Gradients of Real-Life Graphs

Gradients of Real-Life Graphs

Definitions

  • Gradient refers to the slope of a line on a graph. It can demonstrate rate of change or correlation in datasets.
  • The gradient of a line can be positive, negative, zero or undefined.
  • Positive gradient implies an increasing trend.
  • Negative gradient indicates a decreasing trend.
  • Zero gradient signifies a constant, or unchanging, value.
  • Undefined gradient occurs when a vertical line is drawn on the graph.

Calculating Gradient

  • To calculate the gradient of a straight line, use the formula (change in y)/(change in x) or rise/run.
  • Select two points on the line, preferably as far apart as possible for accuracy, and apply the formula.
  • If the line is a curve, find the gradient at a specific point by drawing a tangent to the curve at that point and calculating its gradient.

Linear Graphs

  • Linear graphs have a constant gradient. The graph will be a straight line.
  • Positive gradients slope upwards from left to right, representing direct proportionality. When one variable increases, so does the other.
  • Negative gradients slope downwards from left to right, showing inverse proportionality. When one variable increases, the other decreases.

Non-Linear Graphs

  • Non-linear graphs do not have a constant gradient. The lines curve either upwards or downwards.
  • For these graphs, the gradient varies at different points. For any point, the instantaneous gradient is given by the gradient of the tangent at the point.

Understanding Gradients

  • Gradient is a powerful tool in interpreting real-life situations through graphs. It allows us to see at a glance increasing and decreasing trends in datasets.
  • For example, in a graph plotting a car’s journey, the gradient of the graph at any point is equal to the car’s speed at that moment.

Practice

  • To build a solid understanding on gradients, solve problems from different real-life situations. Try plotting your own graphs and determining the gradients at multiple points. Pay attention to what the gradient conveys about the situation modelled by the graph.