To produce the function f(x) = 2| x−1 | − 3 from the parent function f(x) = |x|, three transformations are required.
Horizontal shift: The function is shifted horizontally by 1 unit to the right. This is done by subtracting 1 from the input value of the function. The transformation is f(x) → f(x − 1).
Vertical stretch: The function is vertically stretched by a factor of 2. This is done by multiplying the output value of the function by 2. The transformation is f(x) → 2f(x − 1).
Vertical shift: The function is vertically shifted down by 3 units. This is done by subtracting 3 from the output value of the function. The transformation is f(x) → 2f(x − 1) − 3.
To find points on the function, we can substitute different values of x into the function and calculate the corresponding y-values. For example:
When x = 0: f(0) = 2|0 − 1| − 3 = -5
When x = 1: f(1) = 2|1 − 1| − 3 = -3
When x = 2: f(2) = 2|2 − 1| − 3 = -1
Using these points, we can plot the graph of the function f(x) = 2| x−1 | − 3. The graph is a V-shaped curve with the vertex at (1,-3), opening upwards. The points we found above can be plotted on the graph as (-1,-5), (1,-3), and (2,-1).