Linear Graphs

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Coordinates

The Cartesian plane has a horizontal x-axis and vertical y-axis meeting at the origin (0,0).
Coordinates are written as (x, y) – x units horizontally, y units vertically.
Positive x: right of origin; negative x: left. Positive y: above; negative y: below.
Example: (2,5) is 2 right, 5 up; (-1,-4) is 1 left, 4 down.
Check the scale on axes – 1 square may not equal 1 unit.

Points on the Cartesian plane

Gradient of a Line

Gradient measures steepness: for 1 unit right, go up (positive) or down (negative) by the gradient.
Gradient = change in y / change in x = rise/run.
Find gradient by drawing a right-angled triangle between two points on the line.
Uphill lines have positive gradient; downhill lines have negative gradient.
Formula: gradient m = (y₂ - y₁) / (x₂ - x₁).
To draw a gradient of a/b: move b units right, a units up (positive) or down (negative).

Gradient of a line: rise/run construction

Equations of Straight Lines (y = mx + c)

General equation: y = mx + c, where m is gradient and c is y-intercept.
y-intercept is where the line crosses the y-axis (x=0).
To find equation from graph: find gradient m (rise/run), read y-intercept c, substitute into y = mx + c.
If y-intercept not visible, substitute a point (x,y) into y = mx + c to solve for c.
Horizontal line: y = c (gradient 0). Vertical line: x = k (gradient undefined).

y = mx + c

Drawing Straight Line Graphs

Use a table of values – choose at least 3 x-values, compute y, plot points, join with a straight line.
Alternatively, start at y-intercept c, then move 1 unit right and m units up (or down if negative).
For fractional gradient a/b, move b units right and a units up/down.
Always plot at least 3 points to check for errors.

Plotting from a table of values

Parallel Lines

Parallel lines have the same gradient but different y-intercepts.
They never intersect.
To find equation of a line parallel to y = mx + c through (x₁,y₁): use y = mx + d, substitute point to find d.

Parallel lines

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Câu hỏi luyện tập

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1.What is the gradient of the line y = 2x - 3?
Easy
- A-3
- B2
- C3
- D-2
2.What is the y-intercept of the line y = 4x - 6?
Easy
- A-4
- B4
- C6
- D-6
3.A line has gradient 3 and passes through (0, 1). What is its equation?
Easy
- Ay = x + 3
- By = 3x + 1
- Cy = -3x + 1
- Dy = 3x - 1
4.What is the gradient of a horizontal line?
Easy
- Aundefined
- B-1
- C0
- D1
5.What are the coordinates of the origin?
Easy
- A(1, 1)
- B(0, 1)
- C(1, 0)
- D(0, 0)
6.Find the gradient of the line passing through (1, 2) and (3, 6).
Easy
- A4
- B1/2
- C3
- D2
7.What is the equation of the line parallel to y = 5x + 6 that passes through (0, -7)?
Easy
- Ay = 5x + 6
- By = -5x - 7
- Cy = 5x + 7
- Dy = 5x - 7
8.A line has gradient -2 and passes through (0, 4). What is its equation?
Easy
- Ay = -2x - 4
- By = -2x + 4
- Cy = 2x + 4
- Dy = 2x - 4

