Which functions have a vertex with a x-value of 0? select three options. f(x) = |x| f(x) = |x| + 3 f(x) = |x + 3| f(x) = |x| − 6 f(x) = |x + 3| – 6

Step-by-step explanation:

The question does not say anything about what the y value can be. So a surprising choice would be f(x)= abs(x) + 3. This gives you a vertex at (0,3).

A second choice (for the same reason) would be y = abs(x) - 6

The third option can by y= abs(x)  which is more in line with what you would expect.

 The correct functions are

(A) f(x) = |x|.

(B) f(x) = |x| + 3.

(D) f(x) = |x| - 6.

Step-by-step explanation:  We are given to select he functions that have a vertex with a x-value of 0.

We know that

the vertex of the function f(x) = |x - a| +b is given by (a, b).

Option (A) : f(x) = |x|.

Here, the vertex is given by (0, 0). So, the x-value of the vertex is 0.

This option is correct.

Option (B) : f(x) = |x| + 3.

Here, the vertex is given by (0, 3). So, the x-value of the vertex is 0.

This option is correct.

Option (C) : f(x) = |x + 3|.

Here, the vertex is given by (-3, 0). So, the x-value of the vertex is -3, not 0.

This option is incorrect.

Option (D) : f(x) = |x| - 6.

Here, the vertex is given by (0, -6). So, the x-value of the vertex is 0.

This option is correct.

Option (E) : f(x) = |x + 3| - 6.

Here, the vertex is given by (-3, -6). So, the x-value of the vertex is -3, not 0.

This option is incorrect.

Thus, the correct functions are

(A) f(x) = |x|.

(B) f(x) = |x| + 3.

(D) f(x) = |x| - 6.

let's cover over the formula of the vertex:

y = a(x – h)^2 + k

in this formula, (h,k) are the vertex, always remember that the positive or negative value of the vertex is based on the sign it has.

for example, if the vertex is -3,-6 then both the x and y value would have the positive sign attached to it and if it were a positive x and y value, then a negative sign would be attached to it.

with that being said, for our problem the x value is a positive one so we are looking for a number with a negative sign next to it, in this case the first and second choice.

however, for the first answer choice it isn't quite part of the formula so we know that the answer choice correctly displays the x value.

to confirm, we can see if the -1 fits correctly with the second answer choice, and as mentioned above a negative number will have a positive sign attached to it. so, for our problem we can see that -1 has been changed to +1, so we can confirm that the second answer choice is correct.

hence, our final answer is y=(x - 1)^2 + 1