Linear Graphs

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टीचर्स के लिए: Linear Graphs (Maths [CIE], Extended) के लिए इस्तेमाल के लिए तैयार लेसन स्लाइड्स, रिवीज़न नोट्स — इन्हें अपने लेसन में इस्तेमाल करें, या टॉपिक को एक इंटरैक्टिव क्लास एक्टिविटी की तरह चलाएं जिसे आपके स्टूडेंट्स लाइव गेम की तरह खेलें।

लेसन नोट्स

Equations of Straight Lines (y = mx + c)

The general equation is y = mx + c, where m is the gradient and c is the y-intercept.
Gradient = rise/run; positive for uphill, negative for downhill.
To find the equation from a graph: find gradient using a triangle, read off y-intercept, substitute into y = mx + c.
If y-intercept is not visible, substitute a known point into y = mx + c and solve for c.
Horizontal lines: y = c; vertical lines: x = k.
Rearrange equations like ax + by = c into y = mx + c to identify gradient and intercept.

y = mx + c

Drawing Straight Line Graphs

Create a table of values for x and y, plot points, and join with a straight line.
Use at least 3 points to ensure accuracy.
Alternatively, start at y-intercept (c) and move up m units for every 1 unit right (down if m negative).
For equations in form ax + by = c, find x-intercept (set y=0) and y-intercept (set x=0), then plot.
If gradient is a fraction a/b, go a units up for every b units right.

Drawing straight line graphs from their equations

Parallel Lines

Parallel lines have the same gradient.
Equation of a line parallel to y = mx + c is y = mx + d (different y-intercept).
To find a parallel line through a given point: use same gradient, substitute point to find d.

Parallel lines

Perpendicular Lines

Perpendicular lines meet at right angles (90°).
Gradients m₁ and m₂ satisfy m₁ × m₂ = -1 (negative reciprocals).
To find a perpendicular gradient: m₂ = -1/m₁.
To find equation of perpendicular line: find negative reciprocal of given gradient, substitute point to find c.
A perpendicular bisector cuts a segment in half at right angles; passes through midpoint.

Perpendicular lines

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साइन अप करें — फ्री लॉग इन

प्रैक्टिस सवाल

फ्री प्रीव्यू — 55 में से 8 सवाल। सभी देखने के लिए साइन अप करें।

1.A straight line, l, has equation y = 5x + 12. What is the gradient of line l?
Easy
- A5
- B12
- C-5
- D-12
2.The line y = 3x − 2 crosses the y-axis at G. What are the coordinates of G?
Easy
- A(0, -2)
- B(0, 2)
- C(-2, 0)
- D(2, 0)
3.Find the coordinates of the point where the line y = 3x − 8 crosses the y-axis.
Easy
- A(0, -8)
- B(0, 8)
- C(-8, 0)
- D(8, 0)
4.Write down the gradient of the line y = 3x − 8.
Easy
- A3
- B-8
- C-3
- D8
5.Find the gradient of a line that is perpendicular to 8y + 4x = 5.
Medium
- A2
- B-2
- C1/2
- D-1/2
6.The equation of line L is 3x − 8y + 20 = 0. Find the gradient of line L.
Medium
- A3/8
- B-3/8
- C8/3
- D-8/3
7.Line L passes through the points (0, -3) and (6, 9). Find the equation of line L.
Medium
- Ay = 2x - 3
- By = 2x + 3
- Cy = -2x - 3
- Dy = -2x + 3
8.Find the gradient of the line that is perpendicular to the line 2y = 3 + 5x.
Medium
- A-2/5
- B2/5
- C-5/2
- D5/2

