Question

Draw the graph of f(x)=[[x - 4]] - 2

Answer 1

See attachment.

Explanation:

Given absolute value function:

`f(x)=|x-4|-2`

The parent absolute value function is f(x) = |x|.

Graph of the parent absolute value function:

• Vertex at (0, 0)
• |y| = -x if x ≤ 0
• |y| = x if x ≥ 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}.`

The graph of the given function is a transformation of the parent function by:

• Translation of 4 units right.
• Translation of 2 units down.

Therefore, the vertex of the given function is (4, -2).

Find two further points on the graph by inputting x-values either side of the x-value of the vertex into the given function:

`\begin{aligned}\implies f(8)&=|8-4|-2\\&=|4|-2\\&=4-2\\&=2\end{aligned}`

`\begin{aligned}\implies f(0)&=|0-4|-2\\&=|-4|-2\\&=4-2\\&=2\end{aligned}`

To draw the graph of the given function:

• Plot the vertex (4, -2).
• Draw a straight line from the vertex through point (8, 2).
• Draw a straight line from the vertex through point (0, 2).