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Unit 1

Chapter 1 - The straight line

  • Distance formula      Distance formula

  • Mid-point formula     mid point formula
  • Gradient formula       gradient formula

    • Gradient - angle formula
    • Gradients of horizontal and vertical linesParallel and perpendicular lines
    • Product of perpendicular gradients =  -1

Forms of the equation of a line:

     y = mx + c               gradient  m  and y-intercept  c

     ax + by + c = 0       linear form - usually with integer coefficients

Finding the equation of a straight line

     y - b = m(x - a)       line has equation m and passes through point (a, b)


Lines in triangles

Median
of a triangle the line from a vertex to the mid-point of the opposite side.

Altitude
of a triangle - the line from a vertex perpendicular to the opposite side.

Perpendicular bisector
of a line - passes through the mid-point of the line at right angles to it.

Intersection
of two lines - solve the equations of the two lines simultaneously.



Chapter 2.1 - Composite and inverse functions

  • Domain and range
  • Composite Functions  f(g(x))
  • Working with fractions
  • Functions with inverses
  • Reflection in the line  y = x

 

Chapter 2.2 - Algebraic functions and graphs

  • Completing the square
    • Maximum and minimum values
    • Working with fractions

  • Sketching the graph of a quadratic equation

  • Graphs of related functions
    • y = -f(x)                  Reflect in x axis
    • y = f(-x)                  Reflect in y axis
    • y = f(x + a)            Move graph a units to the LEFT
    • y = f(x - a)             Move graph a units to the RIGHT
    • y = f(x) + a           Move graph a units UP
    • y = f(x) - a             Move graph a units DOWN
    • y = kf(x)                 Scale graph in y-direction
    • y = f(kx)                 Scale graph in x direction

            Remember to show the images of any given points on the graph.

  • Graphs of exponential functions - sketching them and recognising them
  • Graphs of logarithmic functions - sketching them and recognising them

       Special Logarithms

  • Make use of the two special logs   Special logs
           and the relationship between
           the exponential and logarithmic functions
      Log exponential

 

Chapter 2.3 - Trigonometric functions and radians

  • Degrees and radians
    • Changing between degrees and radians   radians

  • Angles larger than 90°
    • Angles of all sizes 0 to 360° and radian equivalent
    • Using ASTC (All Sinners Take Care) for angles in all 4 quadrants.

  • Exact Values
    • Knowing exact values for sin, cos, tan of 30°, 45°, 60° and radian equivalent

  • Trigonometric Graphs
    • Sketching and recognising trigonometric graphs in the form y = a sin bx + c

  • Equations
    • Solving basic trigonometric equations with solutions in 0 to 360° or radians

 

Chapter 3.1 - Differentiation

  • Differentiation gives the gradient function
    • The gradient function lets you calculate the gradient at any point on the graph.

  • IMPORTANT  - you can only differentiate powers of  x at this point.

  • Rules for differentiation
    • Must be in straight line form - change any surds to index form
    • Multiply by the index and then decrease the index by 1
    • The derivative of x is 1
    • The derivative of a constant is 0
    • Carry through any multipliers - integer or fractional.

  • You should be confident at differentiating:
    • Positive and negative indices
    • Fractional indices
    • Dealing with integer or fractional multipliers

  • You should be able to split fractions into separate powers of x
  • You should be able to multiply out brackets using FOIL.

  • Using Leibniz notation for differentiation  leibniz
    • Find the equation of a tangent to a curve at a given point.
    • Finding the point on a curve when you are given the value of the gradient.

  • Sketch the graph of a derived function
    • Also sketch the graph of a possible function given the derived function graph.


Chapter 3.2 - Using Differentiation

  • Finding stationary points
    • For an SP  f'(x) = 0

  • Maximum and minimum points
    • Nature of stationary points - Table of Signs

  • Maximum and minimum values on closed intervals

  • Problem Solving
    • Making a formula
    • Forming a constraint
    • Finding the maximum or minimun value.

  • The derivative as a rate of change

  • Velocity and acceleration
    • x is displacement, t is time
    • velocity = dx/dt
    • acceleration = dv/dt .
    • maximum height occurs when velocity = 0
    • The common linkage between displacement, velocity and acceleration is t the time of the event.

 

Chapter 4 - Sequences

  • nth term formulae - finding and generating terms.
  • Recurrence relation formulae - finding formula and generating terms

  • Using a recurrence relation to describe a problem

  • Using a geometric sequence
    • growth and decay
    • interest calculations
    • appreciation and depreciation

  • Finite and infinite sequences
    • Finding the limit of an infinite sequence as n tends to infinity

  • Solving problems with the linear recurrence relation
    • Condition for limit of a linear recurrence relation
    • Finding the limit of a recurrence relation

  • Working with recurrence relations algebraically
    • Sequences with the same limits
    • Using simultaneous equations
    • Finding constants
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Unit 2

Chapter 1.1 - Polynomials

  • Nested calculation and synthetic division
    • Divisor, quotient, remainder
    • Dividing a polynomial by (x - a)

  • Remainder Theorem
    • When a polynomial f(x) is divided by (x - h) then the remainder is f(h)

  • Factor Theorem
    • (x - h) is a factor of the polynomial f(x)  if the remainder   f(h) = 0
    • the converse is also true.