और सवाल

9.A is the point (3, 5) and B is the point (1, -7). Find the equation of the line perpendicular to AB that passes through point A. Give your answer in the form y = mx + c.
Hard
- Ay = -1/6 x + 5.5
- By = 6x - 13
- Cy = -6x + 23
- Dy = 1/6 x + 4.5
10.Find the equation of the line that is perpendicular to the line y = 2x - 3 and passes through the point (0, 2).
Medium
- Ay = -1/2 x + 2
- By = 1/2 x + 2
- Cy = -2x + 2
- Dy = 2x + 2
11.The equation of a straight line is 2y = 3x + 4. Find the gradient of this line.
Easy
- A3/2
- B2/3
- C3
- D2
12.A straight line, l, has equation y = 5x + 12. Find the coordinates of the point where line l crosses the x-axis.
Easy
- A(-2.4, 0)
- B(0, 12)
- C(2.4, 0)
- D(0, -2.4)
13.A line joins A(1, 3) to B(5, 8). Find the equation of the line AB in the form y = mx + c.
Medium
- Ay = 1.25x + 1.75
- By = 0.8x + 2.2
- Cy = 1.25x - 1.75
- Dy = 0.8x - 2.2
14.A is the point (7, 2) and B is the point (-5, 8). Find the equation of the line that is perpendicular to AB and that passes through the point (-1, 3). Give your answer in the form y = mx + c.
Hard
- Ay = 2x + 5
- By = -2x + 1
- Cy = 1/2 x + 3.5
- Dy = -1/2 x + 2.5
15.A line from the point (2, 3) is perpendicular to the line y = 1/3 x + 1. The two lines meet at point P. Find the coordinates of P.
Hard
- A(2.4, 1.8)
- B(1.5, 1.5)
- C(2, 3)
- D(3, 2)
16.A is the point (2, 3) and B is the point (7, -5). Find the equation of the line through A that is perpendicular to AB. Give your answer in the form y = mx + c.
Hard
- Ay = 5/8 x + 7/4
- By = -5/8 x + 23/4
- Cy = 8/5 x - 1/5
- Dy = -8/5 x + 31/5
17.What is the gradient of the line with equation y = 4x - 7?
Easy
- A4
- B-7
- C-4
- D7
18.A line has equation y = 2x + 5. What are the coordinates of its y-intercept?
Easy
- A(0, 5)
- B(5, 0)
- C(0, 2)
- D(2, 0)
19.What is the gradient of the line 3y = 6x + 9?
Easy
- A2
- B6
- C3
- D9
20.Line L passes through (0, -2) and (4, 6). What is the equation of L?
Medium
- Ay = 2x - 2
- By = 2x + 2
- Cy = -2x - 2
- Dy = -2x + 2
21.Find the gradient of the line perpendicular to y = 3x + 1.
Easy
- A-1/3
- B1/3
- C-3
- D3
22.Which line is parallel to y = 5x - 2?
Easy
- Ay = 5x + 3
- By = -5x - 2
- Cy = (1/5)x + 1
- Dy = 5
23.The equation of a line is 2y - 4x = 8. What is its gradient?
Easy
- A2
- B-2
- C4
- D-4
24.A line has gradient -2 and passes through (1, 4). What is its equation?
Medium
- Ay = -2x + 6
- By = -2x + 2
- Cy = 2x + 2
- Dy = 2x + 6
25.Which of the following lines is perpendicular to y = (2/3)x + 5?
Medium
- Ay = -(3/2)x + 1
- By = (3/2)x + 1
- Cy = -(2/3)x + 1
- Dy = (2/3)x + 1
26.Line L has equation 3x + 4y = 12. What is its gradient?
Easy
- A-3/4
- B3/4
- C-4/3
- D4/3
27.Find the equation of the line perpendicular to y = 2x - 1 that passes through (0, 3).
Medium
- Ay = -(1/2)x + 3
- By = (1/2)x + 3
- Cy = -2x + 3
- Dy = 2x + 3
28.A line passes through (2, 5) and (6, 9). Find the equation of the line perpendicular to this line that passes through the midpoint of these points.
Hard
- Ay = -x + 11
- By = x + 7
- Cy = -x + 7
- Dy = x + 11
29.The line y = 3x + 2 is perpendicular to another line. What is the gradient of that other line?
Easy
- A-1/3
- B1/3
- C-3
- D3
30.Find the equation of the line parallel to 2y = 4x + 6 that passes through (1, 5).
Medium
- Ay = 2x + 3
- By = 2x + 5
- Cy = 4x + 1
- Dy = 4x + 5
31.A line has equation y = -3x + 4. Find the equation of the line perpendicular to it that passes through (0, -2).
Medium
- Ay = (1/3)x - 2
- By = -(1/3)x - 2
- Cy = 3x - 2
- Dy = -3x - 2
32.Find the equation of the perpendicular bisector of the segment joining (2, 3) and (8, 9).
Hard
- Ay = -x + 11
- By = x + 1
- Cy = -x + 1
- Dy = x + 11
33.A straight line has equation 2y = 6x - 8. What is the gradient of this line?
Easy
- A3
- B6
- C-3
- D2
34.Line L passes through the points (2, 5) and (6, 13). Find the equation of line L.
Medium
- Ay = 2x + 1
- By = 2x + 5
- Cy = 4x - 3
- Dy = 2x - 1
35.Find the equation of the line perpendicular to y = 3x - 2 that passes through the point (0, 4).
Medium
- Ay = -1/3 x + 4
- By = 1/3 x + 4
- Cy = -3x + 4
- Dy = 3x + 4
36.Which of the following lines is parallel to y = 4x - 7?
Easy
- Ay = 4x + 3
- By = -4x + 2
- Cy = 1/4 x + 5
- Dy = 4
37.The line L has equation 3x + 2y = 12. Find the gradient of a line perpendicular to L.
Medium
- A2/3
- B-2/3
- C3/2
- D-3/2
38.A line has equation y = -2x + 5. What are the coordinates of its x-intercept?
Easy
- A(2.5, 0)
- B(0, 5)
- C(-2.5, 0)
- D(5, 0)
39.What is the gradient of a horizontal line?
Easy
- A0
- B1
- Cundefined
- D-1
40.A horizontal line has an equation of the form y = c.
Easy
True or false?
41.What is the gradient of the line y = -3x + 7?
Easy
- A-3
- B3
- C7
- D-7
42.Which of the following lines is parallel to y = 2x + 5?
Easy
- Ay = 2x - 3
- By = -2x + 5
- Cy = 0.5x + 5
- Dy = 2x + 5
43.The lines y = 3x + 1 and y = -1/3 x + 2 are perpendicular.
Easy
True or false?
44.A line passes through (0, 5) and (4, 13). What is its gradient?
Easy
- A2
- B0.5
- C8
- D-2
45.Find the equation of the line that is perpendicular to y = 2x + 1 and passes through (0, 3).
Medium
- Ay = -1/2 x + 3
- By = -2x + 3
- Cy = 1/2 x + 3
- Dy = 2x + 3
46.The equation of a line is 2y = 6x - 8. What is the gradient of a line perpendicular to this line?
Medium
- A-1/3
- B1/3
- C-3
- D3
47.Match each line to its gradient.
Easy
- y = 3x + 2
- y = -x + 4
- y = 0.5x - 1
- y = -2x + 3
- 3
- -1
- 0.5
- -2
48.Arrange the following lines in order of increasing gradient.
Medium
- y = 2x + 1
- y = -3x + 4
- y = x - 2
- y = 0.5x + 3
49.Match each equation of a straight line to its gradient.
Easy
- y = 3x + 2
- y = -2x + 5
- y = 0.5x - 1
- y = 4
- 3
- -2
- 0.5
- 0
50.Find the y-intercept of the line y = 4x - 3.
Easy
51.Fill in the blank: The general equation of a straight line is y = ___ x + c.
Easy
The general equation of a straight line is y = ____ x + c.
52.State the gradient of the line y = 5 - 2x.
Easy
53.Find the gradient of the line that is perpendicular to the line passing through the points (2, 5) and (6, 13).
Medium
54.Find the y-intercept of the line y = 4x - 3.
Easy
- A-3
- B-2
- C-1
- D-4
55.Find the gradient of the line that is perpendicular to the line passing through the points (2, 5) and (6, 13).
Medium
- A-1.5
- B0.5
- C1.5
- D-0.5

