Linear Graphs

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Lesson 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.

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

Slides

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Practice questions

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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

Revision notes

## 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.

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) = -4/8 = -1/2.

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