Question

directions: write your answer in the box.
given the graph of ( f(x) ) and ( f(x + k) ), determine the equation of ( f(x + k) ).
( f(x + k)= )

Explanation:

Step1: Analyze the horizontal shift

The graph of \(y = f(x + k)\) is a horizontal shift of the graph of \(y=f(x)\). If the graph of \(y = f(x)\) (with a key - point at \(x = 0\)) is shifted to the left by \(10\) units to get the graph of \(y=f(x + k)\) (with a key - point at \(x=-10\)).
The rule for a horizontal shift is \(y = f(x - h)\), where \(h\) is the shift. If the shift is to the left by \(a\) units, then \(y=f(x + a)\).
Let \(h=-k\). We know that when \(x=-10\) in \(y = f(x + k)\) corresponds to \(x = 0\) in \(y = f(x)\). Using the formula \(x+ k=0\) (for the corresponding key - points). Substitute \(x=-10\) into \(x + k=0\).

Step2: Solve for \(k\)

We have the equation \(-10 + k=0\).
Add \(10\) to both sides of the equation: \(k = 10\)

Answer:

\(f(x + 10)\)