Coordinate Geometry and Real-Life Graphs
Coordinate Geometry and Real-Life Graphs covers plotting and reading coordinates in all four quadrants, finding the midpoint and gradient of a line segment using the equation y = mx + c, and reading and interpreting real-life graphs such as conversion graphs and travel graphs.
Method
- Remember the convention for plotting coordinates: always read along the x-axis first, then up or down the y-axis (x, y), using negative values correctly in all four quadrants.
- To find the midpoint of two points (x1, y1) and (x2, y2), average the x-coordinates and average the y-coordinates separately.
- To find the gradient between two points, divide the change in y by the change in x: gradient = (y2 - y1) / (x2 - x1).
- In the equation y = mx + c, m is the gradient of the line and c is the y-intercept, the point where the line crosses the y-axis when x = 0.
- To find the equation of a line from a graph, read off the y-intercept directly, then calculate the gradient using two clear points on the line.
- For a distance-time (travel) graph, remember that the gradient represents speed: a steeper line means faster travel, a horizontal (flat) line means the object is stationary, and a negative gradient means returning towards the start.
- For a conversion graph, use a straight line drawn between plotted points to convert between two units, reading values directly off the graph or calculating using the gradient and intercept.
- Always check whether a question asks you to read a value from a graph, where an estimate within a small range is acceptable, or to calculate it exactly using coordinates or a formula.
Worked example
A straight line passes through the points A(2, 3) and B(6, 11). Find (a) the gradient of the line AB, and (b) the equation of the line in the form y = mx + c.
- Find the change in y and the change in x between the two points: change in y = 11 - 3 = 8, change in x = 6 - 2 = 4.
- Calculate the gradient: gradient = 8 / 4 = 2.
- Substitute the gradient and one point, (2, 3), into y = mx + c to find c: 3 = 2 x 2 + c, so 3 = 4 + c.
- Solve for c: c = 3 - 4 = -1. Final answer: gradient = 2, and the equation of the line is y = 2x - 1.
Practice questions
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Q1Write down the coordinates of the point 3 units to the left of the origin and 5 units up.Show answer
Answer: (-3, 5)
Q2Find the midpoint of the line segment joining (-4, 2) and (6, -8).Show answer
Answer: (1, -3) (midpoint x = (-4 + 6) / 2 = 1, midpoint y = (2 + -8) / 2 = -3)
Q3Find the gradient of the line joining (1, 4) and (5, 16).Show answer
Answer: 3 (gradient = (16 - 4) / (5 - 1) = 12 / 4 = 3)
Q4A line has equation y = 4x - 7. Write down its gradient and its y-intercept.Show answer
Answer: Gradient = 4, y-intercept = -7 (the point (0, -7))
Q5A conversion graph shows that 5 miles is equivalent to 8 kilometres. Using this scale, convert 20 miles to kilometres.Show answer
Answer: 32 kilometres (20 miles is 4 times 5 miles, so multiply 8 km by 4: 8 x 4 = 32)
Q6On a distance-time graph, a cyclist's journey is shown as a horizontal line for 15 minutes. What does this tell you about the cyclist during this time?Show answer
Answer: The cyclist was stationary (not moving), for example resting or waiting, since distance from the start is not changing over time.
Q7A straight line passes through (0, 5) and has a gradient of -2. Write down its equation.Show answer
Answer: y = -2x + 5 (the y-intercept is 5 since the line passes through (0, 5), and the gradient is -2)
Q8A train travels 90 km in 1.5 hours at a constant speed, shown as a straight line on a distance-time graph. Calculate the gradient of this line, and state what it represents.Show answer
Answer: 60 (gradient = 90 / 1.5 = 60); it represents the train's speed in kilometres per hour.
Exam-style questions
Written in the style of a 13+ Common Entrance exam paper, with a full mark scheme.
A straight line passes through P(-3, -4) and Q(5, 8). (a) Find the midpoint of PQ. (b) Find the gradient of PQ.
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A plumber charges a call-out fee plus an hourly rate. The total cost, C pounds, for a job lasting h hours is shown on a graph as a straight line passing through (0, 40) and (3, 100). (a) Find the equation of the line in the form C = mh + c. (b) Use your equation to calculate the cost of a job lasting 5 hours.
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A cyclist's journey to and from the shops is shown on a distance-time graph: the line rises steeply for the first 10 minutes, is flat for the next 5 minutes, then falls back to zero over the final 15 minutes. (a) Describe what is happening during the flat section of the graph. (b) Calculate the average speed for the final 15 minutes if the cyclist's distance from home at the start of this section was 3 km.
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Free printable worksheet
Want more practice on paper? Download the coordinate geometry and real-life graphs worksheet pack - 19 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 17 of 13+ Maths Workbook, the whole course as one free printable PDF.
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