Functions and Straight Line Graphs

Functions and Straight Line Graphs

Understanding Functions

  • A function is a special relationship where each input has a single output.
  • It can be thought of as a machine which takes inputs (often denoted by ‘x’) and provides outputs (often denoted by ‘f(x)’ or ‘y’).
  • It’s important to understand the difference between the input (the value you feed into the function) and the output (the result you get out).
  • An example of a function is f(x) = 2x + 3. In this case, for every x you input, the function will output a value that is double the input and then adds three.

Deciphering Straight Line Graphs

Understanding Graph Components

  • A straight line graph can be defined by the equation y = mx + c, where m represents the gradient of the line and c represents the y-intercept.
  • The gradient (m) defines the steepness of the line; a larger absolute value of m will result in a steeper line.
  • The y-intercept (c) is where the line intersects with the y-axis.

Plotting the Graph

  • To plot a straight line, you need at least two points. However, for higher accuracy, it’s recommended to plot more points.
  • Choose some values of x and substitute them into the function to calculate the corresponding y-values.
  • Plot these points (x, y) on a graph and draw the line that passes through these points.

Interpreting the Graph

  • Once the graph is plotted, you can interpret it. If the line slopes upwards from left to right, this indicates a positive correlation, and if it slopes downwards, this indicates a negative correlation.
  • The point where the line crosses the y-axis is the y-intercept; this gives you the base value of the function when x=0.
  • You can also estimate the value of y for any given value of x by finding the corresponding point on the line.

Working with Multiple Functions

  • When working with multiple functions, each function can be plotted on the same set of axes for comparison.
  • This can help to solve problems like finding the intersection points of the functions (the values of x for which the functions are equal).

Key Terms to Remember:

  • Function: a relationship between inputs and outputs
  • Gradient: the slope or steepness of a line
  • Y-intercept: the point where the line crosses the y-axis
  • Positive correlation: when the graph slopes upwards from left to right
  • Negative correlation: when the graph slopes downwards from left to right.