Graphs

Understanding Graphs

  • A graph is a visual way of representing relations between different quantities.
  • The horizontal axis is usually x and the vertical axis is usually y.
  • The point where these axes intersect is known as the origin, denoted as (0,0).
  • Each point on the graph is represented by a coordinate pair (x, y), with x being the horizontal position and y being the vertical position.
  • A straight line graph has an equation of the form y = mx + c, where m is the slope and c is the y-intercept.
  • If the slope or ‘m’ is positive, the line will slope upwards from left to right, and if negative, it will slope downwards.
  • The y-intercept ‘c’ is the point where the line cuts the y-axis.
  • Curved graphs can represent more complex equations, such as quadratic (y = ax² + bx + c), cubic (y = ax³ + bx² + cx + d), or circular equations (x² + y² = r²).

Plotting and Reading Graphs

  • When plotting a graph, it’s necessary to ‘plot’ or mark positions for a range of x values and then draw the line or curve of best fit.
  • The positions where the graph intersects the axes are called the intercepts.
  • The sign of y for a certain x-coordinate determines which quadrant the point lies in.
  • Reading the graph involves interpreting the information it provides. This might be the gradient of a line, the coordinates of an intercept, or the overall trend or pattern.

Transformations of Graphs

  • Transformations refer to the ways in which the graph of a function, f(x), can be altered or moved.
  • A translation of the graph in the plane by the vector (a, b) changes the function to g(x) = f(x - a) + b.
  • A scale factor changes the graph either vertically (effecting the y-values) or horizontally (effecting the x-values).
  • Reflections can occur either in the x-axis (where y becomes -y) or the y-axis (where x becomes -x).

Applications of Graphs

  • Graph interpretation is essential for solving a host of problems in mathematics and real-world situations.
  • Straight line graphs are used to model situations with a constant rate of change, while curved graphs are often employed for non-linear situations.
  • Graphs also help solving equations (finding the roots) and inequalities.
  • Graph sketching is a vital skill, allowing us to represent complex functions and their behaviours visually.