What is the shape obtained when the equation representing a parabola is differentiated.

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justaguide eNotes educator| Certified Educator

The shape of the graph of the equation obtained when the equation of a parabola is differentiated has to be represented.

Consider the general equation of a parabola opening upwards. y = (x - h)^2 + k

Differentiating the equation gives `dy/dx` = 2*(x - k) = 2x - 2k.

y = 2x - 2k is the equation of a straight line. This represents the slope at each point of the parabola. It is seen that for certain values of x, y is positive and for certain values x is negative. This is the case as for every parabola at certain points the slope of the tangent is positive and at others it is negative.

As an illustration take the parabola y = x^2 + 5. The equation of the parabola and that of the line representing the derivative, y = 2x has been plotted below.

The sign of the value of y for the line changes as the slope of the tangent to the parabola changes.