Question

complete the square to rewrite y=x^2+8x+3 in vertex form and then identify the minimum y-value of the function

Answer 1

The vertex-form of the quadratic equation is given by y = (x + 4)² - 13, and the minimum value of y is of -13.

What is the equation of a parabola given it’s vertex?

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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Answer 2

The minimum y-value of the function for the parabolic equation is -13.

What is the vertex of a parabolic equation in quadratic form?

The vertex for an up-down facing parabola of the form y = ax² + bx + c is $\mathbf{x_v= -\dfrac{b}{2a}}$

From the given equation:

y = x² + 8x + 3

The parameters for the parabola are:

a = 1, b = 8, c = 3

$\mathbf{x_v= -\dfrac{8}{2\times 1}}$

$\mathbf{x_v= -4}$

If we replace the value of $\mathbf{x_v= -4}$ to find the minimum value of $\mathbf{y_v}$ value, we have:

$\mathbf{y_v = -13}$

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.

Learn more about the vertex of a parabolic equation here:

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