SOLUTION: a curve is defined by y=(x-1)(ax2 + bx +c)where a,b and c are all constants. the curve touches the x-axis at the point where x=1 and the line y=3x + 6 at the point (-2,0) (a) find

Algebra ->  Proportions  -> Lessons -> SOLUTION: a curve is defined by y=(x-1)(ax2 + bx +c)where a,b and c are all constants. the curve touches the x-axis at the point where x=1 and the line y=3x + 6 at the point (-2,0) (a) find       Log On


   

Question 1132242: a curve is defined by y=(x-1)(ax2 + bx +c)where a,b and c are all constants. the curve touches the x-axis at the point where x=1 and the line y=3x + 6 at the point (-2,0) (a) find the values of a,b and c (b) sketch the curve and fined the area of the finite region bounded by the curve and the lines x=1 and x=4
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


(1) The curve touches the x-axis at x=1 --> x=1 is a double root --> (x-1) is a factor twice

(2) The curve passes through (-2,0) --> x = -2 is a root --> (x+2) is a factor

So the polynomial is

f%28x%29+=+a%28x-1%29%5E2%28x%2B2%29+=+a%28x%5E3-3x%2B2%29

(3) The graph touches the line y=3x+6 at (-2,0) --> the slope (derivative of the function) evaluated at x=-2 is 3:

df%2Fdx+=+a%283x%5E2-3%29
a%283%28-2%29%5E2-3%29+=+3
a%2812-3%29+=+3
9a+=+3
a+=+1%2F3

The polynomial is



ANSWER (a): In the form (x-1)(ax^2+bx+c), a = 1/3, b = 1/3, c = -2/3

A graph (f(x) red; y=3x+6 green), showing the curve touching the x-axis at x=1 and touching the line y=3x+6 at (-2,0)...

graph%28400%2C400%2C-4%2C4%2C-10%2C10%2C%281%2F3%29%28x-1%29%5E2%28x%2B2%29%2C3x%2B6%29

ANSWER (b(2)): area under the curve from x=1 to x=4

Taking the constant 1/3 outside the integral, we have

1/3 of integral from 1 to 4 of ((x^3-3x+2)dx) is

1/3 of
= 63/4