Question

select the correct answer.
which statement is correct with respect to ( f(x) = -3|x - 1| + 12 )?

a. the v - shaped graph opens upward, and its vertex lies at ((-3, 1)).

b. the v - shaped graph opens downward, and its vertex lies at ((-1, 3)).

c. the v - shaped graph opens upward, and its vertex lies at ((1, -12)).

d. the v - shaped graph opens downward, and its vertex lies at ((1, 12)).

Explanation:

Step1: Recall the vertex form of absolute value function

The general form of an absolute value function is \( f(x) = a|x - h| + k \), where \((h, k)\) is the vertex and the sign of \(a\) determines the direction the graph opens. If \(a>0\), the graph opens upward; if \(a<0\), the graph opens downward.

Step2: Identify \(a\), \(h\), and \(k\) from the given function

For the function \( f(x) = -3|x - 1| + 12 \), we have \(a=-3\), \(h = 1\), and \(k = 12\).

Step3: Determine the direction of the graph

Since \(a=-3<0\), the graph opens downward.

Step4: Determine the vertex

The vertex of the function is \((h, k)=(1, 12)\).

Answer:

D. The V - shaped graph opens downward, and its vertex lies at (1, 12)