Question

the black graph is y = f(x). choose the equation for the red graph. remember the big idea of opposites. y = f(x - 3) y + 3 = f(x) y - 3 = f(x) y = f(x + 3)

Explanation:

Step1: Recall function - translation rules

For a function \(y = f(x)\), a vertical translation \(k\) units up is given by \(y - k=f(x)\), and a vertical translation \(k\) units down is given by \(y + k = f(x)\); a horizontal translation \(h\) units to the right is given by \(y=f(x - h)\) and a horizontal translation \(h\) units to the left is given by \(y=f(x + h)\).

Step2: Analyze the graph

The black - graph \(y = f(x)\) has its vertex at \((0,0)\) and the red - graph has its vertex at \((0,3)\). The red graph is a vertical translation of the black graph 3 units up.

Step3: Determine the equation

Using the vertical - translation rule \(y - k=f(x)\) where \(k = 3\), we get \(y-3=f(x)\).

Answer:

\(y - 3 = f(x)\)