tangents on a parabola?

Question

I don't get this at all... Please help! http://img29.picoodle.com/img/img29/9/10/7/f_pg2001m_3879c12.jpg

Answer 1

Y = 1 - X^2

equilateral triangles have 60 degree corners in each corner of the triangle. since the triangle is bisected, we'll use 30 degrees from 90

0 - 30 = -60

60 degrees as far as rise and fall and rate of change, is:

Tan(-60)

then what we do is set this equal to the derivative of the parabola.

Y = -X^2 + 1 => -2X

-2X = Tan(+/-60)

X = +/-.866

Plug this into the function for the parabola to find Y.

Y = -X^2 + 1
Y = -(+/-.866)^2 + 1 = .25

So, the points you are looking for are:

(-.866, .25) ; (.866, .25)

Answer 2

The zeros on the parabola are (-1,0) and (1,0). Therefore the distance between them is 2. Since it is an equilateral triangle, the sides need to both be 2 units long...

The slopes of P and Q are 2 and -2 respectively...

this is because the line P intersects the y-axis at 2 ( because the x value is 0 and the side must be 2 units long). The rise is subsequently 2 and the run is 1, cause the line must reach 0 horizontally...

So the slope of P is 2 and the slope of Q is -2 (for the same reason)...

Now take the derivative of the original equation, y = 1-x^2.

y' = -2x

Set y' equal to -2 and 2 to solve for the x-coordinate at which the tangent line hits the graph...

2 = -2x
x= -1

This is the x-coordinate at which line P hits the graph...

-2 = -2x
x= 1

This is the x-coordinate at which line Q hits the graph...


Now plug both those points into the original equation to get the points of tangency...

y = 1 - (-1)^2 = 1- 1 = 0

y = 1 - (1)^2 = 1-1 = 0

So the two points of tangency are (1,0) and (-1,0).

Answer 3

To simplify the explanation even more, a line tangent to any curve at any point is given by the statement dy/dx = tan theta. It is not true that the tangent touches a curve in only one place. The curve only need be differentiable to obtain its tangent at any point along the curve, the line itself may cross or even be tangential in multiple locations on the curve.

The above answer to your specific question is correct, according to the information on the site you gave us....

J.

http://www.jrichardjacobs.net

"The speed of the brain is inversely proportional to the speed of the mouth squared."

Answer 4

Tangent lines on a parabola refer to a line touching the curve at only one place.

Answer 5

y = 1-x^2
dy/dx = -2x
The slope must be Tan60 = sqrt(3)
so -2x = sqrt(3) --> x = +/- sqrt(3)/2
So points are (-sqrt(3)/2, 1/4) and (sqrt(3)/2, 1/4)