thinkin prob about finding a, b, and c... i cant think

[chrisbyers2005]

find numbers a,b,and c that guarantee that the graph of the function....

f(x)= ax^2 + bx + c

will have x-intercepts at (0,0) and (5,0) and a tangent line with slope 1 at x=2



help guys! need it pronto.... thanks a bunch

 

[soroban]

Hello, chrisbyers2005!

Find numbers a,b,and c that guarantee that the graph of the function: f(x) = ax² + bx + c

will have x-intercepts at (0,0) and (5,0) and a tangent line with slope 1 at x=2

They gave us three "facts" . . . plug them in.

An x-intercept at (0,0) means: when x = 0, f(x) = 0.
. . f(0) .= .a·0² + b·0 + c .= .0 . ---> . c = 0

An x-intercepts at (5,0) means: when x = 5, f(x) = 0.
. . f(5) .= .a·5² + b·5 + c .= .0 . ---> . 25a + 5b .= .0 . ---> . 5a + b .= .0 . [1]

A tangent line with slope 1 at x = 2 means: f '(2) = 1
. . f '(x) = 2ax + b . ---> . f '(2) = 2a(2) + b = 1 . ---> . 4a + b .= .1 .[2]

Solve the system: . [1] . 5a + b .= .0
. . . . . . . . . . . . . . . .[2] . 4a + b .= .1

. . and get: .a = -1, b = 5

The function is: .f(x) .= .-x² + 5x