Graphs in practical situations Revision Notes

Graphs represent relationships between variables in practical situations. The gradient (or slope) of a graph at any point measures the rate at which one quantity changes relative to another.

For straight-line graphs, the gradient is constant and calculated by the change in vertical values divided by the change in horizontal values between two points:

Gradient = \( \frac{\text{change in } y}{\text{change in } x} \) = \( \frac{y_2 - y_1}{x_2 - x_1} \). For example, in a distance-time graph, the gradient represents speed (distance per unit time). A steeper gradient means a higher speed.

• For curved graphs, the gradient varies at each point.
• To find the gradient at a specific point on a curve, draw a tangent line touching the curve only at that point.
• Estimate the gradient of the tangent by choosing two points on the tangent line and applying the gradient formula.
• This gradient represents the instantaneous rate of change at that point.

To create a graph from data, first identify the variables for the horizontal (x-axis) and vertical (y-axis) axes.

Plot each data point accurately using the given values.

Join points with straight lines if the data represents discrete intervals or linear changes, or with a smooth curve if the data suggests continuous variation.

• Label axes with variable names and units.
• Choose an appropriate scale to use the graph space effectively.
• Check plotted points for accuracy before joining.
• Example: Given time (s) and distance (m) data, plot points and join to form a distance-time graph.

Distance-time graphs show how distance changes over time. The gradient at any point gives the speed at that moment.

Speed-time graphs show how speed changes over time. The gradient of a speed-time graph represents acceleration (rate of change of speed).

Positive gradient means acceleration (speeding up), negative gradient means deceleration (slowing down), and zero gradient means constant speed.

• Calculate gradient of straight segments by \( \frac{\Delta y}{\Delta x} \).
• For curves, estimate gradient using tangents.
• Example: A speed-time graph with a straight line from 0 to 10 m/s over 5 seconds has acceleration \( = \frac{10 - 0}{5 - 0} = 2 \text{ m/s}^2 \).

The area under a speed-time graph represents the distance travelled during the time interval.

When the graph consists of linear segments (straight lines), the area can be calculated using simple geometric shapes such as rectangles and triangles.

Calculate areas of each shape and sum them to find total distance.

• Area of rectangle = base × height (time × speed).
• Area of triangle = \( \frac{1}{2} \times \text{base} \times \text{height} \).
• Example: For a speed-time graph with speed increasing linearly from 0 to 10 m/s over 5 s, the area is a triangle with base 5 s and height 10 m/s, so distance = \( \frac{1}{2} \times 5 \times 10 = 25 \) m.

Travel graphs represent journeys involving different speeds, stops, or changes in direction, often shown as distance-time graphs with flat sections indicating stops.

Conversion graphs allow conversion between units by reading corresponding values on the graph, such as converting miles to kilometres.

Interpreting these graphs requires understanding the meaning of gradients and flat sections in context.

• Flat sections in distance-time graphs mean no movement (speed = 0).
• Steeper sections indicate faster travel.
• Conversion graphs are read by locating a value on one axis and finding the corresponding value on the other axis.
• Example: A travel graph showing a car stopped for 10 minutes then moving at constant speed can be analysed to find total distance and duration.