Graphs, Coordinate Geometry and Direct Proportion
Graphs, Coordinate Geometry and Direct Proportion
The last algebra topic connects equations to pictures. It contributes 1 to 2 questions in the Part-III maths paper, usually on quadrant signs, on reading a plotted point, or on telling direct proportion from inverse. The facts are few, but the sign conventions have to be exact.
The Cartesian Plane
Two perpendicular number lines meet at the origin, written (0, 0). The horizontal line is the x-axis and the vertical one the y-axis. A point is written as (x, y), with the x-coordinate always first.
| Quadrant | x sign | y sign | Example |
|---|---|---|---|
| First | + | + | (3, 4) |
| Second | − | + | (−3, 4) |
| Third | − | − | (−3, −4) |
| Fourth | + | − | (3, −4) |
- Any point on the x-axis has y = 0, as in (5, 0).
- Any point on the y-axis has x = 0, as in (0, 5).
- The origin belongs to no quadrant.
- (3, 4) and (4, 3) are different points — order matters.
✗ (0, 5) lies on the x-axis | ✓ (0, 5) lies on the y-axis, because its x-coordinate is 0
Graph of a Linear Equation
An equation of the form y = mx + c always graphs as a straight line, which is why such equations are called linear. Two points are enough to draw it.
- y = x is a straight line through the origin at 45 degrees.
- y = 3 is a horizontal line; x = 3 is a vertical line. These two are commonly reversed.
- The point where the line meets the y-axis is found by putting x = 0.
Direct and Inverse Proportion
| Point | Direct proportion | Inverse proportion |
|---|---|---|
| Relationship | y increases as x increases | y decreases as x increases |
| Rule | y ÷ x is constant | x × y is constant |
| Written as | y = kx | xy = k |
| Graph | Straight line through origin | Curve (hyperbola) |
| Example | Cost and number of items | Speed and time for a fixed distance |
The test is mechanical: check whether the ratio stays constant or the product stays constant. If four workers take twelve days, eight workers take six days — the product stays at forty-eight, so it is inverse.
✗ More workers means more days, so it is direct proportion | ✓ More workers means fewer days — the product stays constant, so it is inverse
Reading Graphs in the Exam
- Read the axis labels and units before anything else.
- For a distance-time graph, a steeper line means greater speed and a horizontal line means rest.
- A straight line through the origin means the two quantities are in direct proportion.
The distance-time reading is worth extra attention because it links this topic to motion in the physics half of the same Part-III paper.
समानुपाती में अनुपात स्थिर, व्युत्क्रमानुपाती में गुणनफल स्थिर।
TRE pointer: Quadrant sign questions are almost guaranteed, and the standard trap is a point like (−3, 4), where candidates read the negative and jump to the third quadrant instead of the second. The y = 3 horizontal, x = 3 vertical pair is the second most reversed item. Because you will teach graph reading to class 8, expect a question set on a distance-time graph rather than on plain coordinates — and with −1/3 negative marking, read the axes before you read the options.
60-Second Recap
- A point is (x, y), x always first; (3,4) and (4,3) are different points.
- Quadrants in order: (+,+), (−,+), (−,−), (+,−).
- Points on the x-axis have y = 0; points on the y-axis have x = 0; the origin is in no quadrant.
- y = mx + c always graphs as a straight line; y = 3 is horizontal, x = 3 is vertical.
- Direct proportion keeps the ratio constant; inverse proportion keeps the product constant.
- On a distance-time graph, a steeper line means faster and a flat line means at rest.
Frequently Asked Questions
In which quadrant does the point minus three, four lie?
The second quadrant, because the x-coordinate is negative and the y-coordinate is positive. The order of quadrants is anticlockwise from the top right, so the signs run plus-plus, minus-plus, minus-minus, plus-minus. Reading only the first negative and answering third is the usual error.
How do I tell direct proportion from inverse proportion?
Check what stays constant. If dividing one quantity by the other always gives the same value, it is direct. If multiplying them always gives the same value, it is inverse. More workers taking fewer days keeps the product constant, so that relationship is inverse.
Is y equals three a horizontal or a vertical line?
Horizontal. Every point on it has a y-coordinate of three while x can be anything, so the line runs parallel to the x-axis. By the same reasoning x equals three is vertical. This pair is reversed by candidates more often than almost any other graph fact.
What does a horizontal segment on a distance-time graph mean?
That the object is at rest, since time is passing but distance is not changing. A steeper segment means a higher speed, because more distance is covered in the same time. This reading connects directly to the motion topic in the science half of Part-III.
Next step: Algebra rewards pattern recognition, and patterns only stick under time pressure. Practise this chapter in the BPSC TRE 4.0 mock test on ExamAtlas — five options, −1/3 negative marking, and the same Part-III depth.