point of intersection

**williamb** · February 22nd 2009, 03:07 PM

Consider the function f(x) = -2 x^2 + 9 x + 11

(a) find [f(x+h) - f(x)]/h in the form __x+ __h+___
(I found -4x-2h+9, which was correct)
(b) Using the result of (a), find the derivative of f at x.
(I found -4x+9, also correct)

***(c) Find the point of intersection of the two tangent lines to the parabola y = -2x^2 + 9x + 11 at the points (3,20) and (-4,-57)

How do i do this? What are the two tangent lines they are talking about?
I don't know where to start on this bad boy, thanks so much

**skeeter** · February 22nd 2009, 03:45 PM

note that $(3,20)$ and $(-4,-57)$ are both on the curve $y = -2x^2 + 9x + 11$

find the tangent line equation to the curve at each point, then set them equal to find where they intersect.

**Amanda H** · February 22nd 2009, 03:50 PM

To find the tangent to a line you need to find the gradient of the tangent which is done by first differentiating the equation of the parabola:

$y = -2x^2 + 9x + 11$
$\frac{dy}{dx} = -4x + 9$
To find the gradient at that point you need to put in the value of the x co-ordinate at the point of the tangent. In this case one of the values would be x = 3
$m = - 4*3 + 9$
$m = -3$

Finally to find the equation of the tangent you now use the co-ordinates and the gradient you have just found and substitute those into the equation of a line:
$y - b = m(x - a)$
$y - 20 = -3(x - 3)$
$y = -3x + 29$

This is the equation of the tangent at (3,20). To find the tangent at (-4, -57) you need to repeat this process using (-4,-57) instead of (3,20).

Arrange the equation of the second to equal y. You can then say both y values are the same so equate both tangents and solve to find the x value. Finally put this x value into either of the equations for the tangents and you will have the y value. This is your point of intersection.

I'll give you the chance to work out the second tangent yourself by following the same process I used. If you need me to explain further just ask.

**williamb** · February 22nd 2009, 04:35 PM

how do you differentiate y= -2x^(2) + 9x+11?

**Amanda H** · February 22nd 2009, 04:42 PM

I've done it above. See the part

Originally Posted by Amanda H

$\frac{dy}{dx} = -4x + 9$

**skeeter** · February 22nd 2009, 04:43 PM

Originally Posted by williamb

Consider the function f(x) = -2 x^2 + 9 x + 11

(a) find [f(x+h) - f(x)]/h in the form __x+ __h+___
(I found -4x-2h+9, which was correct)
(b) Using the result of (a), find the derivative of f at x.
(I found -4x+9, also correct)

you already found the derivative, or didn't you notice?

**williamb** · February 22nd 2009, 04:57 PM

i have the point -1/2 ,30.5
which is correct

thank you both so much =]

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