Question

Which functions have a vertex with a x-value of 0? Select three options.

Of(x) = lxl
Of(x) = |x| +3
Of(x) = x + 31
Of(x)= |x|-6
Of(x)= x + 31-6

Answer 1

Options 1, 2, 4

Explanation:

When you have a value inside the absolute value signs, that is moving the x value right or left. For example, f(x) = |x+3|, that is moving the x value left by 3.

When you have a value outside the absolute value signs, that is moving the y value up or down. For example, f(x) = |x| + 3, that is moving the y value up by 3.

Since you want to find the equations that have an x value of 0 for the vertex, you can move the y value all you want but cannot move the x value.

Options 1, 2, and 4, are the only ones that don't move the x values and has a vertex of 0.

Answer 2

f(x) = |x|

f(x) = |x| + 3

f(x) = |x| - 6

Explanation:

The parent function for all the given functions is the modulus function f(x)=|x|.

A modulus function gives the absolute value of a number or variable.

The absolute value of a number is its positive numerical value.

Therefore, the range of f(x)=|x| is more than or equal to zero.

The graph of f(x)=|x| is:

• Line y = x where x ≥ 0
• Line y = -x where x ≤ 0
• Vertex at (0, 0)

Translations

`\textsf{For} \; a > 0:`

`f(x+a) \implies f(x) \: \textsf{translated $a$ units left}`

`f(x-a) \implies f(x) \: \textsf{translated $a$ units right}`

`f(x)+a \implies f(x) \: \textsf{translated $a$ units up}`

`f(x)-a \implies f(x) \: \textsf{translated $a$ units down}`

For a modulus function to have a vertex with an x-value of zero after translation, the function can only be translated up or down. If it was translated left or right, the x-value of the vertex would no longer be zero.

Therefore:

f(x) = |x| → No translation. Vertex at (0, 0).

f(x) = |x| + 3 → Translated 3 units up. Vertex at (0, 3).

f(x) = |x + 3| → Translated 3 units left. Vertex at (-3, 0).

f(x) = |x| - 6 → Translated 6 units down. Vertex at (0, -6).

f(x) = |x + 3| - 6 → Translated 3 units left and 6 units down. Vertex at (-3, -6).

So the functions that have a vertex with an x-value of zero are:

• f(x) = |x|
• f(x) = |x| + 3
• f(x) = |x| - 6