Gradient change
Starting to the left of a maximum the gradient changes from ' + ' to ' - 'with increasing 'x'.
Starting to the left of a minimum, the gradient changes from ' - ' to ' + 'with increasing 'x'.
At the point of maximum or minimum the gradient is zero.
Example Show that the curve y = x2 has a minimum at (0,0).
Locating the point of maximum or minimum
The x-value at a maximum or minimum is found by differentiating the function and putting it equal to zero.
The y-value is then found by substituting the 'x' into the original equation.
Example
Find the coordinates of the greatest or least value of the function:
Curve Sketching
The power of 'x' gives a hint to the general shape of a curve.
Together with the point of maximum or minimum, where the curve crosses the axes at y=0 and x=0 gives further points.
Example
Sketch the curve y = x2 +3x +2 from the example above, given that there is a minimum at (-1.5,-0.25).
factorising and putting y=0 to find where the curve crosses the x-axis,
(x+1)(x+2)=0
x=-1 and x=-2
so the curve crosses the x-axis at (-1,0) and (-2,0)
putting x=0 to find where the curve crosses the y-axis
y=2
so the curve crosses the y-axis at (0,2)
