Answer to Question #166766 in Mechanical Engineering for Adrian Garcia

Question

Make the parabola y=ax²+bx+c pass through (0,-2) and the tangent to the line at 3x+y+3=0 at (-1,0)

Answer 1

Substitute (x, y) = (0, -2) in the equation y=ax² +bx +c

-2=a (0)² +b (0) + c

C=-2

Substitute (x, y) = (-1, 0) in the equation y=ax² +bx +c

0=a (-1)¹ +b (-1) +c

0=a – b + c

b=a + c…(i)

From the equation y=ax² +bx +c, y'=2ax+b

From the equation y=-3x-3 at (-1,0) the slope is -3

Plugging y'=-3 and x=-1 in the equation y'=2ax+b

-3=2a (-1) +b

-3=-2a+b

b=-3 +2a…(ii)

c=-2…(iii)

Substituting the value of c, (-2) in the equation b=a + c

b=a-2…(iv)

equating equation (ii) to (iv)

-3+2a= a-2

2a-a=3-2

a=1

b=a-2

b=1-2

b=-1

Plugging the values of a, b, and c in the equation y=ax² +bx +c

y=x²-x -2