# 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.