रिवीज़न नोट्स

## Linear Graphs - Quick Revision ### Equation of a Straight Line - **General form**: \(y = mx + c\) - \(m\) = gradient (steepness, positive uphill, negative downhill) - \(c\) = y-intercept (where line crosses y-axis) - **Finding equation from graph**: 1. Find gradient using \(m = \frac{\text{rise}}{\text{run}}\) (draw a triangle) 2. Read off y-intercept \(c\) 3. Substitute into \(y = mx + c\) - **If y-intercept not visible**: substitute a point \((x,y)\) into \(y = mx + c\) and solve for \(c\) - **Horizontal line**: \(y = c\) - **Vertical line**: \(x = k\) ### Drawing Straight Line Graphs - **Using table of values**: choose x-values (e.g. 0, 5, 10), compute y, plot points, join with ruler - **Using gradient and intercept**: start at y-intercept, move 1 unit right and m units up (or down if negative) - **If equation not in \(y = mx + c\)**: rearrange first, e.g. \(ax + by = c\) → \(y = -\frac{a}{b}x + \frac{c}{b}\) - **Alternative for \(ax + by = c\)**: find x-intercept (y=0) and y-intercept (x=0), plot and join ### Parallel Lines - **Same gradient** (\(m_1 = m_2\)) - Equation parallel to \(y = mx + c\) is \(y = mx + d\) (different y-intercept) - To find parallel line through a point: use same gradient, substitute point to find \(d\) ### Perpendicular Lines - **Gradients multiply to -1**: \(m_1 \times m_2 = -1\) - Gradients are **negative reciprocals**: \(m_2 = -\frac{1}{m_1}\) - To find perpendicular line through a point: 1. Find gradient of given line 2. Compute negative reciprocal 3. Substitute point into \(y = mx + c\) to find \(c\) - **Perpendicular bisector**: passes through midpoint of segment, gradient is negative reciprocal of segment's gradient ### Key Tips - Always plot at least 3 points when drawing - Check if lines are parallel/perpendicular by comparing gradients after rearranging to \(y = mx + c\) - For perpendicular lines, ensure you use the negative reciprocal correctly

Gradient (m): The steepness of a line, calculated as rise/run. Positive for uphill, negative for downhill.
y-intercept (c): The y-coordinate where the line crosses the y-axis (x=0).
Equation of a straight line: y = mx + c, where m is the gradient and c is the y-intercept.
Parallel lines: Lines with the same gradient that never intersect.
Perpendicular lines: Lines that meet at right angles (90°); their gradients multiply to -1.
Negative reciprocal: For a gradient m, the negative reciprocal is -1/m. Used to find perpendicular gradients.
Perpendicular bisector: A line that cuts a segment into two equal parts at a right angle; passes through the midpoint and has gradient negative reciprocal of the segment.

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