Question

absolute value functions and translations
complete
2 3 4 5 6 7 8 9 10
(f(x) = |x|)
(f(x) = |x|)
(f(x) = |x|)
(f(x) = |x|)

Explanation:

Identify the parent function and transformation

Using the Absolute Value Functions knowledge point

$$
LATEXBLOCK0
$$

Determine the vertex shift

Using the Absolute Value Functions knowledge point

$$
LATEXBLOCK1
$$

Analyze the range of the function

Using the Absolute Value Functions knowledge point

$$
LATEXBLOCK2
$$

Match with the given options

The student's query asks for a conceptual explanation of why the range of \(g(x) = |x - 12| - 2\) is \(y \ge -2\).
The absolute value expression \(|x - 12|\) represents a distance on the number line, which is always non-negative (\(\ge 0\)).
Subtracting \(2\) from this non-negative value shifts the minimum value of the function from \(0\) down to \(-2\).
Therefore, the lowest value the function can output is \(-2\), and it can produce any value greater than or equal to \(-2\).

Answer:

Question 1

• (A) The graph is shifted 12 units to the right and 2 units up, so the vertex is at (12, 2).
• (B) The absolute value term \(|x - 12|\) is always greater than or equal to 0. Subtracting 2 shifts the minimum value down to -2, making the range \(y \ge -2\). (Correct answer)
• (C) The range is restricted because the domain of the function is restricted to positive numbers.
• (D) The graph is reflected across the x-axis, which changes the direction of the inequality.