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) - 2 )
( circ y = f(x + 5) - 2 )
( circ y = f(x - 5) + 2 )
( circ y = f(x + 5) + 2 )

Explanation:

To solve this, we analyze the transformations of the function \( y = f(x) \) to get the dotted graph. Let's recall the rules of function transformations:

Step 1: Horizontal Shift

• The transformation \( y = f(x - h) \) shifts the graph of \( y = f(x) \) horizontally. If \( h>0 \), it shifts to the right by \( h \) units; if \( h<0 \), it shifts to the left by \( |h| \) units.
• The transformation \( y = f(x + h) \) is equivalent to \( y = f(x - (-h)) \), so it shifts the graph to the left by \( h \) units (when \( h>0 \)).

Step 2: Vertical Shift

• The transformation \( y = f(x) + k \) shifts the graph of \( y = f(x) \) vertically. If \( k>0 \), it shifts up by \( k \) units; if \( k<0 \), it shifts down by \( |k| \) units.

Step 3: Analyze the Given Options

Looking at the graph (even though the exact graph details are a bit unclear from the image, we can infer from the options):

• Let's assume the dotted graph is a transformation of \( y = f(x) \).
• For horizontal shift: If the dotted graph is shifted 5 units to the right, we use \( y = f(x - 5) \). For vertical shift: If it's shifted 2 units up, we use \( +2 \).

Wait, actually, let's re - evaluate. Wait, maybe I made a mistake earlier. Wait, the standard form:

• Horizontal shift: \( y = f(x - h) \): shift right by \( h \), \( y = f(x + h) \): shift left by \( h \)
• Vertical shift: \( y = f(x)+k \): shift up by \( k \), \( y = f(x)-k \): shift down by \( k \)

Now, looking at the options:

Option 1: \( y = f(x - 5)-2 \): shift right 5, down 2

Option 2: \( y = f(x + 5)-2 \): shift left 5, down 2

Option 3: \( y = f(x - 5)+2 \): shift right 5, up 2

Option 4: \( y = f(x + 5)+2 \): shift left 5, up 2

But since the original graph (solid) and the dotted graph (we can infer from typical problems like this) - usually, if the dotted graph is shifted 5 units to the right and 2 units up, the transformation is \( y = f(x - 5)+2 \)

Wait, maybe the correct option is \( y = f(x - 5)+2 \) (assuming the dotted graph is shifted 5 units right and 2 units up from the solid graph of \( y = f(x) \))

So the answer is the option \( y = f(x - 5)+2 \) (the third option in the given choices, which is \( y = f(x - 5)+2 \))

Answer:

\( y = f(x - 5)+2 \) (the option with this formula, usually the third one in the list of options provided in the question)