The vertex-form of the quadratic equation is given by y = (x + 4)² - 13, and the minimum value of y is of -13.
The equation of a quadratic function, of vertex (h,k), is given by:
y = a(x - h)² + k
In which a is the leading coefficient. If a > 0, k is the minimum value of y, and if a < 0, it is the maximum value.
In this problem, the equation is:
y = x² + 8x + 3.
Completing the squares, the function is:
y = (x + 4)² - 13
13 is subtracted as (x + 4)² = x² + 8x + 16, hence for the equations to be equal 13 has to be subtracted. k = -13 is also the minimum value of y.
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The minimum y-value of the function for the parabolic equation is -13.
The vertex for an up-down facing parabola of the form y = ax² + bx + c is [tex]\mathbf{x_v= -\dfrac{b}{2a}}[/tex]
From the given equation:
y = x² + 8x + 3
The parameters for the parabola are:
a = 1, b = 8, c = 3
[tex]\mathbf{x_v= -\dfrac{8}{2\times 1}}[/tex]
[tex]\mathbf{x_v= -4}[/tex]
If we replace the value of [tex]\mathbf{x_v= -4}[/tex] to find the minimum value of [tex]\mathbf{y_v}[/tex] value, we have:
[tex]\mathbf{y_v = -13}[/tex]
Thus, the parabola vertex is (-4,-13). If a< 0, the vertex is a maximum value and If a>0, the vertex is a minimum value.
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