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Properties of Graphs

Maths: Edexcel IGCSE Higher, topic 3.3.3

Estimating Gradient

For a line that isn't straight, we can estimate roughly what the gradient is.

Gradient at a point

Gradient at a point

  • To estimate the gradient at a point draw a tangent to the curve at that point.
  • Work out the gradient of the tangent.
  • Remember that gradient = change in y ÷ change in x.
Gradient change in y ÷ change in x

Gradient = change in y ÷ change in x

  • To estimate the gradient use gradient = change in y ÷ change in x.
  • But remember it is only an estimate because the curve isn’t straight.

Estimating Area Under a Graph

Straight lines only - exact calculation

Straight lines only - exact calculation

  • To calculate the area under a graph, split the area into trapeziums and rectangles and work out the area of each section.
  • Use the formula:
  • Area (Trapezium) = 1⁄2 × base × (a + b)
  • Where a and b are the sides of the trapezium.
Curved line - estimate

Curved line - estimate

  • If the graph is not made up of straight lines you can estimate the area by splitting the graph approximately into trapeziums and doing the same calculation.
Overestimate or underestimate

Overestimate or underestimate

  • An area estimate is an overestimate when the trapeziums are above the curve and an underestimate when the trapeziums are below the curve.
  • In the example the first and second sections are underestimates and the third is an overestimate.

Gradient and Area of a Graph With Units

A real number is a number/value that shows a quantity (or amount) on a line/scale. For a graph showing two real quantities, the gradient and area also show real quantities.

Gradient

Gradient

  • For any graph the gradient represents: (y-axis units) per (x-axis units).
  • For example:
  • litres per second
  • £ per kg.
Area

Area

  • The area under the graph represents: (y-axis units) × (x-axis units).
  • For a velocity-time graph the area under the graph represents distance travelled because km⁄h × h = km.

Test yourself on Properties of Graphs

  • What is the gradient of a straight line between (10,0) and (12,2)?

  • What is the gradient of a straight line between (-6,0) and (0,12)?

  • Will the triangle underneath the graph between x = 3 & x = 6 be an under or overestimate of the true area?

  • Will the triangle underneath the graph between x = 0 & x = 3 be an under or overestimate of the true area?

  • Will the trapezium approximating the graph between x = 6 & x = 9 be an under or overestimate of the true area?

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