Identify the vertex, axis of symmetry, minimum or maximum, domain, and range of the function ()=−(+)^−
<em><u>Answer:</u></em>
vertex = (-4, -5)
Axis of symmetry = -4
use the (-4, -5) to find the minimum value
![Domain = ( - \infty, \infty ) , [ x | x\ is\ real ]\\\\Range = [ -5, \infty ), y\geq -5](https://tex.z-dn.net/?f=Domain%20%3D%20%28%20-%20%5Cinfty%2C%20%5Cinfty%20%29%20%2C%20%5B%20x%20%7C%20x%5C%20is%5C%20real%20%5D%5C%5C%5C%5CRange%20%3D%20%5B%20-5%2C%20%5Cinfty%20%29%2C%20y%5Cgeq%20-5)
<em><u>Solution:</u></em>
Given function is:
![f(x) = (x+4)^2 - 5](https://tex.z-dn.net/?f=f%28x%29%20%3D%20%28x%2B4%29%5E2%20-%205)
The equation in vertex form is given as:
![y = a(x-h)^2+k](https://tex.z-dn.net/?f=y%20%3D%20a%28x-h%29%5E2%2Bk)
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
![Minimum\ value = f(\frac{-b}{a})](https://tex.z-dn.net/?f=Minimum%5C%20value%20%3D%20f%28%5Cfrac%7B-b%7D%7Ba%7D%29)
![f(x) = x^2 + 8x + 16 - 5\\\\f(x) = x^2 + 8x + 11](https://tex.z-dn.net/?f=f%28x%29%20%3D%20x%5E2%20%2B%208x%20%2B%2016%20-%205%5C%5C%5C%5Cf%28x%29%20%3D%20x%5E2%20%2B%208x%20%2B%2011)
![f(x) = ax^2+bx+c](https://tex.z-dn.net/?f=f%28x%29%20%3D%20ax%5E2%2Bbx%2Bc)
On comparing,
a = 1
b = 8
![x = \frac{-b}{2a} = \frac{-8}{2 \times 1} = -4](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%7B-b%7D%7B2a%7D%20%3D%20%5Cfrac%7B-8%7D%7B2%20%5Ctimes%201%7D%20%3D%20-4)
![f(-4) = (-4)^2 + 8(-4) + 11 = 16 - 32 + 11 = -5](https://tex.z-dn.net/?f=f%28-4%29%20%3D%20%28-4%29%5E2%20%2B%208%28-4%29%20%2B%2011%20%3D%2016%20-%2032%20%2B%2011%20%3D%20-5)
Thus, use the (-4, -5) to find the minimum value
Domain and range
![f(x) = (x+4)^2 - 5](https://tex.z-dn.net/?f=f%28x%29%20%3D%20%28x%2B4%29%5E2%20-%205)
The domain is the input values shown on the x-axis
The range is the set of possible output values f(x)
Therefore,
![Domain = ( - \infty, \infty ) , [ x | x\ is\ real ]\\\\Range = [ -5, \infty ), y\geq -5](https://tex.z-dn.net/?f=Domain%20%3D%20%28%20-%20%5Cinfty%2C%20%5Cinfty%20%29%20%2C%20%5B%20x%20%7C%20x%5C%20is%5C%20real%20%5D%5C%5C%5C%5CRange%20%3D%20%5B%20-5%2C%20%5Cinfty%20%29%2C%20y%5Cgeq%20-5)