  • Applications
    • Finding factors of polynomial expressions
    • Solving polynomial equations (mainly cubics)

  • Approximate roots of f(x) = 0   -  Iteration

 

Chapter 1.2 - Quadratic Theory

  • Solving quadratic equations
    • by looking at the graph
    • factorisation
    • completing the square
    • the quadratic formula

  • The discriminant
    • > 0   roots are real and distinct
    • = 0   roots are identical
    • < 0   no real roots

  • Tangents to curves
    • when you solve the equations simultaneously
      • a line is a tangent to a curve, if you have equal roots.
      • a line intersects a curve, if you have real and distinct roots.
      • a line does not touch or intersect a curve, if you have no real roots.

  • Quadratic Inequalities
    • By sketching the graph of a quadratic, you can determine the region for an inequality:
      • for  f(x) < 0, this is the values of x where the curve is below the x-axis.
      • for  f(x) > 0, this is the values of x where the curve is above the x-axis.
      • f(x) = 0, this is the values of x where the curve is on the x-axis.

Chapter 2 - Integration

  • Integration is the reverse of differentiation
    • We can solve differential equations

  • General Solution
    • Integration gives rise to a constant of integration C
    • because any constant when differentiated becomes zero.

  • Particular Solution
    • This is when we know a point on the original curve
    • We can calculate the constant C
    • We call this extra information a boundary condition
    • If the variable of the function is t (time)
      • then our boundary condition is often the initial condition
      • when t = 0, the value of the function may be (not always) = 0
      • and again we can calculate C.

  • IMPORTANT  - you can only inetgrate powers of  x at this point.

  • Rules for Integration
    • Must be in straight line form - change any surds to index form
    • Increase the index by 1 and divide by the new index. Integrating a constant k gives kx
    • Carry through any multipliers - integer or fractional

  • You should be confident at integrating:
    • Positive and negative indices
    • Fractional indices
    • Dealing with integer or fractional multipliers

  • You should be able to split fractions into separate powers of x
  • You should be able to multiply out brackets using FOIL.


  • Area under a curve
    • y = f(x)  between the lines x = a and x = b and the x-axis is    integral
    • This is evaluated as

  • The area below the x-axis is numerically negative and so must be treated separately.


  • Area between two curves
    • The area between y = f(x)  and y = g(x) is   
                   two curves
            where g(x) is the top curve.
    • This automatically deals with any negative areas.
    • Remember that g(x) is the TOP curve.
    • You should also combine g(x) - f(x) into a simplified expression before integrating.

 

Chapter 3.2 - Compound Angles

  • Related Angles
    • s in (-A) = -sin A
    • cos (-A) = cos A
    • sin (90° - A) = cos A
    • cos (90° - A) = sin A
    • sin (180° - A) = sin A
    • cos (180° - A) = -cos A

  • Sin-cos-tan formula   

    • sin cos tan

    • Trig Pythagoras

  • Compound Angles
    • cos(A + B) = cos A cos B  - sin A sin B
    • cos (A - B) = cos A cos B + sin A sin Bsin (A + B) = sin A cos B + cos A sin B
    • sin (A - B) = sin A cos B - cos A sin B

  • Useful tips:

    • If you are given sin or cos as a fraction then sketch a right angled triangle
      • Use Pythagoras to calculate the remaining side. This will give you all the ratios.
      • Use exact values where possible from the exact value table

    • If you cannot remember these - learn how to derive them yourself.
      • 30° and 60° from an equilateral triangle of side 2, then draw in axis of symmetry.
      • 45° from a square of side 1, then draw in a diagonal.

    • To find the sin 75° you can write this as sin(45° + 30°)
    • You can obtain many other values in this way e.g. sin 15° = sin(45° - 30°)


  • Double Angles

      •   double angles

  • Solving Equations with double angles

    • Change sin 2A to 2 sin A cos A   -  using formula above

    • If cos 2A present, then choose an appropriate equation
      depending on whether the other term is sin or cos,
      because you want a quadratic equation in sin or cos.

    • Factorise the quadratic equation
      • Two brackets - if there are 3 terms
      • Common factor if only 2 terms.

    • Find solution for each factor - remember there are usually two of them.
      • Using ASTC
      • If factor is -1, 0 or 1 then sketch the sin or cos graph to get the solution.

    • Take care if the value is > 1 or  < -1 in which case that solution is discarded.


We have not yet covered the final chapter in Unit 2 - The circle.

We will do this after the prelim

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