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Increasing & Decreasing Functions

Recall that upward sloping straight lines have a positive gradient whereas downward sloping straight lines have a negative gradient. The same applies to curves. Gradients on a curve are always changing but an upward sloping curve has a positive gradient and a downward sloping curve has a negative gradient.

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Recall the graph of y=x^2. You will notice that for positive x, the graph has a positive gradient; for negative x the graph has a negative gradient; and for x=0 the gradient is also 0.

This can be seen from the gradient function \frac{dy}{dx}=2x. Find out more about differentiating. 2x is positive when x is positive, negative when x is negative and 0 when x is 0.

Example 1Find the range of values of x for which the graph of y=x^2-5x+4 has a positive gradient.

Differentiating y gives \frac{dy}{dx}=2x-5. This is positive when 2x-5\textgreater 0, i.e. when 2x is greater than 5. This gives the solution x\textgreater 2.5.

Example 2 Explain why the graph of y=x^3 is never negative.

You can see from the graph of x cubed that it never has a negative gradient but we show it using differentiation.

\frac{dy}{dx}=3x^2 which is 3 lots of a square number. Irrespective of the value of x that is put in, this number will always be positive.