# Equation of parabola joined smoothly to the line y=2x at a point.

• Jun 20th 2010, 07:32 AM
Mr Rayon
Equation of parabola joined smoothly to the line y=2x at a point.
Find the equation of the parabola joined smoothly to the straight line y=2x at the point P(2,4) as shown below. The parabola passes through the point(5,1).

$\displaystyle y = a(x - b)^2 + c$
$\displaystyle y = a(x - 2)^2 + 4$
$\displaystyle 1 = a(5 - 2)^2 + 4$
$\displaystyle 1 = a(3)^2 + 4$
$\displaystyle 1 = 9a + 4$
$\displaystyle a = \frac{-1}{3}$

$\displaystyle y = \frac{-1}{3}(x - 2)^2 + 4$
• Jun 20th 2010, 07:50 AM
p0oint
The solution here:

Spoiler:

$\displaystyle y=ax^2+bx+c$

$\displaystyle 1=25a+5b+c$

$\displaystyle y'(x)=2ax+b$

$\displaystyle 2=4a+b$

$\displaystyle 4=4a+2b+c$
• Jun 20th 2010, 08:25 AM
Mr Rayon
Is this right?
a = -9/36
b = 3/4
c = -9

Can't be bothered showing working out.
• Jun 20th 2010, 08:55 AM
Unknown008
Firstly, you are given two points, (2, 4) and (5, 1), and you are given the gradient at the point (2, 4) which is 2.

Let's say that your equation has the form:

y = ax^2 + bx + c.

Plug in the coordinates to have a pair of equations:

$\displaystyle (4) = a(2)^2 + b(2) + c$ --> $\displaystyle 4 = 4a + 2b + c$ -I

$\displaystyle (1) = a(5)^2 + b(5) + c$ --> $\displaystyle 1 = 25a + 5b + c$ -II

Now the gradient on the curve is given by the derivative of the curve.

$\displaystyle y' = 2ax + b$

Bud in the value of x and the value of y':

$\displaystyle (2) = 2a(2) + b$

$\displaystyle 2 = 4a + b$ -III

Now, eliminate 'c' from the first two equations (II - I)to get:

$\displaystyle -3 = 21a + 3b$

Reduce to:

$\displaystyle -1 = 7a + b$ -IV

Now, use simultaneous solving to get the values of a and b from equations III and IV.

you should get a = -1, b = 6 and c = -4
• Jun 20th 2010, 09:07 AM
p0oint
a = -9/36
b = 3/4
are not satisfying 4a+b=2

$\displaystyle -9/4+3/4 \neq 2$
• Jun 20th 2010, 05:14 PM
mr fantastic
Quote:

Originally Posted by Mr Rayon
Is this right?
a = -9/36
b = 3/4
c = -9

Can't be bothered showing working out.

And yet you want people to be bothered working the problem from scratch to check these answers.

If you want help, it is in your own interests to show your work so that people can more easily help you. eg. You might have made a mistake in your very first line of working, this could then be easily pointed out and people wouldn't have to waste time doing their own calculations from scratch.