Model with straight line graphs

Modelling with straight line graphs is an essential skill in mathematics. Straight line graphs are often used to represent real-life situations where there is a linear relationship between two variables. By understanding how to interpret and draw straight line graphs, we can solve problems involving rates of change, distances, and other linear relationships.

1. Equation of a Straight Line:

The general equation of a straight line is:

\[ y = mx + c \]

Where:

• \(y\) is the dependent variable (the output),
• \(x\) is the independent variable (the input),
• \(m\) is the slope (gradient) of the line, and
• \(c\) is the y-intercept (the value of \(y\) when \(x = 0\)).

The slope \(m\) represents the rate of change of \(y\) with respect to \(x\), i.e., how much \(y\) increases or decreases as \(x\) changes. The y-intercept \(c\) shows where the line crosses the y-axis.

2. Modelling with Straight Line Graphs:

In real-life problems, straight line graphs can be used to model situations where one quantity depends linearly on another. For example, the total cost of an item might increase with the number of units purchased, or the distance travelled might increase with time.

Example 1: A taxi company charges a base fare of £2 and an additional £1 per mile. The total cost \(C\) for \(x\) miles can be modelled by the equation:

\[ C = x + 2 \]

Here, \(x\) represents the number of miles travelled, and \(C\) represents the total cost. The slope of the line is \(1\), which shows that the cost increases by £1 for each additional mile. The y-intercept is \(2\), meaning that the base fare is £2 when no miles are travelled.

3. Finding Points on the Graph:

To plot a straight line, we can find two or more points that satisfy the equation of the line. These points are then plotted on a graph, and the line is drawn through them. For example, consider the equation \(y = 2x + 1\). We can find two points by substituting different values of \(x\):

• When \(x = 0\), \(y = 2(0) + 1 = 1\), so one point is \((0, 1)\).
• When \(x = 1\), \(y = 2(1) + 1 = 3\), so another point is \((1, 3)\).

Plotting these points and drawing a line through them gives the graph of the equation \(y = 2x + 1\).

4. Interpretation of the Graph:

Once we have plotted the straight line graph, we can use it to interpret the relationship between the variables. The slope tells us how steep the line is, which can represent the rate of change. The y-intercept tells us the starting value when \(x = 0\). The graph can also help us to find values of one variable for a given value of the other variable by reading the graph.

Example 2: A car travels at a constant speed, and its distance from a starting point over time is given by the equation:

\[ d = 60t \]

Where \(d\) is the distance travelled in miles, and \(t\) is the time in hours. The slope of the line is \(60\), meaning that for every hour, the car travels 60 miles. The y-intercept is \(0\), meaning that at time \(t = 0\), the car has travelled 0 miles. The graph of this equation will be a straight line passing through the origin, with a slope of 60.

5. Summary:

• The general equation of a straight line is \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.
• In modelling, straight line graphs can represent linear relationships between variables, such as cost, distance, or speed.
• By finding points that satisfy the equation and plotting them on a graph, you can draw the straight line and interpret the relationship between the variables.