The graph of f (x) = |x| is reflected across the x-axis and translated right 8 units. Which statement about the transformed function is correct?

1. The domain is the same as the parent function's domain.

2. The graph opens in the same direction as the parent function.

3. The range is the same as the parent function's range.

4. The vertex is the same as the parent function's vertex.

Option 1. is true

Step-by-step explanation:

our original function is f (x) = IxI

let us do the transformation to this

step 1: Reflect it across x axis, hence it becomes f (x) = -IxI

now we shift it 8 units towards + ve x axis

Hence our new function is f (x) = -Ix-8I

let us analyse the options one by one

1. Domain, Earlier domain was all real numbers, now also f (x) = -Ix-8I is defined for all real numbers hence domain is same.

2. Opening direction: f (x) = -IxI opens up, f (x) = -Ix-8I opens down as it was reflected along x axis.

3. Range of f (x) = IxI is all positive real numbers, Range of f (x) = -Ix-8I is all negative real numbers

4. Vertex of f (x) = IxI is (0,0) and vertex of f (x) = -Ix-8I is (8,0)

Hence we see that only first option is true

1

Step-by-step explanation:

The graph f (x) = |x| with vertex (0,0) becomes g (x) = - |x-8| whose vertex is (8,0). This means the range of the parent function y>0 becomes y<0. This is so because the new graph opens downward unlike the parent function which opens up. The domain does stay the same. Statement 1 is correct.