How can I find the point on the line y = 2x + 4 that is closest to the origin?

To find the point on the line y = 2x + 4 that is closest to the origin (0, 0), we can use the concept of distance between points in a Cartesian plane.

The distance d from any point (x, y) to the origin is given by:

d = √(x² + y²)

In our case, since the point must lie on the line, we can substitute y from the line equation into the distance formula. Substituting y = 2x + 4, we get:

d = √(x² + (2x + 4)²)

Now, we can simplify the expression under the square root:

d = √(x² + (4x² + 16x + 16))

d = √(5x² + 16x + 16)

To minimize distance d, we can minimize the function 5x² + 16x + 16 (the square root function is increasing, so minimizing the inside suffices). We can find the vertex of this quadratic equation using the vertex formula:

x = -b/2a where a = 5 and b = 16.

Now, substituting:

x = -16 / (2 * 5) = -16 / 10 = -1.6

Now, we can find the corresponding y-coordinate using the line equation:

y = 2(-1.6) + 4 = -3.2 + 4 = 0.8

Thus, the point on the line y = 2x + 4 that is closest to the origin is (-1.6, 0.8).

In conclusion, the closest point on the line to the origin is (-1.6, 0.8).