Question

question
the graph of $y = f(x)$ is the solid black graph below. which function represents the dotted graph?
answer
$\circ$ $y = f(-x) + 5$
$\circ$ $y = -f(x + 5)$
$\circ$ $y = -f(x - 5)$
$\circ$ $y = f(-x) - 5$

Explanation:

To determine the function for the dotted graph, we analyze transformations:

1. Reflection over the y - axis: The transformation \(y = f(-x)\) reflects the graph of \(y = f(x)\) over the \(y\) - axis.
2. Vertical shift down: The transformation \(y=f(-x)-5\) shifts the graph of \(y = f(-x)\) down by 5 units.

Looking at the graph, the dotted graph appears to be a reflection over the \(y\) - axis and a vertical shift down from the solid graph \(y = f(x)\).

• For \(y = f(-x)+5\), it would be a reflection over the \(y\) - axis and a shift up, which does not match.
• For \(y=-f(x + 5)\), it involves a reflection over the \(x\) - axis and a horizontal shift left, which is not the case here.
• For \(y=-f(x - 5)\), it involves a reflection over the \(x\) - axis and a horizontal shift right, which is incorrect.
• For \(y = f(-x)-5\), it is a reflection over the \(y\) - axis and a vertical shift down, which matches the appearance of the dotted graph.

Answer:

\(y = f(-x)-5\)