Gradients of straight-line graphs - Intermediate and Higher tier
Finding the gradient
The gradient of a straight line describes the slope or steepness of the line.
\(\text{Gradient =}~\frac{change~in~y}{change~in~x}\)
To determine the gradient of a line:
- choose any two points on the line
- draw a right-angled triangle from one to the other, using the line as the hypotenuse
- determine the height and width of the triangle
- gradient = height ÷ width
Example
The triangle goes from 2 to 8 on the \(\text{y}\)-axis, so has a height of 6. It goes from 1 to 3 on the \(\text{x}\)-axis, so has a width of 2.
\(\text{Gradient =}~\frac{6}{2}~=~{3}\)
Question
What is the gradient of this straight line?
\(\text{Gradient =}~\frac{2}{1}~=~{2}\)
Positive and negative gradients
Gradients can be positive or negative, depending on the slant of the line.
This line has a positive gradient, because going from the left to right in the direction of the \(\text{x}\)-axis, the \(\text{y}\) values increase.
This line has a negative gradient, because going from the left to right in the direction of the \(\text{x}\)-axis, the \(\text{y}\) values decrease.
Example
We use the formula:
\(\text{Gradient}~=~\frac{change~in~y}{change~in~x}\)
The triangle goes from 8 to 4 on the \(\text{y}\)-axis, so the change in \(\text{y}\) is -4. It goes from 1 to 3 on the \(\text{x}\)-axis so the change is \(\text{x}\) is 2.
\(\text{Gradient =}~\frac{-4}{2}~=~{-2}\)
Question
What is the gradient of the following line?
\(\text{Gradient =}~\frac{-1}{2}~=~{-0.5}\)
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