Question

Which statement is correct with respect to f(x) = -3|x − 1| + 12?

Answer 1

The answer should be D if not it is C

Answer 2

Let's start from the parent function $y = |x|$ and see which transformation we have applied. The parent function's graph is open upards, and its vertes lies at $(0,0)$

The first transformation is $|x| \to |x-1|$. This is a tranformation of the form $f(x)\to f(x+k)$. This kind of transformation shift the graph horizontally, k units to the left if k is positive, k units to the right if k is negative. In this case $k = -1$, so the function is shifted one unit to the right. The new vertex lies at $(1,0)$

The second transformation is $|x-1| \to -3|x-1|$. This is a tranformation of the form $f(x)\to kf(x)$. This kind of transformation stretch the graph vertically, compressing it if k is between 0 and 1, expanding it if k is greater than 1. Moreover, if k is negative, the function is reflected along the x axis. In this case $k = -3$, so the function is reflected and stretched. This means that now the graph opens downwards, the vertex still lies at $(1,0)$.

Finally, we have $-3|x-1| \to -3|x-1|+12$. This is a tranformation of the form $f(x)\to f(x)+k$. This kind of transformation translates the graph vertically, k units up if k is positive, k units down if k is negative. In this case, $k = 12$, so the graph is shifted 12 units up. The graph still opens downwards, while the new vertex lies at $(1,12)$