How to Determine the Equation of a Parabola with Given Points and Gradient?

[Kafka]

Hi there,

I have to complete the following question, but I have no idea how to approach it (there are four other parts to the question that I can't complete until I get the first bit). I have made numerous attempts, but am not sure how to get the a part of the equation.

All help would be appreciated!

Thanks,
Kafka

Question
A curve f(x) is defined by the equation : y = ax² + bx + c, where a, b and c are constants.

The curve crosses the y-axis at the point (0,4). At this point the gradient of the graph is -5.

The curve crosses the x-axis at point (1,0).

(i) Find the values of a, b, and c and write down the equation of the curve

Attempt

Sub point (0.4) into equation to get c (x=0,y=4):

y = ax² + bx + c
4 = 0 + 0 + c
c = 4

If gradient at point (0,4) is -5, then dy/dx must be equal to -5.

dy/dx = 2ax + b
-5 = 2ax + b
-5 = 2a(0) + b
-5 = 0 + b
b = -5

*not sure about the bit below*

c=-5, b=4, so sub these into equation of curve and use a point to find a

y = ax² + bx + c

at (1,0) x=1, y=0

y = ax² + bx + c
0 = a1² + (-5 x 1) + 4
0 = a -5 + 4
0 = a - 1
a = 1 ?

 

[Hootenanny]

Looks good to me

 

[robphy]

Now if only PHYSICS students will recognize this as a projectile problem...

 

[HallsofIvy]

Kafka, you did that exactly right.

It does, however, have nothing to do with "differential equations" so I am moving it to the Calculus section.