GCSE Maths (AQA)

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Advanced Proportionality Graphs

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Ratio, Proportion and Rates of Change Ratio, Proportion and Rates of Change

Advanced Proportionality Graphs

4:16 Advanced Proportionality
Spec R13 R14
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  • Graphs can be drawn to show the relation between a variable and a variable it is proportional to a power or root of.
  • If two variables are proportional to each other, when one variable is multiplied, the other variable is multiplied by the same number.
  • Algebraically, this can be written as y equals k x squared.
  • The graph of y equals k x squared is a simple quadratic.
  • As x increases, y increases, and the more x increases, the faster y increases.
  • If y is proportional to x cubed, this can be written as y is kx cubed.
  • When this equation is graphed, we get a simple cubic.
  • Like a quadratic graph, as x increases, y increases. The larger the value of x, the more quickly y is increasing.
  • This effect is even more noticeable with a cubic graph than a quadratic.
  • You need to know the general shape and properties of all of the graphs encountered in this topic.
  • However, you won't need to tell the difference between two similar looking graphs.
  • For example, you need to know that a graph where y is proportional to x squared has a general shape.
  • But you won't need to tell the difference between a quadratic and a cubic graph since they look so similar.
  • If y is proportional to the square root of x, We can write the equation y is k root x.
  • The graph of y is k root x looks like this.
  • As x increases, y increases, but the larger the value of x, the more slowly y is increasing.
  • If y is proportional to the cube root of x, we can write the equation y is k times the cube root of x.
  • This gives us a very similar looking graph.
  • As x increases, y increases. The larger the value of x, the slower y increases, causing the graph to flatten out.
  • The graph of y is k times the cube root of x is even flatter than the graph of y is k times the square root of x.
  • Graphs can also be drawn to show the relation between a variable and a variable it is inversely proportional to a power or root of.
  • If two variables are inversely proportional, when one variable is multiplied by a number, the other variable is divided by that same number.
  • If y is inversely proportional to x squared, this means we can write the equation y is k over x squared.
  • The graph of this equation resembles a reciprocal graph that rises more sharply near the y axis.
  • If cubed, we can write the equation y is k over x cubed.
  • This graph also resembles the reciprocal graph, but it rises even more sharply near the y axis.
  • If y is inversely proportional to the square root of x, we have the equation y is k over root x.
  • Once again, this gives us a graph which looks like a reciprocal graph, but it doesn't go towards the x axis as quickly.
  • If y is inversely proportional to the cube root of x, we have the equation y is k over the cube root of x.
  • Again, we get something that looks like a reciprocal graph, but it goes towards the x axis even more slowly.

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