Coordinates and straight-line graphs

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Notes de leçon

Coordinates in All Four Quadrants

A point is located by an ordered pair (x, y): x first (across), y second (up or down).
The x-axis is horizontal and the y-axis is vertical; they cross at the origin (0, 0).
The axes divide the plane into four quadrants.
In the first quadrant x and y are both positive, e.g. (3, 2).
In the second quadrant x is negative and y positive, e.g. (−3, 2).
In the third quadrant both are negative, e.g. (−3, −2).
In the fourth quadrant x is positive and y negative, e.g. (3, −2).

Midpoint of a Line Segment

The midpoint is the point exactly halfway between two endpoints.
For endpoints (x₁, y₁) and (x₂, y₂), the midpoint is \left(\frac{x₁ + x₂}{2},\ \frac{y₁ + y₂}{2}\right).
In words: average the x-coordinates and average the y-coordinates.
Example: the midpoint of (2, 3) and (6, 7) is \left(\frac{2+6}{2},\ \frac{3+7}{2}\right) = (4, 5).
The midpoint lies on the segment, so it is a good check that your answer is sensible.

Plotting Straight-Line Graphs from a Table of Values

Choose several x-values and work out the matching y-values using the equation.
Record the pairs in a table of values, one column for x and one for y.
Plot each (x, y) pair as a point on the coordinate grid.
Join the points with a straight line using a ruler.
Extend the line beyond the plotted points to show the pattern continues.
Check a third point: if it does not lie on the line, re-check your calculations.

Horizontal and Vertical Lines

A line of the form y = b is horizontal and crosses the y-axis at (0, b).
Every point on y = b has the same y-coordinate, b.
A line of the form x = a is vertical and crosses the x-axis at (a, 0).
Every point on x = a has the same x-coordinate, a.
Example: y = 3 is horizontal through (0, 3); x = −2 is vertical through (−2, 0).
The x-axis itself is the line y = 0, and the y-axis is the line x = 0.

The Equation y = mx + c

A straight-line graph can be written in the form y = mx + c.
Here m is the gradient (how steep the line is) and c is the y-intercept (where it crosses the y-axis).
The point (0, c) is always on the line.
If m > 0 the line slopes upwards from left to right.
If m < 0 the line slopes downwards from left to right.
If m = 0 the line is horizontal, y = c.
Example: y = 2x + 1 has gradient 2 and crosses the y-axis at (0, 1).

Finding the Gradient Between Two Points

Gradient measures steepness: m = \frac{\text{change in } y}{\text{change in } x}.
For points (x₁, y₁) and (x₂, y₂): m = \frac{y₂ - y₁}{x₂ - x₁}.
The change in y is the rise, and the change in x is the run.
Example: through (1, 2) and (4, 11), m = \frac{11 - 2}{4 - 1} = \frac{9}{3} = 3.
A positive gradient means the line rises; a negative gradient means it falls.
The gradient is the same between any two points on the same straight line.

Parallel Lines

Parallel lines never meet and stay the same distance apart.
Parallel lines have the same gradient.
Example: y = 3x + 1 and y = 3x − 4 are parallel because both have gradient 3.
They have different y-intercepts, so they are distinct lines.
If two lines have different gradients, they are not parallel.

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1.In the equation y = mx + c, what does m represent?
Easy
- AThe gradient of the line
- BThe y-intercept of the line
- CThe x-intercept of the line
- DThe value of y when x = 0
2.In the equation y = mx + c, what does c represent?
Easy
- AThe gradient of the line
- BThe y-intercept of the line
- CThe x-intercept of the line
- DThe steepness of the line
3.Which of these is the equation of a horizontal line?
Easy
- Ax = 3
- By = 3
- Cy = 3x
- Dy = x + 3
4.Which of these is the equation of a vertical line?
Easy
- Ay = -4
- Bx = -4
- Cy = -4x
- Dy = x - 4
5.A line has gradient 5 and passes through the point (0, -2). What is its equation?
Medium
- Ay = 5x - 2
- By = -2x + 5
- Cy = 5x + 2
- Dy = 2x - 5
6.What is the midpoint of the line segment joining A(2, 5) and B(8, 1)?
Medium
- A(5, 3)
- B(10, 6)
- C(6, 4)
- D(3, 2)
7.A line has gradient -3 and passes through the point (1, 4). What is its equation?
Medium
- Ay = -3x + 7
- By = -3x + 4
- Cy = 3x + 7
- Dy = -3x - 7
8.Which of these lines is parallel to y = 4x - 1?
Medium
- Ay = 4x + 9
- By = -4x - 1
- Cy = (1/4)x + 3
- Dy = x - 1

