Answer:
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
