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U Linear functions
3 ESO Mathematics
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A8.1 Plotting Linear graphs
Contents
A8 Linear and real-life graphs
A8.2 Gradients and intercepts
A8.3 Analytical expression
A8.5 Distance-time graphs
A8.6 Speed-time graphs
A8.4 Parallel lines
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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.
It is very recommendable to add the points of intersection to your table.
xy = 2x + 5
–3 –2 –1 0 1 2 3
For example,
y = 2x + 5
y
x
–1 1 3 5 7 9 11
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Plotting graphs of linear functions
In the example, y = 2x + 5:
1) If you make x = 0, you obtain y: y = 2 · 0 + 5 y = 5
2) If you make y = 0, you obtain x: 0 = 2x + 5 2x = -5 x = -5 / 2
y = 2x + 5
y
x
You can find the points of intersectionvery easily.
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Plotting graphs of linear functions
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A8.2 Gradients and intercepts
Contents
A8.3 Analytical expression
A8 Linear and real-life graphs
A8.1 Linear graphs
A8.5 Distance-time graphs
A8.6 Speed-time graphs
A8.4 Parallel lines
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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 by specified.
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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,
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Calculating gradients
• A8.3 Parallel and perpendicular lines• A8.3 Parallel and perpendicular lines
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A8.3 Analytical expression
Contents
A8.2 Gradients and intercepts
A8 Linear and real-life graphs
A8.1 Linear graphs
A8.4 Parallel lines
A8.5 Distance-time graphs
A8.6 Speed-time graphs
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Investigating linear graphs
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The general equation of a straight line
The general equation of a straight line can be written as:
y = mx + c
The value of m tells us the gradient of the line.
The value of c tells us where the line crosses the y-axis.
This is called the y-intercept and it has the coordinate (0, c).
For example, the line y = 3x + 4 has a gradient of 3 and crosses the y-axis at the point (0, 4).
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The gradient and the y-intercept
Complete this table:
equation gradient y-intercept
y = 3x + 4
y = – 5
y = 2 – 3x
1
–2
3 (0, 4)
(0, –5)
–3 (0, 2)
y = x
y = –2x – 7
x2
12
(0, 0)
(0, –7)
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What is the equation of the line?
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Match the equations to the graphs
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Rearranging equations into the form y = mx + c
Sometimes the equation of a straight line graph is not given in the form y = mx + c.
The equation of a straight line is 2y + x = 4.Find the gradient and the y-intercept of the line.
Rearrange the equation by performing the same operations on both sides,
2y + x = 4
y = – x + 212
2y = –x + 4subtract x from both sides:
y =–x + 4
2divide both sides by 2:
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Rearranging equations into the form y = mx + c
Sometimes the equation of a straight line graph is not given in the form y = mx + c.
The equation of a straight line is 2y + x = 4.Find the gradient and the y-intercept of the line.
Once the equation is in the form y = mx + c we can determine the value of the gradient and the y-intercept.
So the gradient of the line is 12
– and the y-intercept is 2.
y = – x + 212
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Substituting values into equations
A line with the equation y = mx + 5 passes through the point (3, 11).
What is the value of m?
To solve this problem we can substitute x = 3 and y = 11 into the equation y = mx + 5.
This gives us, 11 = 3m + 5
6 = 3msubtract 5 from both sides:
2 = mdivide both sides by 3:
m = 2
The equation of the line is therefore y = 2x + 5.
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A8.4 Parallel lines
Contents
A8.2 Gradients and intercepts
A8 Linear and real-life graphs
A8.1 Linear graphs
A8.5 Distance-time graphs
A8.6 Speed-time graphs
A8.3 Analytical expression
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Investigating parallel lines
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Parallel lines
If two lines have the same gradient they are parallel.If two lines have the same gradient they are parallel.
Show that the lines 2y + 6x = 1 and y = –3x + 4 are parallel.
We can show this by rearranging the first equation so that it is in the form y = mx + c.
2y = –6x + 1subtract 6x from both sides:
y =–6x + 1
2divide both sides by 2:
2y + 6x = 1
y = –3x + ½
The gradient m is –3 for both lines and so they are parallel.
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Matching parallel lines
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A8.5 Distance-time graphs
Contents
A8.3 Analytical expression
A8.2 Gradients and intercepts
A8.1 Linear graphs
A8 Linear and real-life graphs
A8.6 Speed-time graphs
A8.4 Parallel lines
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Formulae relating distance, time and speed
It is important to remember how distance, time and speed are related.
Using a formula triangle can help,
distance = speed × timedistance = speed × time
DISTANCE
SPEED TIME
time =distance
speed
speed =distance
time
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Distance-time graphs
In a distance-time graph the horizontal axis shows time and the vertical axis shows distance.
For example, John takes his car to visit a friend. There are three parts to the journey:
John drives at constant speed for 30 minutes until he reaches his friend’s house 20 miles away.
He stays at his friend’s house for 45 minutes.
He then drives home at a constant speed and arrives 45 minutes later.
0
time (mins)
dist
ance
(m
iles)
15 30 45 60 75 90 105 120
5
10
15
20
0
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Finding speed from distance-time graphs
How do we calculate speed?
Speed is calculated by dividing distance by time.
The steeper the line, the faster the object is moving.
time
dis
tan
ce
In a distance-time graph this is given by the gradient of the graph.
change in distance
change in time
gradient =change in distance
change in time
= speed
A zero gradient means that the object is not moving.
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Interpreting distance-time graphs
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Distance-time graphs
When a distance-time graph is linear the objects involved are moving at a constant speed.
Most real-life objects do not always move at constant speed, however. It is more likely that they will speed up and slow down during the journey.
Increase in speed over time is called acceleration.
acceleration =change in speed
time
It is measured in metres per second per second or m/s2.
When speed decreases over time is often is called deceleration.
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Distance-time graphs
Distance-time graphs that show acceleration or deceleration are curved. For example,
This distance-time graph shows an object decelerating from constant speed before coming to rest.
time
dis
tan
ce
This distance-time graph shows an object accelerating from rest before continuing at a constant speed.
time
dis
tan
ce
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A8.6 Speed-time graphs
Contents
A8.3 Parallel and perpendicular lines
A8.2 Gradients and intercepts
A8.1 Linear graphs
A8 Linear and real-life graphs
A8.5 Distance-time graphs
A8.4 Interpreting real-life graphs
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Speed-time graphs
Travel graphs can also be used to show change in speed over time.
For example, this graph shows a car accelerating steadily from rest to a speed of 20 m/s.
0
time (s)
spee
d (m
/s)
5 10 15 20 25 30 35 40
5
10
15
20
0
It then continues at a constant speed for 15 seconds.
The brakes are then applied and it decelerates steadily to a stop.
The car is moving in the same direction throughout.
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Finding acceleration from speed-time graphs
Acceleration is calculated by dividing speed by time.
The steeper the line, the greater the acceleration.
time
spe
ed
In a speed-time graph this is given by the gradient of the graph.
gradient =change in speedchange in time
= acceleration
A zero gradient means that the object is moving at a constant speed.
change in speed
change in time
A negative gradient means that the object is decelerating.
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Interpreting speed-time graphs
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