Final answer:
Brielle is correct; the function representing the graph is f(x)=2|x|. The determining factor is that f(x)=2|-x| does not correctly map x=-1 to y=2 unlike f(x)=2|x|, which does, maintaining symmetry about the y-axis.
Step-by-step explanation:
The student has presented the graph of an absolute value function that contains the points (1,2), (0,0), and (-1,2). We need to determine whether the function representing this graph is f(x) = 2|-x|, as Shawn suggests, or f(x) = 2|x|, as Brielle suggests. First, we can assess the graph's symmetry. An absolute value function is typically symmetrical about the y-axis, which means it should look the same to the left and right of the y-axis.
Let's evaluate f(x) = 2|-x| and f(x) = 2|x| at the defined points. When x equals 0 for both functions, f(x) equals 0, which matches with the given point (0,0). If we plug in x equals 1 into f(x) = 2|-x|, we get f(1) = 2|-1| = 2, and for f(x) = 2|x|, we get f(1) = 2|1| = 2. Both functions yield the correct y-value at x equals 1, which is again consistent with the point (1,2).
However, when we plug in x equals -1 into f(x) = 2|-x|, we get f(-1) = 2|-(-1)| = 2 * 2 = 4, which does not match the given point (-1,2). On the other hand, for f(x) = 2|x|, plugging in x equals -1 gives us f(-1) = 2|-1| = 2, which is the correct y-value.
Therefore, Brielle is correct, and the function that represents the graph is f(x) = 2|x|, which correctly maps both positive and negative x-values to the appropriate y-values as per the points given.