To determine the coordinates of the minimum point of the curve [tex]\( y = (x + 10)^2 + 4 \)[/tex], we should analyze the structure of the equation. This given equation is in the form of a quadratic function in vertex form, which is typically expressed as [tex]\( y = a(x - h)^2 + k \)[/tex]. In this form, the vertex of the parabola, which represents the minimum or maximum point, is located at the coordinates [tex]\((h, k)\)[/tex].
For the quadratic function [tex]\( y = (x + 10)^2 + 4 \)[/tex]:
1. Identify the constants inside the equation:
- The term [tex]\((x + 10)^2\)[/tex] indicates that the horizontal shift is to the left by 10 units. By comparison with [tex]\((x - h)\)[/tex], we see that [tex]\( h = -10 \)[/tex].
- The constant term outside the square, [tex]\( + 4 \)[/tex], represents a vertical shift upwards by 4 units. This means [tex]\( k = 4 \)[/tex].
2. Combining these observations, the vertex (minimum point) of the quadratic function [tex]\( y = (x + 10)^2 + 4 \)[/tex] is at the coordinates [tex]\( (-10, 4) \)[/tex].
Thus, the coordinates of the minimum point of the curve are [tex]\( (-10, 4) \)[/tex].
