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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
a downwards slope
Negative gradient
y
x
an upwards slope
Positive gradient
If a line is vertical, its gradient cannot be specified.
y
x
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the gradient =change in ychange in x
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:
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
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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:
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).
y = mx + c
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The gradient and the y-intercept
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Rearranging to y = mx + c
Sometimes the equation of a straight line graph is not given in the form y = mx + c.
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:
The equation of a straight line is 2y + x = 4. Find the gradient and the y-intercept of the line.
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Once the equation is in the form y = mx + c we can determine the value of the gradient and the y-intercept.
c = 2 and so the y-intercept is (0, 2).
y = – x + 212
Rearranging to y = mx + c
12
–m = so the gradient of the line is .12
–
x
y
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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What is the equation of the line?
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Match the equations to the graphs