Plus de questions

9.A straight line passes through the points (2, 3) and (6, 11). What is its gradient?
Medium
- A2
- B4
- C8
- D1/2
10.Which of the following statements about the line y = -2x + 6 are true? (select all that apply)
Medium
- AThe gradient is -2
- BThe y-intercept is 6
- CThe line slopes downwards from left to right
- DThe line passes through the origin
- EThe line is parallel to y = 2x + 6
11.Which of the following points lie on the line y = 3x - 1? (select all that apply)
Medium
- A(0, -1)
- B(1, 2)
- C(2, 5)
- D(3, 8)
- E(4, 10)
12.Which of the following lines pass through the point (0, 0)? (select all that apply)
Medium
- Ay = 5x
- By = x
- Cy = -x
- Dy = x + 1
- Ey = 2x - 3
13.The point (-3, 4) lies in the second quadrant.
Easy
True or false?
14.The line y = 2 passes through the point (5, 2).
Easy
True or false?
15.The lines y = 3x + 1 and y = 3x - 4 are parallel.
Medium
True or false?
16.The line x = 7 crosses the y-axis at the point (0, 7).
Easy
True or false?
17.Match each equation to the correct description of its graph.
Medium
- y = 5
- x = -2
- y = 3x + 1
- y = -x
- A horizontal line
- A vertical line
- A line with gradient 3
- A line sloping downwards
18.Match each line to its y-intercept.
Medium
- y = 2x + 7
- y = x - 3
- y = 4x
- y = -5x + 1
- 7
- -3
- 0
- 1
19.Order these y-values from smallest to largest for the line y = 3x + 2 when x = -2, -1, 0, 1.
Medium
- -4
- -1
- 2
- 5
20.Which point lies on this line?
Medium
- A(2, 2)
- B(2, 6)
- C(0, 4)
- D(1, 4)
21.Which line is parallel to y = 3x - 4?
Medium
- Ay = 3x + 8
- By = -3x - 4
- Cy = (1/3)x + 8
- Dy = 4x - 3
22.The table gives the points (0, 1), (1, 3), (2, 5) and (3, 7). What is the gradient of the line through these points?
Medium
- A2
- B1
- C3
- D1/2
23.The point (5, 11) lies on the line y = 2x + 1.
Medium
True or false?
24.What is the equation of the line through P and Q?
Medium
- Ay = 2x
- By = 2x + 1
- Cy = 4x - 2
- Dy = x + 1
25.Match each line equation to the description of its graph.
Easy
- y = 5
- x = 5
- y = 5x
- y = x + 5
- horizontal line
- vertical line
- passes through the origin
- crosses the y-axis at (0, 5)
26.Order these lines by their gradient, from smallest (most negative) to largest.
Medium
- y = -4x + 1
- y = -x + 7
- y = 2x - 3
- y = 6x + 2
27.Select all the lines that are parallel to y = 2x + 3.
Medium
- Ay = 2x - 7
- By = -2x + 3
- Cy = 2x
- Dy = 1/2 x + 3
- Ey = 3x + 2
28.The line y = -x + 4 is drawn on a grid. Select all statements that are true.
Medium
- AIt passes through (0, 4).
- BIt passes through (4, 0).
- CIts gradient is 1.
- DIt slopes downwards from left to right.
- EIt passes through (2, 2).
29.Match each pair of points to the gradient of the line joining them.
Medium
- (1, 2) and (3, 8)
- (0, 5) and (2, 1)
- (2, 3) and (6, 3)
- (1, 1) and (4, 7)
- 3
- -2
- 0
30.Order these y-values from smallest to largest for the line y = -2x + 5 when x = -1, 0, 2, 4.
Medium
- -3
- 5
- 1
- -1
31.Select all the points that lie on the line y = -3x + 2.
Medium
- A(0, 2)
- B(1, -1)
- C(2, -4)
- D(-1, 5)
- E(3, -7)
32.What is the gradient of the line that passes through the points (2, 1) and (6, 9)?
Medium
- A2
- B-2
- C1/2
- D4
