Question

Consider the function f(x) = |x|. Write a function rule for g(x) which is a translation of f(x) down 4 units.

Answer 1

To translate the function f(x) = |x| down by 4 units, the new function will be g(x) = |x| - 4. This function g(x) shifts the graph of f(x) down on the y-axis by 4 units, without changing the x-values.

Step-by-step explanation:

To translate the function f(x) = |x| down by 4 units, we need to subtract 4 from the function's value. This means that for any given x, the corresponding value of g(x) (the translated function) will be 4 less than what we would get from f(x). Therefore, the function rule for g(x) is:

g(x) = |x| - 4

This will effectively shift the entire graph of f(x) downwards on the y-axis by 4 units. It's important to understand that this transformation does not affect the x-values; it only changes the y-values. Thus, the shape of the graph remains the same, just positioned lower.

To verify this, consider any point on the graph of f(x). For example, when x is 5, f(x) is 5. Under our new function, g(x) at x equals 5 would be 1 (which is 5-4). Similarly, for x equal to -3, while f(x) would be 3, g(x) would then be -1 (which is 3-4).

Answer 2

The function f(x) = |x| represents the absolute value of x. To obtain a function rule for g(x) which is a translation of f(x) down 4 units, we simply subtract 4 from the original function f(x).

So, the function rule for g(x) is:

g(x) = |x| - 4

This rule represents a vertical translation of the graph of f(x) down 4 units.