Question

Identify the vertex, axis of symmetry, maximum or minimum and domain and range of each function

Answer 1

Answer: For first function vertex is (-2,-6), axis of symmetry is x = -2, minimum at (-2,-6), domain is all real numbers and range is $[-6,\infty)$. For second function vertex is (-1,5), axis of symmetry is x = -1, minimum at (-1,5), domain is all real numbers and range is $[5,\infty)$.

Explanation:

The standard for of the parabola is,

$y=a(x-h)^2+k$

Where (h,k) is vertex and vertex is the extreme point of a parabola. The axis of symmetry is y=h. If a>0 then f(x) is minimum at (h,k) and if a<0 then f(x) is maximum at (h,k).

The first function is,

$f(x)=4(x+2)^2-6$

On comparing this equation with standard form of parabola we get,

$h=-2,k=-6,a=4> 0$

Therefore, the vertex is (-2,-6), axis of symmetry is x = -2, minimum at (-2,-6).

The function defined for all values of x, so the domain of f(x) is all real numbers. Since the minimum value of the function is -6, therefore range must be greater than or equal to -6.

The second function is,

$f(x)=10(x-1)^2+5$

On comparing this equation with standard form of parabola we get,

$h=-1,k=5,a=10> 0$

Therefore, the vertex is (-1,5), axis of symmetry is x = -1, minimum at (-1,5).

The function defined for all values of x, so the domain of f(x) is all real numbers. Since the minimum value of the function is 5, therefore range must be greater than or equal to 5.