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.

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).