Question

f(x) = 2|x - 1| - 3
a. graph of a v - shaped function with vertex, axes labeled
b. graph of a line - like function with a corner, axes labeled
c. graph of a v - shaped function with vertex, axes labeled
d. partially shown graph of a v - shaped function, axes labeled

Explanation:

Step1: Identify 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, when \(x\geq1\)), the function can be written as \(y = 2(x - 1)-3=2x-2 - 3=2x-5\), and the slope of this line is \(m = 2\) (positive). When \(x\lt h\) (when \(x\lt1\)), the function can be written as \(y=-2(x - 1)-3=-2x + 2-3=-2x-1\), and the slope of this line is \(m=-2\) (negative). Also, since \(|a| = 2>1\), the graph of the function is vertically stretched compared to the parent function \(y = |x|\).

Step3: Analyze the options

• Option A: The vertex of the graph in option A is at \((1, - 3)\)? Let's check the coordinates. The vertex in option A seems to be at \((1,-3)\)? Wait, no, looking at the grid, the vertex in option A is at \((1, - 3)\)? Wait, the graph in option A has a vertex that is not at \((1,-3)\). Wait, maybe I made a mistake. Wait, let's re - evaluate. Wait, the function \(f(x)=2|x - 1|-3\). Let's find the value of the function at \(x = 1\): \(f(1)=2|1 - 1|-3=2\times0-3=-3\). So the vertex is \((1,-3)\). Now, let's check the slope. For \(x>1\), when \(x = 2\), \(f(2)=2|2 - 1|-3=2\times1-3=-1\). When \(x = 3\), \(f(3)=2|3 - 1|-3=4 - 3 = 1\). So the line for \(x\geq1\) has a positive slope. For \(x\lt1\), when \(x = 0\), \(f(0)=2|0 - 1|-3=2 - 3=-1\). When \(x=-1\), \(f(-1)=2|-1 - 1|-3=4 - 3 = 1\). The graph in option A has a vertex at \((1,-3)\) (wait, no, looking at the grid, the vertex in option A is at \((1,-3)\)? Wait, the first graph (option A) has a vertex at \((1, - 3)\)? Wait, the y - coordinate of the vertex in option A: the graph in option A has a vertex that is 3 units below the x - axis at \(x = 1\). Also, the slopes: when \(x>1\), the line is increasing with a slope of 2 (since for each unit increase in \(x\), \(y\) increases by 2), and when \(x\lt1\), the line is decreasing with a slope of - 2. And the vertex is at \((1,-3)\).

• Option B: The graph in option B is a line with a small slope (close to 0) and the vertex is not at \((1,-3)\), so it does not match.

• Option C: The slope of the lines in option C is not correct (the slopes are too small, and the vertex is not at \((1,-3)\)).

• Option D: The graph is not clear, but from the other options, option A is the only one with the correct vertex \((1,-3)\) and the correct slopes (positive slope for \(x\geq1\) and negative slope for \(x\lt1\)) and vertical stretch.

Answer:

A. The graph with vertex at \((1, - 3)\), a line with positive slope for \(x\geq1\) and negative slope for \(x\lt1\), and vertically stretched (slope of 2 and - 2)