algebra

• There are conventions used in algebra to speed things up
• x * y


  Can be written as xy (no need to write the 'times')
• x * x * x

  Can be written as x^3
• x / y


  Can be written as x/y
• Putting terms into an algebraic expression

  1. Replace the variable with the given value

  2. Calculate the expression
• Collecting like terms


  1. Rewrite in order

  2. Highlight and group like terms
• Multiplying a number in front of a bracket


  Multiply the number by everything inside the bracket
• Factorising an algebraic expression

  1. Look for common factors

  2. Put common factors outside the bracket
• Multiplying two algebraic expressions

  1. Draw lines to organise the multiplication

  2. Multiply each term in one expression by each term in the other
• Simplifying an algebraic expression

  1. Collect like terms

  2. Factorise

  3. Deal with brackets
• Rearranging an equation to make a variable the subject

  Isolate the variable on one side of the equation
• Square root (√)


  A way of writing a long number more simply
• Multiplying terms with indices

  Add the indices
• Dividing terms with indices

  Subtract the indices
• Evaluating a function


  Replace the variable with the given value and calculate
• Coordinates


  Represented as (x, y)
• Vectors


  Represented as (x, y)
• Plotting points on a graph

  1. Go across x units then up y units

  2. Use a sharp pencil and mark with a clear cross
• Plotting a line from an equation


  1. Calculate some points that satisfy the equation

  2. Plot the points and join them with a ruler
• Gradient


  The steepness of a line, calculated as rise/run
• 1. intercept


  The point where a line crosses the y-axis
• Parallel lines have the same gradient
• Roots of a quadratic

  The x-intercepts of the graph
• Turning point of a quadratic

  The coordinates of the minimum/maximum point
• Graphs of linear functions

  • Steeper with larger coefficient
  • Shallower with smaller coefficient
  • Shifted up/down by constant term
• Graphs of quadratic functions

  • Parabolic shape
  • Wider/narrower with smaller/larger coefficient
  • Shifted left/right by constant term
• Graph of 1/x


  • Hyperbolic shape
• Plotting graphs


  1. Use accurate points marked with clear crosses

  2. Axis must have clear scale, title and units

  3. Draw smooth, confident line
• Distance-time and velocity-time graphs look the same but represent different things
• Distance-time graph


  • Constant speed shown by straight line
  • Stationary shown by horizontal line
  • Acceleration/deceleration shown by changing gradient
• Velocity-time graph


  • Constant speed shown by horizontal line
  • Acceleration/deceleration shown by changing gradient
• On a distance-time graph, the middle section where distance is not progressing indicates the object is staying still
• On a distance-time graph, the steady speed in the last section is slower than the steady speed in the first section
• On a velocity-time graph, the middle flat section indicates the object is moving at a steady speed, not zero speed
• On a velocity-time graph, the slope of the acceleration sections is shallower in the last section compared to the first, indicating less rapid acceleration
• Solving linear simultaneous equations from a graph

  1. Draw the points on the graph

  2. Identify where the lines cross over

  3. This gives the solution
• Solving a linear and quadratic equation from a graph

  1. Draw the lines on the graph

  2. Identify the points where they cross over

  3. This gives the two solutions
• Solving a quadratic equation by factorising

  1. Write the equation in bracket form

  2. Find two numbers that multiply to give the constant term and add/subtract to give the coefficient of x

  3. Set each bracket equal to 0 to find the solutions
• There are multiple valid methods to solve simultaneous equations, such as substitution, addition/subtraction, and graphical
• Solving simultaneous equations by substitution

  1. Rearrange one equation to make one variable the subject

  2. Substitute this into the other equation

  3. Solve for one variable

  4. Substitute back into the original equation to find the other variable