Straight-Line Graphs

Understanding Straight-Line Graphs

  • A straight-line graph can be described using the equation y = mx + c, often known as the slope-intercept form.
  • m refers to the slope or gradient of the line.
  • The slope or gradient determines how steep the line is — a positive slope indicates a line sloping upwards, a negative slope means the line slopes downwards.
  • The intercept c is where the line crosses the y-axis. This point is often called the y-intercept.

Plotting Straight-Line Graphs

  • When plotting a straight-line graph, first plot the y-intercept on the y-axis (vertical line).
  • From there, follow the slope to plot your next point. For example, if the slope is 2, then for every one unit you go to the right, you go up by two units.
  • Once two points are plotted, you can draw your straight line all the way across the graph.
  • For GCSE standard straight-line graphs, you don’t usually need more than two points, but the more points the better your line.

Determining the Equation of a Line

  • To determine the equation of a line from a straight-line graph, you need to first find out the slope (m) and the y-intercept (c).
  • To calculate the slope, choose two points on the line. Then, subtract the y-coordinate of the second point from the y-coordinate of the first point. Divide this by the difference of the x-coordinates of the two points. This is written as (y2 - y1) / (x2 - x1).
  • To find the y-intercept, look at where the line crosses the y-axis. The y-coordinate of that point is the y-intercept (c).
  • Once you have m and c, you can put these values into the equation y = mx + c to define your line.

Interpreting Real-World Situations

  • Straight-line graphs can be used to describe real-world situations.
  • For example, a taxi company may charge a flat fee (the y-intercept) plus a rate per mile (the slope).
  • Understanding the slope and y-intercept in this context allows you to calculate total cost for any given length of journey.
  • Be aware that sometimes the real world imposes limits on the values of x or y, creating a line segment rather than a full line.

Working with Negative Slopes

  • Negative slopes indicate that the y-value decreases as the x-value increases.
  • This may occur in situations where one quantity decreases as another one increases, such as the value of a car decreasing with the amount of mileage.
  • To plot a graph with a negative slope, plot the y-intercept as normal, then move to the left for each step up in the y-direction.