NON-LINEAR EQUATIONS

Question

Given the following demand functions, express TR as a function of Q and hence sketch the graphs of TR against Q

1. P = 10 − 4Q

???? Please help

Answer 1

TR = total revenue = PQ = (10-4)Q = 10Q - 4Q^2

plot TR on the vertical or y axis, with Q on the horizontal or x axis

think of it like y=-4x^2 + 10x where TR = y and Q=x

It's a downward opening parabola, with vertex as the point of maximum total revenue

y= -4(x^2 -10x/4 + 25/16) +25/4 (complete the square then add 25/4 to compensate for -25/4)

y = -4(x-5/4)^2 + 25/4. the vertex or maximum point is (5/4, 25/4) when Q = 5/4, TR = 25/4

to find the x intercepts set y=0, to get the break even point of zero profit, zero loss

0 =-4x^2 + 10x = -4x(x-10) roots or zeroes are x=0 and x=5/2, the x intercepts or (0,0) and (5/2,0)

y intercept = 25/4 or (0,25/4)

It's a standard parabola, opening downward. Plot the 2 x intercepts, the y intercept and the vertex and draw a standard parabola through them with a smooth curve. Max TR = 25/4 = 6 1/4 when output or Q = 5/4= 1 1/4