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
The vertex form is [tex](x + 4)^2 - 20[/tex]. The minimum value of the function is -20.
The general form of a line is [tex]ax^2 + bx + c=0[/tex], 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
[tex]\Rightarrow x^2 + 8x - 4 = g(x - h)^2 + k\\ \\ \Rightarrow x^2 + 8x - 4 = g(x^2 - 2xh + h^2) + k\\ \\ \Rightarrow x^2 + 8x - 4 = gx^2 - 2ghx + gh^2 + k[/tex]
Now let's equate the like terms,
[tex]a) \;x^2=gx^2\Rightarrow g=1\\b)\; 8x=-2ghx\Rightarrow -2gh=8\Rightarrow gh=-4\; at \;g=1[/tex],
h=-4
[tex]c) \;-4=gh^2+k\;\Rightarrow -4=(1)(-4)^2+k\;\Rightarrow 16+k=-4\;\Rightarrow k=-20[/tex]
The vertex form is [tex](x + 4)^2 - 20[/tex] and the vertex point is (-4 , -20). The minimum value of the function [tex](x + 4)^2 - 20[/tex] will be at x=-4 as the minimum value of [tex](x+ 4)^2[/tex] will be zero because it is a square function.
∴ The minimum value of the function is -20.
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