Thêm câu hỏi

9.What is the y-intercept of the line y = 3x - 8?
Easy
- A-3
- B3
- C8
- D-8
10.A line passes through (0, 2) and (4, 10). What is its equation?
Medium
- Ay = 2x - 2
- By = 2x + 2
- Cy = 4x + 2
- Dy = x + 2
11.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)
12.A line has equation y = 5x + 12. Find the coordinates of the point where it crosses the x-axis.
Medium
- A(-12/5, 0)
- B(0, -12)
- C(0, 12)
- D(12/5, 0)
13.The point (p, -26) lies on the line y = -3x + 4. Find p.
Medium
- A-22/3
- B-10
- C22/3
- D10
14.A line passes through (2, 1) and is parallel to y = 3x + 7. Find its equation.
Medium
- Ay = 3x + 7
- By = 3x + 1
- Cy = -3x + 1
- Dy = 3x - 5
15.What is the gradient of the line that passes through (0, 3) and (5, 0)?
Easy
- A-3/5
- B5/3
- C3/5
- D-5/3
16.A line has equation y = 2x + 3. What is the gradient of a line parallel to it?
Easy
- A-2
- B2
- C3
- D-3
17.A line passes through the points (2, 5) and (6, 13). What is the equation of the line?
Medium
- Ay = 4x + 1
- By = 2x - 1
- Cy = 2x + 1
- Dy = 4x - 3
18.What is the y-intercept of the line y = -3x + 7?
Easy
- A-7
- B7
- C3
- D-3
19.The line y = 2x - 5 crosses the x-axis at point P. What are the coordinates of P?
Easy
- A(2.5, 0)
- B(-2.5, 0)
- C(0, -5)
- D(0, 5)
20.What is the gradient of a vertical line?
Easy
- A1
- BUndefined
- C0
- DInfinity
21.Which of the following lines is parallel to y = 4x - 2?
Easy
- Ay = -4x + 1
- By = 4x + 3
- Cy = 2x - 2
- Dy = 4x - 2
22.A line has gradient -3 and passes through the point (2, -1). What is its equation?
Medium
- Ay = -3x + 5
- By = 3x - 7
- Cy = -3x + 1
- Dy = -3x - 7
23.What is the gradient of the line passing through (0, 4) and (3, 10)?
Easy
- A2
- B6
- C4
- D3
24.The point (p, 13) lies on the line y = 2x - 5. Find p.
Easy
- A7
- B6
- C4
- D9
25.A line passes through (0, -2) and (5, 3). What is its equation?
Medium
- Ay = x + 2
- By = x - 2
- Cy = -x - 2
- Dy = 5x - 2
26.What is the y-intercept of the line y = 7 - 2x?
Easy
- A-7
- B-2
- C7
- D2
27.Find the coordinates of the point where the line y = 3x + 6 crosses the x-axis.
Easy
- A(2, 0)
- B(-2, 0)
- C(0, 6)
- D(0, -6)
28.A line has equation y = 5x - 3. What is the gradient of a line parallel to it?
Easy
- A-5
- B-3
- C5
- D3
29.The line y = -2x + 8 crosses the y-axis at point Q. What are the coordinates of Q?
Easy
- A(0, -2)
- B(0, 8)
- C(-2, 0)
- D(8, 0)
30.What is the gradient of the line y = 7 - 2x?
Easy
- A2
- B-7
- C-2
- D7
31.What is the y-intercept of the line y = 5x - 3?
Easy
- A5
- B-5
- C3
- D-3
32.Which of the following is the equation of a horizontal line?
Easy
- Ax = 3
- By = 3
- Cy = 3x + 1
- Dy = x
33.A line has gradient 2 and passes through (0, 5). What is its equation?
Easy
- Ay = 5x + 2
- By = 2x - 5
- Cy = 2x + 5
- Dy = 5x - 2
34.What is the equation of the line parallel to y = 3x + 7 that passes through (2, 1)?
Medium
- Ay = 3x - 7
- By = 3x + 7
- Cy = 3x - 5
- Dy = 3x + 1
35.What is the gradient of the line passing through the points (2, 5) and (6, 13)?
Easy
- A4
- B8
- C3
- D2
36.A line has equation y = 3x + 6. What are the coordinates of the point where it crosses the x-axis?
Easy
- A(0, -2)
- B(0, 6)
- C(-2, 0)
- D(2, 0)
37.Find the equation of the line parallel to y = 5x + 6 that passes through (0, -7).
Medium
- Ay = 5x + 1
- By = 5x + 6
- Cy = -5x - 7
- Dy = 5x - 7
38.What is the gradient of the line y = 3x + 7?
Easy
- A3
- B-3
- C7
- D-7
39.What is the y-intercept of the line y = -2x + 5?
Easy
- A-2
- B-5
- C2
- D5
40.A line has gradient 4 and passes through (0, -3). What is its equation?
Easy
- Ay = -4x - 3
- By = 4x - 3
- Cy = -4x + 3
- Dy = 4x + 3
41.What is the gradient of the line passing through (0, 2) and (4, 10)?
Easy
- A0.5
- B2
- C4
- D8
42.What is the equation of the line parallel to y = 3x + 5 that passes through (0, -2)?
Medium
- Ay = 3x - 2
- By = 3x + 2
- Cy = -3x - 2
- Dy = 3x + 5
43.A line passes through (1, 5) and (3, 11). What is its equation?
Medium
- Ay = 2x - 3
- By = 3x - 2
- Cy = 3x + 2
- Dy = 2x + 3
44.Match each equation to its description.
Medium
- y = 2x + 3
- y = -3x + 1
- y = 5
- x = -2
- Gradient 2, y-intercept 3
- Gradient -3, y-intercept 1
- Horizontal line
- Vertical line
45.Arrange the following lines in order of steepness, from least steep to most steep. (Use the absolute value of gradient.)
Medium
- y = 2x + 1
- y = -3x + 4
- y = 0.5x - 2
- y = -x + 3
46.Which of the following lines are parallel to y = 4x - 2? (Select all that apply.)
Medium
- Ay = 4x + 5
- By = -4x + 3
- Cy = 4x - 7
- Dy = 2x - 2
- Ey = 4x
47.The line y = 3x + 2 passes through the point (1, 5).
Easy
True or false?
48.Find the gradient of the line that passes through the points (0, 0) and (4, 12).
Easy
49.Complete the sentence.
Easy
The equation of a straight line is y = mx + c, where m is the ____ and c is the ____.
50.A line has gradient -2 and passes through the point (3, 1). Find the equation of the line in the form y = mx + c.
Medium
51.Find the gradient of the line that passes through the points (0, 0) and (4, 12).
Easy
- A3
- B5
- C2
- D4

