Question

given the parent function (f(x) = |x|), which of the following is true about this graph?

• the parent function was translated.
• the parent function was reflected.
• the parent function was dilated.
• the parent function was restricted.

Explanation:

⚡ Using what you learned: combining transformations

Step 1: Identify the parent function behavior

The parent function is \( f(x) = |x| \).

• The vertex of the parent function is at the origin \( (0,0) \).
• The parent function opens upwards, forming a "V" shape.

Step 2: Analyze the given graph

• The vertex of the given graph is at \( (0,0) \), which means there is no horizontal or vertical translation.
• The graph opens downwards, forming an upside-down "V" shape.
• The slope of the right branch is \( -3 \) (it passes through \( (0,0) \) and \( (1,-3) \)), which means it has also been vertically stretched (dilated).

Step 3: Match with the options

• The parent function was translated: False, the vertex remains at \( (0,0) \).
• The parent function was reflected: True, because the graph opens downwards instead of upwards (reflected across the x-axis).
• The parent function was dilated: True, because the slope of the branches is steeper than \( 1 \) (vertical stretch by a factor of 3).
• The parent function was restricted: False, the domain is still all real numbers.

Looking at standard single-choice questions of this type, the most fundamental change in orientation shown by the upside-down "V" is the reflection. Let's check if multiple selections are allowed or if "reflected" is the primary intended answer. Since the graph is reflected (\( y = -3|x| \)), "The parent function was reflected" is a true statement.

Answer:

• The parent function was reflected.

(Note: If multiple checkboxes can be selected, The parent function was dilated is also true because the graph has a vertical stretch factor of 3).