© boardworks ltd 2008 1 of 49 a8 linear and real-life graphs maths age 14-16
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© Boardworks Ltd 2008 1 of 49
A8 Linear and real-life graphs
Maths Age 14-16
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A8.1 Linear graphs
Contents
A8 Linear and real-life graphs
A8.2 Gradients and intercepts
A8.3 Parallel and perpendicular lines
A8.4 Interpreting real-life graphs
A8.5 Distance-time graphs
A8.6 Speed-time graphs
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Coordinate pairs
When we write a coordinate, for example,
Together, the x-coordinate and the y-coordinate are called a coordinate pair.
the first number is called the x-coordinate and the second number is called the y-coordinate.
(3, 5)
x-coordinate
(3, 5)
y-coordinate
(3, 5)
the first number is called the x-coordinate and the second number is the y-coordinate.
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Graphs parallel to the y-axis
What do these coordinate pairs have in common?
(2, 3), (2, 1), (2, –2), (2, 4), (2, 0) and (2, –3)?
The x-coordinate in each pair is equal to 2.
Look what happens when these points are plotted on a graph.
x
y All of the points lie on a straight line parallel to the y-axis.
Name five other points that will lie on this line.
This line is called x = 2.x = 2
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Graphs parallel to the y-axis
All graphs of the form x = c,
where c is any number, will be parallel to the y-axis and will cut the x-axis at the point (c, 0).
All graphs of the form x = c,
where c is any number, will be parallel to the y-axis and will cut the x-axis at the point (c, 0).
x
y
x = –3x = –10 x = 4 x = 9
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Graphs parallel to the x-axis
What do these coordinate pairs have in common?
(0, 1), (4, 1), (–2, 1), (2, 1), (1, 1) and (–3, 1)?
The y-coordinate in each pair is equal to 1.
Look at what happens when these points are plotted on a graph.
x
y All of the points lie on a straight line parallel to the x-axis.
This line is called y = 1.
y = 1Name five other points that will lie on this line.
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Graphs parallel to the x-axis
All graphs of the form y = c,
where c is any number, will be parallel to the x-axis and will cut the y-axis at the point (0, c).
All graphs of the form y = c,
where c is any number, will be parallel to the x-axis and will cut the y-axis at the point (0, c).
x
y
y = –2
y = 5
y = –5
y = 3
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Plotting graphs of linear functions
The x-coordinate and the y-coordinate in a coordinate pair can be linked by a function.
What do these coordinate pairs have in common?
(1, –1), (4, 2), (–2, –4), (0, –2), (–1, –3) and (3.5, 1.5)?
In each pair, the y-coordinate is 2 less than the x-coordinate.
These coordinates are linked by the function:
y = x – 2
We can draw a graph of the function y = x – 2 by plotting points that obey this function.
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Plotting graphs of linear functions
Given a function, we can find coordinate points that obey the function by constructing a table of values.
Suppose we want to plot points that obey the function
y = 2x + 5
We can use a table as follows:
x
y = 2x + 5
–3 –2 –1 0 1 2 3
–1
(–3, –1)
1 3 5 7 9 11
(–2, 1) (–1, 3) (0, 5) (1, 7) (2, 9) (3, 11)
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Plotting graphs of linear functions
to draw a graph of y = 2x + 5:
1) Complete a table of values:
2) Plot the points on a coordinate grid.
3) Draw a line through the points.
4) Label the line.
5) Check that other points on the line fit the rule.
For example,
y = 2x + 5
y
x
xy = 2x + 5
–3 –2 –1 0 1 2 3–1 1 3 5 7 9 11
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Plotting graphs of linear functions
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A8.2 Gradients and intercepts
Contents
A8.3 Parallel and perpendicular lines
A8 Linear and real-life graphs
A8.1 Linear graphs
A8.4 Interpreting real-life graphs
A8.5 Distance-time graphs
A8.6 Speed-time graphs
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Gradients of straight-line graphs
The gradient of a line is a measure of how steep the line is.
y
x
a horizontal line
Zero gradient
y
x
a downwards slope
Negative gradient
y
x
an upwards slope
Positive gradient
The gradient of a line can be positive, negative or zero if, moving from left to right, we have
If a line is vertical, its gradient cannot be specified.
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Calculating gradients
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Finding the gradient from two given points
If we are given any two points (x1, y1) and (x2, y2) on a line we can calculate the gradient of the line as follows,
the gradient =change in ychange in x
the gradient =y2 – y1
x2 – x1
x
y
x2 – x1
(x1, y1)
(x2, y2)
y2 – y1
Draw a right-angled triangle between the two points on the line as follows,