Gradients
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The gradient of a straight line is a measure of how steep it is. The gradient of a straight line is constant for any point on the line.
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The gradient of a curve at any point is given by the gradient of the tangent at that point. The gradient of a curve is different at each point on the curve.
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The Gradient Function of y = x²
Consider the curve y = x². Investigate the gradient at various points. (Worked out by finding the slope of the tangents as above.)
The gradients themselves form a function:
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x
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Gradient
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-2
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-4
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-1
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-2
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0
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0
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1
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2
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2
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4
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The equation of this function is y = 2x.
This function is called the gradient function and from it the gradient of the curve y = x² can be found at any point, for any value of x.
The gradient function is sometimes called the derived function or the derivative.
In the next topic, a non-graphical method of finding the derived function will be shown. This process is called differentiation.
Here are three functions (in pink) and their gradient functions (in red).
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Function is y = x²
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Function is y = 2x² + 3x
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Function is y = x³
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Gradient function is y = 2x
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Gradient function is y = 4x + 3
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Gradient function is 3x²
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"Can you see the algebraic connection between the function and the gradient function, before the next topic?
