Question

question
the graph of ( y = f(x) ) is the solid black graph below. which function represents the dotted graph?

answer
( y = f(-x) + 2 )
( y = -f(x - 2) )
( y = f(-x) - 2 )
( y = -f(x + 2) )

Explanation:

To solve this, we analyze function transformations:

Step 1: Recall Transformation Rules

• Reflection over y - axis: Replace \( x \) with \( -x \), giving \( y = f(-x) \).
• Vertical shift up: Add a constant to the function, \( y = f(x)+k \) ( \( k>0 \) shifts up \( k \) units).
• Reflection over x - axis: Multiply the function by \( -1 \), \( y=-f(x) \).
• Horizontal shift: \( y = f(x - h) \) (shift right \( h \) units) or \( y = f(x + h) \) (shift left \( h \) units).

Step 2: Analyze Each Option

1. Option \( y = f(-x)+2 \):

• \( f(-x) \): Reflects \( y = f(x) \) over the \( y \) - axis.
• \( +2 \): Shifts the reflected graph up 2 units. This matches a reflection over \( y \) - axis and vertical shift up.

1. Option \( y=-f(x - 2) \):

• \( -f(x) \): Reflects over \( x \) - axis.
• \( x-2 \): Shifts right 2 units. This involves reflection over \( x \) - axis and horizontal shift, not matching typical dotted graph behavior here.

1. Option \( y = f(-x)-2 \):

• \( -2 \) shifts down 2 units, which is opposite of a likely upward shift.

1. Option \( y=-f(x + 2) \):

• Reflects over \( x \) - axis and shifts left 2 units, not matching the expected transformation.

Answer:

\( y = f(-x)+2 \) (the first option in the answer choices)