Question

which statement is true about the graph of the function \\(y = |x + 4| - 6\\)?

a. shift the graph 4 units left and 6 units up
b. shift the graph 4 units left and 6 units down
c. shift the graph 4 units right and 6 units up
d. shift the graph 6 units right and 4 units down

Explanation:

Identify the parent function and transformation form

The given function is \(y = |x + 4| - 6\). The parent function is the absolute value function:

$$ f(x) = |x| $$

The general transformation form is:

$$ g(x) = a|x - h| + k $$

Determine the horizontal shift

The term inside the absolute value is \(x + 4\), which can be written as \(x - (-4)\).
Since \(h = -4\), the graph is shifted horizontally:

$$ \text{Shift} = 4\text{ units left} $$

Determine the vertical shift

The constant term outside the absolute value is \(-6\).
Since \(k = -6\), the graph is shifted vertically:

$$ \text{Shift} = 6\text{ units down} $$

Answer:

• a. Shift the graph 4 units left and 6 units up
• b. Shift the graph 4 units left and 6 units down (Correct answer)
• c. Shift the graph 4 units right and 6 units up
• d. Shift the graph 6 units right and 4 units down