How do you find the point on the graph of #y = x^2# where the tangent line is parallel to the line #x + 2y = 4#?

Question

Isaiah Allen · Accepted Answer

The answer is:
#(-1/2,1/16)#

Rather than trying to write down an explanation, I thought I would make a quick video to show you how it works.

Hope it helps :)

Aurora Alvarado · Answer

undefined To find the point on the graph of y = x^2 where the tangent line is parallel to the line x + 2y = 4, we need to determine the slope of the line x + 2y = 4.

First, we rearrange the equation x + 2y = 4 to solve for y: 
2y = 4 - x 
y = (4 - x)/2 
y = 2 - (1/2)x

The slope of this line is -1/2.

Next, we find the derivative of the function y = x^2 to determine the slope of the tangent line at any given point on the graph. 
dy/dx = 2x

To find the x-coordinate of the point where the tangent line is parallel to the line x + 2y = 4, we set the derivative equal to the slope of the line: 
2x = -1/2

Solving for x: 
x = -1/4

Substituting this value of x back into the original equation y = x^2, we find the y-coordinate: 
y = (-1/4)^2 
y = 1/16

Therefore, the point on the graph of y = x^2 where the tangent line is parallel to the line x + 2y = 4 is (-1/4, 1/16).