Identify the vertex, axis of symmetry, minimum or maximum, domain, and range of the function ()=−(+)^−
Answer:
vertex = (-4, -5)
Axis of symmetry = -4
use the (-4, -5) to find the minimum value
[tex]Domain = ( - \infty, \infty ) , [ x | x\ is\ real ]\\\\Range = [ -5, \infty ), y\geq -5[/tex]
Solution:
Given function is:
[tex]f(x) = (x+4)^2 - 5[/tex]
The equation in vertex form is given as:
[tex]y = a(x-h)^2+k[/tex]
Where, (h, k) is constant
On comparing give function with vertex form,
h = -4
k = -5
Vertex is (-4 , -5)
Axis of symmetry : x co-ordinate of vertex
Thus, axis of symmetry = -4
The coefficient of x^2 is positive in given function.
Thus the vertex point will be a minimum
[tex]Minimum\ value = f(\frac{-b}{a})[/tex]
[tex]f(x) = x^2 + 8x + 16 - 5\\\\f(x) = x^2 + 8x + 11[/tex]
[tex]f(x) = ax^2+bx+c[/tex]
On comparing,
a = 1
b = 8
[tex]x = \frac{-b}{2a} = \frac{-8}{2 \times 1} = -4[/tex]
[tex]f(-4) = (-4)^2 + 8(-4) + 11 = 16 - 32 + 11 = -5[/tex]
Thus, use the (-4, -5) to find the minimum value
Domain and range
[tex]f(x) = (x+4)^2 - 5[/tex]
The domain is the input values shown on the x-axis
The range is the set of possible output values f(x)
Therefore,
[tex]Domain = ( - \infty, \infty ) , [ x | x\ is\ real ]\\\\Range = [ -5, \infty ), y\geq -5[/tex]
