Complete the square to rewrite y = x2 + 8x – 4 in vertex form, and then identify the minimum y-value of the function

The vertex form is . The minimum value of the function is -20.

The general form of a line is , where a, b, c are constant. Vertex form of a line is g(x - h)^2 + k, where k is constant and (h , k) is the vertex point. Now by equating them we can find the vertex point

Now let's equate the like terms,

,

h=-4

The vertex form is and the vertex point is (-4 , -20). The minimum value of the function will be at x=-4 as the minimum value of will be zero because it is a square function.

∴ The minimum value of the function is -20.

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The vertex form is (x + 4)² - 20

The minimum value of the function is -20

Step-by-step explanation:

* Lets write the general form and the vertex form of the

 quadratic function

- General form ⇒ ax² + bx + c, where a , b , c are constant

- Vertex form ⇒g(x - h)² + k, where g , h , k are constant and

  (h , k) is the vertex point (minimum or maximum)

- By equating them we can find the vertex point

* Lets do that in the problem

∵ y = x² + 8x - 4

∴ x² + 8x - 4 = g(x - h)² + k ⇒ solve the bracket

∴ x² + 8x - 4 = g(x² - 2xh + h²) + k ⇒ open the bracket

∴ x² + 8x - 4 = gx² - 2ghx + gh² + k

* Now lets equate the like terms

- Let x² = gx²

∴ g = 1

- Let 8x = -2ghx ⇒ -2gh = 8 ⇒ -2(1)h = 8 ⇒ -2h = 8 ⇒ ÷ -2 for both sides

∴ h = -4

- let -4 = gh² + k ⇒ -4 = (1)(-4)² + k ⇒ 16 + k = -4 ⇒ -16 for both sides

∴ k = -4 - 16 = -20

* Substitute these values in the equation

∴ x² + 8x - 4 = (x - -4)² + -20

* The vertex form is (x + 4)² - 20

∵ The vertex point is (h , k)

∴ The vertex point is (-4 , -20)

* The minimum value of the function is the value of y

  of the vertex point

∴ The minimum value of the function is -20

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