Question 1127558
<br>
The function is y = -x^2+4x+3<br>
The derivative is -2x+4<br>
The derivative at x=3 is -2(3)+4 = -2; so the slope of the curve at x=3 is -2.<br>
The slope of the normal to the curve at x=3 is the negative reciprocal, 1/2.<br>
The equation of the line with slope 1/2 passing through (3,6) is y = (1/2)x+9/2, or 2y = x+9.<br>
The x-intercept of the normal (set y=0) is (-9,0); the y-intercept (set x=0) is (0,9/2).<br>
The midpoint of the segment with endpoints (-9,0) and (0,9/2) is (-9/2,9/4).<br>
A graph....
{{{graph(400,400,-12,12,-12,12,-x^2+4x+3,(1/2)x+9/2)}}}