Edexcel A Level Maths: Pure:复习笔记2.7.4 Proportional Relationships

Proportional relationships

• Proportional relationships describe a proportional connection between two variables
• This can happen in two ways
  • Direct proportion
    • one variable increases or decreases the other does the same
  • Inverse proportion
    • one variable increases the other decreases and vice versa
• Proportional relationships use the symbol  which means is proportional to

• Both direct and inverse proportion can be represented graphically
  • Direct proportion creates a linear graph where k is the gradient
  • Inverse proportion creates a reciprocal graph

Direct proportion

• means y is proportional to x
• y increases as x does, k determines the rate (gradient)
• by changing this to the equation  we can substitute in given values and solve to find k
  • Note that this means the ratio of x and y is constant k = y / x

  

Inverse proportion

• means y is proportional to  or y is inversely proportional to x
• y decreases as x increases and vice versa, k determines the rate
• by changing this to the equation  we can substitute in given values and solve to find k
  • Note that this means the product of x and y is constant k = xy 

How do I work with proportional relationships?

• Set up your proportional relationship using  then change to = k
• Be clear about what y is proportional to …
  • “… the square of x”
  • “… x plus four” (x + 4)
• Calculate or deduce the value of k from the information given or a graph
• Once you've found k sub it back in to your original proportion equation
• You can now find any values using this proportional relationship
• y = mx + c rearranges to y – c = mx so (y - c) is directly proportional to x
• Proportional relationships are often used in modelling