33.What is the equation of the line with gradient -2 that passes through the point (0, 3)?
Medium
- Ay = -2x + 3
- By = 2x + 3
- Cy = -2x - 3
- Dy = 3x - 2
34.The line y = -4x + 1 passes through the point (2, -7).
Easy
True or false?
35.A straight line has equation y = 6x + 1. State the gradient of the line.
Easy
- A1
- B6
- C6x
- D7
36.A straight line passes through the points (0, 5) and (4, 17). Work out the gradient of the line.
Medium
- A3
- B4
- C-3
- D12
37.The line y = -3x + 8 passes through the point (2, k). Work out the value of k.
Medium
- A2
- B14
- C3
- D-2
38.Work out the gradient of a line that is parallel to y = -5x + 2.
Medium
- A-5
- B5
- C2
- D-2
39.The line y = 4x - 3 is drawn on a coordinate grid. Select all the statements that are true. (select all that apply)
Medium
- AThe gradient is 4.
- BThe line crosses the y-axis at (0, -3).
- CThe line passes through the point (2, 5).
- DThe line passes through the point (1, 1).
- EThe line is parallel to y = -4x - 3.
40.The point (-2, 7) lies in the second quadrant.
Easy
True or false?
41.What is the gradient of any line that is parallel to y = 7x - 2?
Medium
42.A straight line passes through the points (1, 2) and (5, 10). Work out the gradient of the line.
Medium
43.The line y = 4x + 3 crosses the y-axis at the point (0, c). State the value of c.
Medium
44.Work out the y-coordinate of the point on the line y = 2x - 5 where x = 6.
Medium
45.Complete the sentence about the equation of a straight line.
Easy
In the equation y = mx + c, the letter m stands for the ____ and the letter c stands for the ____.
46.Complete the sentence about parallel lines.
Medium
Two straight lines are parallel when they have the same ____.
47.Complete the sentence about horizontal and vertical lines.
Medium
The graph of x = a is a ____ line and the graph of y = b is a ____ line.
48.State the gradient of this line.
Medium
49.State the y-intercept of this line.
Medium
50.The table gives x = 0, 1, 2, 3 with the corresponding y-values. Work out the missing y-value when x = 3.
Medium
51.Work out the gradient of the line PQ.
Medium
52.Work out the x-coordinate of the midpoint of PQ.
Medium
53.Explain how to find the midpoint of the line segment joining two points, and state the midpoint of A(1, 4) and B(7, 10).
Hard
54.The points A(1, 2) and B(7, 10) are joined by a straight line. Without drawing the line, explain how you would work out the gradient of AB, and state its value.
Hard
55.A straight line passes through the points (1, 2) and (5, 10). Work out the gradient of the line.
Medium
- A3
- B4
- C1
- D2
56.The line y = 4x + 3 crosses the y-axis at the point (0, c). State the value of c.
Medium
- A3
- B5
- C4
- D2
57.Work out the y-coordinate of the point on the line y = 2x - 5 where x = 6.
Medium
- A8
- B9
- C7
- D6
58.State the gradient of this line.
Medium
- A4
- B3
- C2
- D5
59.State the y-intercept of this line.
Medium
- A-5
- B-4
- C-2
- D-3
60.The table gives x = 0, 1, 2, 3 with the corresponding y-values. Work out the missing y-value when x = 3.
Medium
- A8
- B6
- C7
- D9
61.Work out the gradient of the line PQ.
Medium
- A4
- B3
- C2
- D1
62.Work out the x-coordinate of the midpoint of PQ.
Medium
- A4
- B5
- C2
- D3

