Question

f(x) = 2 |x - 1| - 3
a. graph of a v - shaped function
b. graph of a function with a different shape
c. graph of a v - shaped function with different vertex position
d. partially shown graph of a v - shaped function

Explanation:

Step1: Find the vertex of the absolute - value function

The general form of an absolute - value function is \(y = a|x - h|+k\), where the vertex is at the point \((h,k)\). For the function \(f(x)=2|x - 1|-3\), we have \(h = 1\) and \(k=-3\). So the vertex of the graph of the function should be at the point \((1,-3)\).

Step2: Analyze the slope of the lines

The coefficient \(a\) in the absolute - value function \(y = a|x - h|+k\) determines the slope of the two linear parts of the graph. When \(x\geq h\) (in our case \(x\geq1\)), the function can be written as \(y = a(x - h)+k\), and when \(x\lt h\) (in our case \(x\lt1\)), the function can be written as \(y=-a(x - h)+k\). Here, \(a = 2\), which means that for \(x\geq1\), the slope of the line is \(m = 2\) (a positive slope, the line goes up from left to right), and for \(x\lt1\), the slope of the line is \(m=- 2\) (a negative slope, the line goes down from left to right).

Step3: Compare with the given graphs

• Let's check the vertex first. The vertex of option A: Looking at the graph of option A, the vertex seems to be at \((1, - 3)\)? Wait, no, looking at the grid, in option A, the vertex is at \((1,-3)\)? Wait, the y - coordinate of the vertex in option A: the graph has a vertex at (1, - 3)? Wait, the first graph (option A) has a vertex at (1, - 3)? Wait, let's re - examine. The function \(y = 2|x - 1|-3\): when \(x = 1\), \(y=-3\). Now, let's check the slopes. For \(x\geq1\), the slope is 2, so the line should be steeper (since \(|a| = 2\) is greater than 1).
• Option B: The vertex of option B is not at \((1,-3)\), and the slopes of the lines are not consistent with \(a = 2\).
• Option C: The vertex of option C is not at \((1,-3)\), and the slopes of the lines are not consistent with \(a = 2\).
• Option A: The vertex is at \((1,-3)\), and the slopes of the two lines (one with slope \(2\) for \(x\geq1\) and one with slope \(- 2\) for \(x\lt1\)) match the properties of the function \(f(x)=2|x - 1|-3\).

Answer:

A. The graph with vertex at \((1, - 3)\) and slopes of 2 (for \(x\geq1\)) and - 2 (for \(x\lt1\))