) 2 – 3 = 0 (x – 2) 2 – 7 = 0 (x – 2) 2 = 7 x – 2 = ± 7 x = 2 + 7 or x = 2 – 7 § Using the formula to solve ax2 + bx + c = 0 x = – b ± b 2 – 4ac 2a E.g Solve x2 - 4x - 3 = 0 x = – ( – 4) ± ( – 4) 2 – 4 ´ 1 ´ ( – 3) 2 ´ 1 = 4 ± 28 2 = 2 ± 7 § The graph of y = ax2 + bx + c crosses the y axis at y = c It crosses or touches the x-axis if the equation has real solutions The DISCRIMINANT of ax2 + bx + c = 0 is the expression b2 – 4ac If b2 – 4ac >0 there are 2 real distinct roots If b2 – 4ac = 0 there is one repeated root If b2 – 4ac < 0 there are no real roots Graphs of Quadratic Functions § The graph of any quadratic expression in x is called a PARABOLA § The graph of y – q = k(x - p)2 is a TRANSLATION of the graph y = kx2 In VECTOR notation this translation can be described as ú û ù ê ë é p q The equation can also be written as y = k(x – p)2 + q The VERTEX of the graph is (p,q) The LINE OF SYMMETRY is x = p 3 2x 2 + 4x + 5 = 2(x + 1) 2 + 3 Vertex (-1,3) Line of symmetry x = -1 y x1 2 3 4 5– 1– 2– 3– 4– 5 1 2 3 4 5 6 7 8 9 10 – 1 – 2 – 3 – 4 Translation of y=2x2 ú û ù ê ë é-1 3 www.mathsbox.org.uk
- 4. 4 Simultaneous Equations § Simultaneous equations can be solved by substitution to eliminate one of the variables Solve the simultanoeus equations y = 2x – 7 and x 2 + xy + 2 = 0 y = 7 + 2x so x 2 + x(7 + 2x) + 2 = 0 3x 2 + 7x + 2 = 0 (3x + 1)(x + 2) = 0 x = – 1 3 y = 6 1 3 or x = – 2 y = 3 § A pair of simultaneous equations can be represented as graphs and the solutions interpreted as points of intersection. If they lead to a quadratic equation then the DISCRIMINANT tells you the geometrical relationship between the graphs of the functions b2 – 4ac < 0 no points of intersection b2 – 4ac = 0 1 point of intersection b2 – 4ac > 0 2 points of intersection 5 Inequalities Linear Inequality § Can be solved like a linear equation except Multiplying or dividing by a negative value reverses the direction of the inequality sign e.g Solve – 3x + 10 £ 4 x – 3x + 10 £ 4 – 3x £ – 6 ³ 2 Quadratic Inequality § Can be solved by either a graphical or algebraic approach. e.g. solve the inequality x2 + 4x – 5 <0 Algebraic x2 + 4x – 5 < 0 factorising gives (x + 5)(x –1) < 0 Using a sign diagram x + 5 - - - 0 + + + + + + + x – 1 - - - - - - - 0 + + + (x + 5)(x – 1) + + + 0 - - - 0 + + + The product is negative for – 5 < x < 1 4 y x1 2– 1– 2– 3– 4– 5– 6– 7 2 – 2 – 4 – 6 – 8 – 10 Graphical The curve lies below the x–axis for –5 < x < 1 www.mathsbox.org.uk
- 5. 6 Polynomials Translation of graphs To find the equation of a curve after a translation of ú replace x with (x-p) and û ù ê ë é p q replace y with (y - p) e.g The graph of y = x3 is translated by ú û ù ê ë é 3 -1 The equation for the new graph is y=(x - 3)3 -1 Polynomial Functions A polynomial is an expression which can be written in the form a + bx + cx2 + dx3 + ex4 + fx5 (a, b, c..are constants) y x1 2 – 1 3 4 5– 1– 2– 3– 4– 5 1 2 3 4 5 – 2 – 3 – 4 – 5 · Polynomials can be divided to give a QUOTIENT and REMAINDER x2 -3x + 7 x + 2 x3 - x2 + x + 15 x3 +2x2 -3x2 + x Qutoient -3x2 - 6x 7x + 15 7x + 14 1 Remainder · REMAINDER THEOREM When P(x) is divided by (x - a) the remainder is P(a) · FACTOR THEOREM If P(a) = 0 then (x – a) is a factor of P(x) e.g. The polynomial f(x) = hx3 -10x2 + kx + 26 has a factor of (x - 2) When the polynomial is divided by (x+1) the remainder is 15. Find the values of h and k. Using the factor theorem f(2) = 0 8h -40 + 2k + 26 = 0 8h +2k = 14 Using the remainder theorem f(-1) = 15 -h -10 –k + 26 = 14 h + k = 2 Solving simultaneously k = 2 –h 8h + 2(2 –h) = 14 6h + 4 = 14 5 www.mathsbox.org.uk
