Linear Graphs

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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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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

更多題目

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

複習筆記

## 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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