# Thread: find a function problem

1. ## find a function problem

The problem says..

Q. Find a function y=ax2+bx+c whose graph has an x-intercept of 1, y-intercept of -3, and a tangent line with a slope of 1 at the y-intercept.

I can't even understand what this question is asking for let alone solving it. Can any one help?

Thanks!

2. ## Re: find a function problem

y = ax^2+bx + c ------- (1)
It is given the x intercept is . That means when y = 0 x = 1
Hence from ( 1 ) we get a + b + c = 0 ----- ( 2 )
Again we have y intercept = -3. That means when x = 0 y = -3.
Again from (1) we get c = -3. ------- ( 3 )
Now it says that the slope of tangent is 1 at y intercept. It means slope of tangent = 1 at ( 0,-3)
But the slope of tangent to a curve is given by dy/dx. Thus fro ( 1)
dy/dx = 2ax + b at ( 0,-3 ), dy/dx = 1. Thus we have b = 1 ----- ( 4 )
Now from (1), (2) and (3) we get a = 2, b=1 and c = -3.
Thus by plugging in these values in (1) we get the function as
2x^2 + x – 3 = 0

3. ## Re: find a function problem

We are told to find:

$y(x)=ax^2+bx+c$

where:

$y(0)=-3$

$y(1)=0$

$y'(0)=1$

From these, you should find 3 linear equations in $a,b,c$ from which you may determine their values. Can you proceed?

4. ## Re: find a function problem

Originally Posted by ibdutt
y = ax^2+bx + c ------- (1)
It is given the x intercept is . That means when y = 0 x = 1
Hence from ( 1 ) we get a + b + c = 0 ----- ( 2 )
Again we have y intercept = -3. That means when x = 0 y = -3.
Again from (1) we get c = -3. ------- ( 3 )
Now it says that the slope of tangent is 1 at y intercept. It means slope of tangent = 1 at ( 0,-3)
But the slope of tangent to a curve is given by dy/dx. Thus fro ( 1)
dy/dx = 2ax + b at ( 0,-3 ), dy/dx = 1. Thus we have b = 1 ----- ( 4 )
Now from (1), (2) and (3) we get a = 2, b=1 and c = -3.
Thus by plugging in these values in (1) we get the function as
2x^2 + x – 3 = 0
Our goal here is to help with the problems, not provide full solutions...giving full solutions is much less helpful than engaging the student in the process.

5. ## Re: find a function problem

when some one asks to find the function this means finding the values of the constants ?

6. ## Re: find a function problem

Originally Posted by ameerulislam
when some one asks to find the function this means finding the values of the constants ?
Yes, you could consider the constants the parameters to be found. The quadratic $ax^2+bx+c$ is general, and you are asked to find the one specific quadratic that satisfies the given conditions, which means you need to determine that values of a, b and c.

7. ## Re: find a function problem

ok what is the reason of saying this line..
a tangent line with a slope of 1 at the y-intercept. why specific intercept?

what would have happened if it said
a tangent line with a slope of 1 at the x-intercept. Or is it even possible?

8. ## Re: find a function problem

We know the value of x at the y-intercept, but we don't know the value of x at the x-intercept, as this would depend on the parameters via the quadratic formula.

9. ## Re: find a function problem

Originally Posted by MarkFL
We know the value of x at the y-intercept, but we don't know the value of x at the x-intercept, as this would depend on the parameters via the quadratic formula.

Isn't the value of x is given at x-intercept? which is 1 here, right?

10. ## Re: find a function problem

Originally Posted by ameerulislam

Isn't the value of x is given at x-intercept? which is 1 here, right?
No, the slope of the tangent line is given as being 1 at the y-intercept, which means we are given:

$\left.\frac{dy}{dx} \right|_{x=0}=1$

11. ## Re: find a function problem

Another question:-

if slope is given let say slope =0

and y=x^2

and equation is x^3+y^3-3xy=0

Can we figure out the the point(s) in the graph where slopes are Zero?

And the point (0,0) is on which quadrant?