Notes de révision

## Coordinates and Straight-Line Graphs ### Coordinates - Written as **(x, y)**: x first (across), y second (up/down). - Four quadrants: (+,+), (−,+), (−,−), (+,−). - **Midpoint** of a segment joining (x_1, y_1) and (x_2, y_2): \(\left(\frac{x_1 + x_2}{2},\ \frac{y_1 + y_2}{2}\right)\) — average the x's, average the y's. ### Plotting straight lines - Make a **table of values**, substitute each x into the equation, plot the points, join with a ruler. - **Horizontal line:** y = b (constant function, gradient 0). - **Vertical line:** x = a (not a function of x). ### y = mx + c - m = **gradient** (slope), c = **y-intercept** (where the line crosses the y-axis). - If m > 0 the line slopes upwards; if m < 0 it slopes downwards. ### Finding the gradient between two points \[m = \frac{\text{change in } y}{\text{change in } x} = \frac{y_2 - y_1}{x_2 - x_1}\] ### Parallel lines - Parallel lines have the **same gradient** (same m, different c). ### Quick checks - To test if a point lies on a line, substitute its x and y into the equation. - The graph of y = mx + c is a straight (non-vertical) line; it passes through the origin only when c = 0.

Coordinates: An ordered pair (x, y) giving a position on the grid: x is the horizontal distance, y the vertical distance.
Quadrant: One of the four regions of the coordinate plane, separated by the axes.
Midpoint: The point exactly halfway between two points; found by averaging their x-coordinates and their y-coordinates.
Gradient (m): The steepness of a line: change in y divided by change in x between two points.
y-intercept (c): The value of y where the line crosses the y-axis (where x = 0).
Linear function: A function whose graph is a straight line, of the form f(x) = ax + b where a and b are constants.
Constant function: A function of degree zero, e.g. y = b; its graph is a horizontal line.
Parallel lines: Lines that never meet; they have the same gradient.

Cartes mémo

In the coordinate plane, what do the coordinates (x, y) of a point tell you?

x tells you the horizontal position (across) and y tells you the vertical position (up or down). Together they locate the point exactly.

How are the four quadrants of a coordinate grid numbered?

Starting from the top-right and going anticlockwise: first quadrant (+, +), second (−, +), third (−, −), fourth (+, −).

What is the midpoint of the line segment joining (x₁, y₁) and (x₂, y₂)?

\left(\frac{x₁ + x₂}{2},\ \frac{y₁ + y₂}{2}\right) — the average of the x-coordinates and the average of the y-coordinates.

Find the midpoint of the segment joining (2, 3) and (8, 11).

x = (2 + 8)/2 = 5 and y = (3 + 11)/2 = 7, so the midpoint is (5, 7).

What does the graph of x = a look like, and which axis does it cross?

It is a vertical line through the point (a, 0) on the x-axis. Every point on it has x-coordinate a.

What does the graph of y = b look like, and which axis does it cross?

It is a horizontal line through the point (0, b) on the y-axis. Every point on it has y-coordinate b.

What is the general equation of a straight-line graph, and what do m and c represent?

y = mx + c, where m is the gradient (steepness) and c is the y-intercept (where the line crosses the y-axis).

How do you find the gradient of a line between two points (x₁, y₁) and (x₂, y₂)?

m = \frac{y₂ - y₁}{x₂ - x₁} — the change in y divided by the change in x.

Find the gradient of the line through (1, 2) and (4, 11).

m = (11 − 2)/(4 − 1) = 9/3 = 3.

If a line has a positive gradient (m > 0), how does its graph slope?

It slopes upwards from left to right — as x increases, y increases.

If a line has a negative gradient (m < 0), how does its graph slope?

It slopes downwards from left to right — as x increases, y decreases.

What is the gradient of a horizontal line such as y = 5?

The gradient is 0, because y does not change as x increases.

Why does a vertical line such as x = 3 have no gradient value?

The change in x is 0, so the gradient calculation would divide by zero. Its gradient is undefined.

What is true about the gradients of parallel lines?

Parallel lines have the same gradient. Only their y-intercepts (the value of c) differ.

How do you plot a straight-line graph from a table of values?

Substitute each x-value into the equation to get the matching y-value, write the (x, y) pairs, plot them as points, then join them with a straight line.

For y = 2x + 1, what is y when x = 3?

y = 2 × 3 + 1 = 7, so the point (3, 7) lies on the line.

How do you find the y-intercept of a line from its equation y = mx + c?

It is the value c, giving the point (0, c) where the line crosses the y-axis.

What is the y-intercept of the line y = 4x − 3?

c = −3, so the line crosses the y-axis at (0, −3).

How can you tell whether a point lies on a given straight line?

Substitute the point's x and y into the equation. If the two sides are equal, the point lies on the line.

Does the point (2, 9) lie on the line y = 3x + 2?

Yes: 3 × 2 + 2 = 8, which is not 9 — so no, it does not lie on the line.

What is the equation of the line with gradient 5 that passes through the origin?

y = 5x, since the y-intercept c = 0.

In the equation y = mx + c, what happens to the line if c = 0?

The line passes through the origin (0, 0), so it is a straight line through the centre of the grid.

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