Question

Describe how the graph of f(x)=|x| can be transformed to the graph of g(x)=−14|x−1|+1.

A. g(x) is reflected and vertically compressed, the graph shifts to the right 1 unit and then up 1 unit.
B. g(x) is reflected and vertically stretched, the graph shifts to the right 1 unit and then down 1 unit.
C. g(x) is reflected and vertically stretched, the graph shifts left 1 unit and then up 1 unit.
D. g(x) is reflected and vertically compressed, the graph shifts left 1 unit and then down 1 unit.

Answer 1

N/A, but A if anything

Explanation:

Let's start with the change within the absolute value.

In f(x), x is turned into x-1. There is no other way to add the -1 into the absolute value.

Therefore, our first transformation is f(x) -> f(x-1). In f(x+C), the graph is moved to the left by c units. Therefore, as C = -1, the graph is shifted to the right by 1 unit.

Next, we can move to the 14. Given that f(x-1) = |x-1|, we can see that we multiply 14 to f(x). Cf(x) vertically stretches a function if C is greater than 0, which 14 is, so the graph is vertically stretched.

After that, we have the negative. This happens outside the absolute values, so it is reflected in -f(x), not f(-x). This reflects a function across the x axis, but all we need to know for this question is that it is reflected.

Finally, we have the +1. This happens outside the absolute values, so it is reflected in f(x) + C, not f(x+C). The value of C moves the graph up by C units, and the +1 therefore moves the graph up by 1 unit.

We thus know 4 things:

1. The graph goes up by 1 units

2. The graph is reflected across the x axis

3. The graph is stretched vertically

4. The graph is shifted to the right by 1 unit

None of the answers given fit these, but A comes the closest.