Question

For the function f(x) = (x − 2)2 + 4, identify the vertex, domain, and range.

a. The vertex is (–2, 4), the domain is all real numbers, and the range is y ≥ 4.
b. The vertex is (–2, 4), the domain is all real numbers, and the range is y ≤ 4.
c. The vertex is (2, 4), the domain is all real numbers, and the range is y ≤ 4.
d.The vertex is (2, 4), the domain is all real numbers, and the range is y ≥ 4.

Answer 1

The vertex is (2, 4), the domain is all real numbers, and the range is y ≥ 4.

Explanation:

The equation of the parabola is
`f(x)=(x-2)^2+4`

The vertex form of the parabola is given by

`f(x)=a(x-h)^2+k`, here (h,k) is the vertex.

Comparing given equation with the vertex form of the parabola, we get

h = 2, k = 4

Hence, the vertex of the parabola is (h,k) = (2,4)

Now, domain is the set of x values for which the function is defined. The given function is defined for all real values of x.

Hence, domain is all real numbers.

Range is the set of y values for which the function is defined.

Since, here a = 1>0 hence it is a upward parabola and the vertex is the minimum point of this parabola.

Since, vertex is (2,4) hence, y values never less than 4.

Hence, range is y ≥ 4.

D is the correct options.

Answer 2

I think the correct answer from the choices listed above is option D. For the function f(x) = (x − 2)2 + 4, the vertex is (2, 4), the domain is all real numbers, and the range is y ≥ 4. The domain is all x-values available for the function and the range are the y values and in this case it should be greater than or equal to 4. Hope this answers the question.