Which statement about the range is true if an absolute value graph opens downward?

Which statement about the range is true if an absolute value graph opens downward?

• A. All real numbers
• B. y >= k
• C. y <= k
• D. y <= k, if the graph opens downward

Answer: (D)

If an absolute value graph opens downward, the vertex sits at the top of the graph and represents the highest y-value. In the standard form y = a|x − h| + k, the vertex is (h, k), and a negative a makes the graph open downward. Because the arms go downward from that top point, y can never exceed k, but it can go as low as you like as x moves away from h. So the range is all y-values less than or equal to k. For example, y = −|x| + 3 has its highest point at y = 3, and smaller y-values as you move away from the vertex. The statement that matches this is that the range is y ≤ k, given the graph opens downward.

If an absolute value graph opens downward, the vertex sits at the top of the graph and represents the highest y-value. In the standard form y = a|x − h| + k, the vertex is (h, k), and a negative a makes the graph open downward. Because the arms go downward from that top point, y can never exceed k, but it can go as low as you like as x moves away from h. So the range is all y-values less than or equal to k. For example, y = −|x| + 3 has its highest point at y = 3, and smaller y-values as you move away from the vertex. The statement that matches this is that the range is y ≤ k, given the graph opens downward.