- 6. · A circle with centre (a,b) and radius r has the equation (x - a)2 +( y - b)2 =r2 e.g. A circle has equation x2 + y2 + 2x – 6y= 0 Find the radius of the circle and the coordinates of its centre. x 2 + 2x + y 2 – 6y = 0 (x + 1) 2 – 1 + (y – 3) 2 – 9 = 0 (x + 1) 2 + (y – 3) 2 = 10 Centre (1, 3) radius = √10 · A line from the centre of a circle to where a tangent touches the circle is perpendicular to the tangent. A perpendicular to a tangent is called a NORMAL. e.g. C(-2,1) is the centre of a circle and S(-4,5) is a point on the circumference. Find the equations of the normal and the tangent to the circle at S. C(-2,1) S (-4,5) Gradient of SC is 1 – 5 – 2( – 4) = – 4 2 = – 2 Equation of SC y = – 2x + 7 Gradient of the tangent = – 1 – 2 = 1 2 Equation of y = 1 x + 7 2 · Solving simultaneously the equations of a line and a circle results in a quadratic equation. b2 - 4ac > 0 the line intersects the circle b2 - 4ac = 0 the line is a tangent to the circle b2 - 4ac < 0 the line fails to meet the circle 8 Rates of Change · The gradient of a curve is defined as the gradient of the tangent Gradient is denoted dx dy if y is given as a function of x Gradient is denoted by f ’(x) if the function is given as f(x) · The process of finding dy dx or f’(x) is known as DIFFERENTIATING · Derivatives f(x) = x n f '(x) = nx n – 1 f(x) = a f '(x) = 0 6 www.mathsbox.org.uk Equation of a Circle · A circle with centre (0,0) and radius r has the equation x2 +y2 =r2
- 7. dx y = x 3 + 4x 2 – 3x + 6 dy = 3x 2 + 8x – 3 9 Using Differentiation · If the value of dy dx is positive at x = a, then y is increasing at x = a · If the value of dy dx is negative at x = a, then y is decreasing at x = a · Points where dy dx = 0 are called stationary points Minimum and Maximum Points (Stationary Points) Stationary points can be investigated y x1 2 3 4 5– 1– 2– 3– 4– 5 1 2 3 4 5 – 1 – 2 – 3 – 4 – 5 + ve- ve GRADIENT GRADIENT 0 - ve+ ve Local Minimum Local Maximum 0 · by calculating the gradients close to the point (see above) · by differentiating again to find d 2 or f “(x) 2 y dx o d 2 > 0 then the point is a local minimum 2 y dx o d 2 2 y dx < 0 then the point is a local maximum Optimisation Problems Optimisation means getting the best result. It might mean maximising (e.g. profit) or minimising (e.g. costs) 10 Integration § Integration is the reverse of differentiation 7 www.mathsbox.org.uk
- 8. ò x n dx = x n + 1 n + 1 + c Constant of integration e.g. Given that f '(x) = 8x 3 – 6x and that f(2) = 9 find f(x) f(x) = ò 8x 3 – 6x dx = 8x – 6x + c 4 4 2 2 = 2x 4 – 3x 2 + c To find c use f(2) = 9 So f(x) = 2x4 – 3x2 –11 32 – 12 + c = 9 c = – 11 11 Area Under a Graph § The are under the graph of y = f(x) between x= a and x = b is found by evaluating the definite integral ò b f(x) dx a e.g. Calculate the area under the graph of y = 4x – x3 between the lines x = 0 and x = 2 y x1 2 3 4 5– 1 1 2 3 4 5 – 1 ò0 2 4x – x 3 dx = = 2x 2 – x 4 4 = (8 – 4) – (0 – 0) = 4 § An area BELOW the x–axis has a NEGATIVE VALUE 8 www.mathsbox.org.uk