Linear Graphs

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Notes

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.

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).

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).

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.

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.

Coordinate Grid with Points

Gradient of a Line

Equation y = mx + c

Parallel Lines

Practice questions

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1.What is the gradient of the line $y = 2x - 3$ ?
Easy
- A2
- B-3
- C3
- D-2
2.What is the y-intercept of the line $y = 4x - 6$ ?
Easy
- A-6
- B4
- C6
- D-4
3.A line has gradient 3 and passes through (0, 1). What is its equation?
Easy
- A $y = 3x + 1$
- B $y = 3x - 1$
- C $y = -3x + 1$
- D $y = x + 3$
4.What is the gradient of a horizontal line?
Easy
- A0
- B1
- Cundefined
- D-1
5.What are the coordinates of the origin?
Easy
- A(0, 0)
- B(1, 1)
- C(0, 1)
- D(1, 0)
6.Find the gradient of the line passing through (1, 2) and (3, 6).
Medium
- A2
- B4
- C $\frac{1}{2}$
- D3
7.What is the equation of the line parallel to $y = 5x + 6$ that passes through (0, -7)?
Medium
- A $y = 5x - 7$
- B $y = 5x + 6$
- C $y = -5x - 7$
- D $y = 5x + 7$
8.A line has gradient -2 and passes through (0, 4). What is its equation?
Medium
- A $y = -2x + 4$
- B $y = 2x + 4$
- C $y = -2x - 4$
- D $y = 2x - 4$

Summary

## 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.

Key terms

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.

Slides

Coordinates

The Cartesian plane has an x-axis (horizontal) and y-axis (vertical).
Coordinates are written as (x, y) – 'along the corridor, up the stairs'.
The origin is (0, 0). Positive x is right, negative left; positive y is up, negative down.

What are the coordinates of the point 3 units left and 2 units up from the origin?

(3, 2)
(-3, 2)
(2, -3)
(-3, -2)

Gradient of a Line

Gradient measures steepness: gradient = rise / run.
Positive gradient: line goes uphill (bottom left to top right).
Negative gradient: line goes downhill (top left to bottom right).
For two points (x₁,y₁) and (x₂,y₂): gradient = (y₂ - y₁) / (x₂ - x₁).

A line with gradient 3 means for every 1 unit to the right, you $go ____$ units up.

Equation of a Straight Line: y = mx + c

m is the gradient, c is the y-intercept (where the line crosses the y-axis).
To find the equation from a graph: find m using rise/run, then read off c.
If c is not visible, substitute a point on the line into $y = mx + c$ to solve for c.

What is the gradient of the line $y = 5x - 2$ ?

5
-2
2
-5

Drawing Straight Line Graphs

Create a table of values by choosing x-values and calculating y.
Plot at least 3 points and join them with a straight line.
Alternatively, start at the y-intercept (c) and use the gradient to find other points.

Which of these are true about drawing straight line graphs?

You should plot at least 3 points.
The gradient tells you how many units to go up for 1 unit right.
A negative gradient means the line goes uphill.

Parallel Lines

Parallel lines have the same gradient.
If a line is parallel to $y = mx + c$ , its equation is $y = mx + d (different y-intercept)$ .
To find d, substitute a point on the new line into $y = mx + d$ .

A line parallel to $y = 4x + 1$ has gradient ____.

Finding the Equation from a Graph

Identify two points on the line to calculate the gradient $m = rise/run$ .
Read the y-intercept c where the line crosses the y-axis.
Write the equation as $y = mx + c$ .

A line passes through (0, 3) and (2, 7). What is its equation?

$y = 2x + 3$
$y = 4x + 3$
$y = 2x - 3$
$y = 4x - 3$

Horizontal and Vertical Lines

Horizontal line: $y = c (gradient 0)$ .
Vertical line: $x = k (gradient undefined)$ .
For example, $y = 4$ is horizontal; $x = -2$ is vertical.

Which of these are true?

The line $y = 3$ is horizontal.
The line $x = 5$ has gradient 0.
A vertical line has equation $x =$ constant.

Using the Gradient Formula

For points (x₁, y₁) and (x₂, y₂), gradient = (y₂ - y₁) / (x₂ - x₁).
Be careful with negative coordinates.
Always check if the line is uphill (positive) or downhill (negative).

The gradient between (1, 2) and (4, 11) $is ____$ .

Finding Parallel Line Equations

Use the same gradient m as the given line.
Let the new equation be $y = mx + d$ .
Substitute the given point to find d.

What is the equation of the line parallel to $y = 3x + 2$ passing through (0, 5)?

$y = 3x + 5$
$y = 3x + 2$
$y = 5x + 3$
$y = 3x - 5$

Flashcards

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 $= (0-4)/(10-2) = -\frac{4}{8} = -\frac{1}{2}$ .

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