Coordinates and straight-line graphs

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Coordinates and the gradient

Points are written (x, y): across first, then up or down. The axes cross at the origin (0, 0).
The gradient measures steepness: m = \frac{\text{change in } y}{\text{change in } x} = \frac{y₂ - y₁}{x₂ - x₁}.
A line sloping up from left to right has positive gradient; sloping down, negative; horizontal, zero.

Gradient from two points

The equation \(y = mx + c\)

Every non-vertical straight line can be written as y = mx + c, where m is the gradient and c is the y-intercept. A vertical line has equation x = k and undefined gradient.
y = 2x + 3 has gradient 2 and crosses the y-axis at (0, 3).
Horizontal lines are y = k; vertical lines are x = k.

Reading y = 2x + 3

Drawing a line

Make a table of values: substitute at least three x-values and plot the points.
Or use the gradient and intercept: start at c on the y-axis, then move across 1 and up m.

Finding the equation of a line

Through two points: work out the gradient m, then substitute one point into y = mx + c to find c.
Through (1, 5) and (3, 9): m = \frac{9 - 5}{3 - 1} = 2; 5 = 2(1) + c gives c = 3, so y = 2x + 3.

Line through two points

Midpoints and lengths

The midpoint of (x₁, y₁) and (x₂, y₂) is \left(\frac{x₁ + x₂}{2}, \frac{y₁ + y₂}{2}\right).
The length of a line segment comes from Pythagoras' theorem.

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Übungsfragen

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1.What are the coordinates of the origin?
Easy
- A(1, 1)
- B(0, 0)
- C(0, 1)
- D(1, 0)
2.What is the gradient of y = 4x - 1?
Easy
- A-1
- B4
- C1/4
- D3
3.Where does y = 3x + 7 cross the y-axis?
Easy
- A(7, 0)
- B(0, 7)
- C(0, 3)
- D(3, 7)
4.Which line is horizontal?
Easy
- Ax = 4
- By = 4
- Cy = 4x
- Dy = x + 4
5.Which line is vertical?
Easy
- Ay = 3
- Bx = 3
- Cy = 3x
- Dy = x
6.Find the gradient of the line through (0, 1) and (4, 9).
Easy
- A2
- B4
- C1/2
- D8
7.Find the gradient of the line through (2, 7) and (5, 1).
Easy
- A2
- B-2
- C1/2
- D-1/2
8.Which point lies on y = 2x + 1?
Easy
- A(3, 6)
- B(3, 7)
- C(2, 6)
- D(1, 4)

Mehr Fragen

9.What is the gradient of y = 5 - 2x?
Easy
- A5
- B-2
- C2
- D-5
10.Find the midpoint of (2, 4) and (8, 10).
Easy
- A(5, 7)
- B(6, 6)
- C(10, 14)
- D(3, 3)
11.A line has gradient 3 and passes through (0, -2). What is its equation?
Medium
- Ay = 3x + 2
- By = 3x - 2
- Cy = -2x + 3
- Dy = 3 - 2x
12.Find the equation of the line through (1, 5) and (3, 9).
Medium
- Ay = 2x + 3
- By = 2x + 5
- Cy = 3x + 2
- Dy = 4x + 1
13.Find the equation of the line through (0, 4) with gradient -1.
Medium
- Ay = x + 4
- By = -x + 4
- Cy = -x - 4
- Dy = 4x - 1
14.The line y = 2x - 6 crosses the x-axis at:
Medium
- A(-6, 0)
- B(3, 0)
- C(6, 0)
- D(0, -6)
15.Does (4, 9) lie on y = 2x + 1?
Medium
- AYes
- BNo, y should be 8
- CNo, y should be 10
- DNo, y should be 7
16.Find the equation of the line through (-1, 3) and (2, -3).
Medium
- Ay = -2x + 1
- By = 2x + 5
- Cy = -2x - 1
- Dy = -3x
17.The midpoint of A(1, 3) and B is (4, 5). What are the coordinates of B?
Medium
- A(7, 7)
- B(2.5, 4)
- C(3, 2)
- D(5, 8)
18.Find the length of the line from (1, 2) to (4, 6).
Medium
- A3
- B4
- C5
- D7
19.What is the y-intercept of 2y = 6x + 8?
Easy
- A8
- B4
- C3
- D6
20.Which table of values belongs to y = x + 2?
Medium
- Ax = 0, 1, 2; y = 2, 3, 4
- Bx = 0, 1, 2; y = 0, 2, 4
- Cx = 0, 1, 2; y = 2, 4, 6
- Dx = 0, 1, 2; y = 1, 2, 3
21.A line has a negative gradient. Which description is correct?
Easy
- AIt slopes up to the right
- BIt slopes down to the right
- CIt is horizontal
- DIt is vertical
22.What is the gradient of a line perpendicular to y = 2x?
Medium
- A2
- B-2
- C-\frac{1}{2}
- D\frac{1}{2}
23.The point (3, 5) is 3 units across and 5 units up from the origin.
Easy
True or false?
24.In y = mx + c, m is the gradient.
Easy
True or false?
25.The line y = 4 is vertical.
Easy
True or false?
26.The line y = x passes through the origin.
Easy
True or false?
27.Distinct non-vertical parallel lines have the same gradient.
Medium
True or false?
28.Which lines have a gradient of 3?
Medium
- Ay = 3x + 1
- By = x + 3
- Cy = 3x - 5
- Dy = \frac{x}{3}
- Ey = 3 - x
29.Which points lie on y = 2x - 1?
Medium
- A(0, -1)
- B(1, 1)
- C(2, 4)
- D(3, 5)
- E(4, 6)
30.What is the gradient of the line y = x?
Easy
- A0
- B1
- C-1
- Dundefined
31.Which line passes through the origin?
Easy
- Ay = 2x + 1
- By = 3x
- Cy = x - 2
- Dy = 5
32.Find the equation of the line through (2, 3) and (4, 7).
Medium

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