the graph of f(x) = |x| is translated 6 units to the right and 2 units up to form a new function. which…

the graph of f(x) = |x| is translated 6 units to the right and 2 units up to form a new function. which statement about the range of both functions is true?\nthe range is the same for both functions: {y | y is a real number}.\nthe range is the same for both functions: {y | y ≥ 0}.\nthe range changes from {y | y ≥ 0} to {y | y ≥ 2}.\nthe range changes from {y | y ≥ 0} to {y | y ≥ 6}.

the graph of f(x) = |x| is translated 6 units to the right and 2 units up to form a new function. which statement about the range of both functions is true?\nthe range is the same for both functions: {y | y is a real number}.\nthe range is the same for both functions: {y | y ≥ 0}.\nthe range changes from {y | y ≥ 0} to {y | y ≥ 2}.\nthe range changes from {y | y ≥ 0} to {y | y ≥ 6}.

Answer

Explanation:

Step1: Recall range of original absolute - value function

The function $f(x)=|x|$ has a range of ${y|y\geq0}$ because the absolute - value of any real number $x$ is non - negative.

Step2: Analyze the effect of translation on the range

A translation of 6 units to the right and 2 units up of the function $y = f(x)$ gives a new function $g(x)=|x - 6|+2$. The vertical translation of 2 units up shifts the entire graph of the function vertically. For the new function $g(x)$, the minimum value of $g(x)$ occurs when $|x - 6| = 0$ (i.e., $x = 6$), and $g(6)=2$. So the range of the new function is ${y|y\geq2}$.

Answer:

The range changes from ${y|y\geq0}$ to ${y|y\geq2}$.