Ghi chú ôn tập

## Linear Graphs - Night Before Exam Summary ### Coordinates - Cartesian plane: x-axis (horizontal), y-axis (vertical), origin (0,0). - Coordinates: (x, y) – "along the corridor, up the stairs". - Check scale: 1 square may not be 1 unit. ### Gradient of a Line - Gradient (m) = steepness. Positive = uphill, negative = downhill. - Formula: m = (change in y)/(change in x) = rise/run. - For two points (x1,y1) and (x2,y2): m = (y2 - y1)/(x2 - x1). - To draw a gradient: e.g., m = 2/3 means 3 right, 2 up; m = -5 means 1 right, 5 down. ### Equation of a Straight Line: y = mx + c - m = gradient, c = y-intercept (where line cuts y-axis). - To find equation from graph: 1. Find gradient (rise/run). 2. Read 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 - Use table of values: choose at least 3 x-values, compute y, plot points, draw line. - Or start at y-intercept (c), then use gradient: for m = a/b, go b right, a up (or down if negative). ### Parallel Lines - Parallel lines have the same gradient. - Equation of line parallel to y = mx + c is y = mx + d (different y-intercept). - To find d: substitute the given point into y = mx + d and solve. ### Key Formulas - Gradient: m = (y2 - y1)/(x2 - x1) - Line equation: y = mx + c - Parallel lines: same m.

Gradient: A measure of steepness of a line; calculated as rise/run or (y2 - y1)/(x2 - x1). Positive for uphill, negative for downhill.
y-intercept: The point where a line crosses the y-axis; denoted by c in y = mx + c.
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.
Cartesian plane: A two-dimensional grid with x-axis (horizontal) and y-axis (vertical), meeting at the origin (0,0).
Coordinates: A pair (x, y) describing a point's location: x units horizontally, y units vertically from the origin.

Thẻ ghi nhớ

What are coordinates?

Coordinates are a pair of numbers (x, y) that describe the location of a point on the Cartesian plane. x is the horizontal position, y is the vertical position.

What is the origin?

The origin is the point (0, 0) where the x-axis and y-axis meet.

How do you remember the order of coordinates?

'Along the corridor, up the stairs' helps to remember horizontal (x) then vertical (y).

What is the gradient of a line?

The gradient is a measure of how steep a straight line is. It is calculated as rise/run (change in y over change in x).

How do you find the gradient of a line from a graph?

Choose two points on the line, draw a right-angled triangle, then gradient = rise/run. Positive for uphill, negative for downhill.

What does a positive gradient look like?

A positive gradient means the line goes upwards (uphill) from bottom left to top right.

What does a negative gradient look like?

A negative gradient means the line goes downwards (downhill) from top left to bottom right.

What is the general equation of a straight line?

y = mx + c, where m is the gradient and c is the y-intercept.

What is the y-intercept?

The y-intercept is the value where the line cuts the y-axis (the point where x = 0).

How do you find the equation of a straight line from a graph?

Find the gradient (m) using rise/run and read off the y-intercept (c). Then substitute into y = mx + c.

How do you find the equation of a line given its gradient and a point?

Substitute the gradient (m) into y = mx + c, then substitute the coordinates of the point to solve for c.

What is the equation of a horizontal line?

A horizontal line has equation y = c, where c is the y-intercept.

What is the equation of a vertical line?

A vertical line has equation x = k, where k is the x-intercept.

How do you draw a straight line from a table of values?

Plot the (x, y) points from the table and join them with a straight line using a ruler.

How do you draw a straight line using y = mx + c without a table?

Start at the y-intercept (0, c). Then, for every 1 unit right, go up m units (down if m is negative). Join the points.

What are parallel lines?

Parallel lines are straight lines that have the same gradient. They never meet.

How do you find the equation of a line parallel to a given line?

Use the same gradient (m) as the given line. Then use a point to find the new y-intercept (c).

Find the gradient of the line y = 2x - 3.

The gradient is 2.

Find the y-intercept of the line y = 4x - 6.

The y-intercept is -6.

Find the equation of the line parallel to y = 5x + 6 that passes through (0, -7).

y = 5x - 7.

Find the coordinates of the point where the line y = 3x - 8 crosses the y-axis.

(0, -8).

Find the gradient of the line passing through (0,2) and (1,5).

Gradient = (5-2)/(1-0) = 3.

Write down the equation of the line with gradient -4 and y-intercept 3.

y = -4x + 3.

What is the gradient of a line that goes through (2,4) and (10,0)?

Gradient = (0-4)/(10-2) = -4/8 = -1/2.

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