Grade 10 Graphs: Short Notes

10 ශ්‍රේණිය · 20 පරිච්ඡේදය · ප්‍රස්තාර

94%

of Grade 10 term tests include it (51 of 54)

594

marks in past papers (97 questions)

12 min

full lesson with quick checks

🎯 What you'll learn

Find the gradient and intercept of a line from its equation, its graph or two points
Write the equation of a line, including a parallel line
Make a table and draw the graph of y = ax² + b
Read the turning point, maximum or minimum, axis of symmetry and roots
Give ranges where a quadratic is positive, negative, increasing or decreasing, and shift its graph

Lines and curves around you

A three-wheeler meter in Galle starts at Rs 100 and adds Rs 80 for every kilometre. Plot fare against distance and you get a perfectly straight line. Now watch a six at Galle International Stadium: the ball rises, turns and falls along a smooth curve. This chapter is about reading both.

y = mx + c is a straight line

Any function y = mx + c has a straight-line graph.• m is the gradient: how much y changes when x goes up by 1.• c is the intercept: where the line cuts the y-axis, at (0, c).The fare above is y = 80x + 100: gradient 80 (rupees per km), intercept 100.

Reading gradient from a graph

On y = 2x + 1, go from (1, 3) to (3, 7): across 2, up 4. Gradient = 4 ÷ 2 = 2. The line cuts the y-axis at (0, 1), so c = 1.

🧮 Gradient from two points

Use it for any two points (x₁, y₁) and (x₂, y₂) on the line. Change in y goes on top. Then find c by putting one point into y = mx + c.

m = (y₂ − y₁) / (x₂ − x₁)

What the sign of m tells you

• m > 0: the line rises to the right (acute angle with the positive x-axis).• m < 0: it falls to the right (obtuse angle).• m = 0: it is flat, y = c, parallel to the x-axis.• Parallel lines have equal gradients: y = 3x − 2 and y = 3x + 5 are parallel.

✏️ Equation from two points

A straight line passes through (−1, −1) and (2, 5). Find its gradient, its intercept and its equation.

1m = (5 − (−1)) / (2 − (−1)) = 6 / 3 = 2
2Put (2, 5) into y = 2x + c: 5 = 4 + c, so c = 1.
3Check with (−1, −1): 2(−1) + 1 = −1 ✓

✅ m = 2, c = 1, y = 2x + 1

⚠️ Upside down or wrong sign

Examiners keep seeing two slips: putting the x-change on top (getting ½ instead of 2), and subtracting in different orders top and bottom, which flips the sign. Always y on top, and start both subtractions from the same point. Then look: a line falling to the right must have m < 0.

✏️ Equation not in y = mx + c form

For the line 2y = 6x − 4, find the gradient and intercept. Then write the line parallel to it that cuts the y-axis at (0, 5).

1Divide by 2 first: y = 3x − 2.
2Gradient m = 3, intercept c = −2.
3Parallel line: same m = 3, new c = 5.
4y = 3x + 5

✅ m = 3, c = −2; y = 3x + 5

Quadratic graphs: y = ax² + b

Make a table of values, plot the points and join them with a smooth curve (a parabola), never with a ruler.• a > 0: U-shaped, with a minimum value b.• a < 0: upside-down U, with a maximum value b.• Turning point (0, b). Axis of symmetry: x = 0 (the y-axis).

Reading the curve

y = x² − 4: minimum value −4 at the turning point (0, −4); axis of symmetry x = 0 (dashed); it meets the x-axis at x = −2 and x = 2 (the roots of x² − 4 = 0).

Positive, negative, increasing, decreasing

• Positive: curve above the x-axis. For y = x² − 4: x < −2 or x > 2.• Negative: below the x-axis: −2 < x < 2.• Decreasing: y falls as x moves right: x < 0.• Increasing: y rises as x moves right: x > 0.Combine them carefully: 'negative and increasing' is 0 < x < 2.

⚠️ (−2)² is +4

When filling tables, many students write (−2)² = −4 and the curve comes out wrong. A negative number squared is positive: (−2)² = 4, so in y = 3 − x² at x = −2, y = 3 − 4 = −1. Your table should be symmetric: equal y values for x = 2 and x = −2.

📌 Remember

• y = mx + c: m = gradient, c = intercept at (0, c).• m = (y₂ − y₁)/(x₂ − x₁); parallel lines share m.• y = ax² + b: a > 0 minimum b, a < 0 maximum b; turning point (0, b); axis x = 0.• Roots: where the curve meets the x-axis. Positive above it, negative below.• Moving up k: add k to the whole function.

🧾 Quick revision sheet

In y = mx + c, what are m and c?	m is the gradient, c is the y-intercept.
Gradient from two points	m = (y₂ − y₁)/(x₂ − x₁)
Gradients of parallel lines?	Equal.
A line falling to the right has…	a negative gradient.
Turning point of y = ax² + b	(0, b): minimum if a > 0, maximum if a < 0.
Axis of symmetry of y = ax² + b	x = 0 (the y-axis).
Roots of y = x² − 9 from the graph	Where it meets the x-axis: x = −3 and x = 3.
Move y = x² − 1 up 4 units	y = x² + 3