Question

So, here are my instructions for these problems: Write an equation of the absolute value graph that has the following properties based on the graph of y=|x| I would like to focus on part C. Reflected over the x-axis , horizontal shrink of 1/2 translated 7 down. I had two answers for this but I’m not sure which one is correct.

Answer 1

When we horizontally shrink a function by a factor k, we multiply the variable x, in the expression of the function, by k:

y = f(x) -> y' = f(kx)

When the factor k is greater than 1, that will represent in fact a horizontal shrink of the graph. But, since the factor we need to apply is less than 1, the result of horizontally shrinking the graph will effectively be a horizontal stretch.

First, let's reflect the function y = |x| over the x-axis. This changes the sing of y:

$y=-|x|$

Now, we need to multiply the variable x by 1/2 to horizontally shrink the graph by 1/2:

$y=-|\frac{1}{2}x|$

Now, we need to translate the graph 7 units down. So, we need to subtract 7 from the final expression for y:

$y=-|\frac{1}{2}x|-7$

Notice the result of those transformations:

Notice that, strictly, a horizontal shrink of 1/2 is actually a horizontally stretch of 2.

Now, if what the exercise really means is that the graph is in fact horizontally shrunk, then you need to divide the variable x by 1/2. This results in the function

$\begin{gathered} y=-|\frac{x}{\frac{1}{2}}|-7 \\ \\ y=-|2x|-7 \end{gathered}$