An Analysis of the Function f(x) = 2|x| and its Graph: Understanding Slopes and Vertex

f(x) = 2|x|

The given function is f(x) = 2|x|

The given function is f(x) = 2|x|.

In this function, the variable x is inside the absolute value sign, therefore we need to consider two cases: one when the value inside the absolute value is positive, and one when it is negative.

Case 1: When x is positive (x > 0)
For positive values of x, the absolute value sign is unnecessary, as the value inside the absolute value is already positive. So, in this case, f(x) = 2x.

Case 2: When x is negative (x < 0) For negative values of x, the absolute value sign changes the sign of the value inside the absolute value. So, in this case, f(x) = 2(-x) = -2x. To summarize the two cases: When x > 0, f(x) = 2x.
When x < 0, f(x) = -2x. Graphically, the function f(x) = 2|x| will have a V-shape, opening upwards. On the positive side of the x-axis (x > 0), the function will be a straight line with a positive slope (m = 2), starting from the origin and extending to the right with no end.

On the negative side of the x-axis (x < 0), the function will again be a straight line with a negative slope (m = -2), starting from the origin and extending to the left with no end. The point (0, 0) is the vertex of the V-shape, which is the lowest point of the graph. In summary, the function f(x) = 2|x| has a V-shaped graph, opening upwards, with the vertex at (0, 0). On the positive side, the slope is positive (m = 2), and on the negative side, the slope is negative (m = -2).

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