A normal to a curve - Math Central

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Question from Samantha, a student:

The function f is defined by f:x --> -0.5x^2 + 2x + 2.5
Let N be the normal to the curve at the point where the graph intercepts the y-axis. Show that the equation of N may be written as y = -0.5x + 2.5.
Let g:x--> -0.5x + 2.5
(i) find the solutions of f(x) = g(x)
(ii) hence find the coordinates of the other point of intersection of the normal and the curve

Hi Samantha,

The graph of f:x --> -0.5x² + 2x + 2.5 intercepts the y-axis when x = 0 and hence the y-intercept is (0, 2.5) = (0, 5/2). The slope m of the tangent at this point is the derivative of f, f':x --> -x + 2, at x = 0 and hence m = 2. The slope of the normal to the curve at this point is -1/m = -1/2 and hence g is the line with slope -1/2 and through the point (0, 5/2).

Solve the equation f(x) = g(x) and you will find two solutions. One is the value you already know, x = 0, and the other is the x-coordinate of the second point where the line g(x) intercepts the curve f(x).

Penny

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