Question

question
the graph of $y = f(x)$ is the solid black graph below. which function represents the dotted graph?
answer
$\circ$ $y = f(x - 1) - 5$
$\circ$ $y = f(x + 1) - 5$
$\circ$ $y = f(x + 1) + 5$
$\circ$ $y = f(x - 1) + 5$

Explanation:

Step1: Analyze horizontal shift

The solid graph is \( y = f(x) \). For the dotted graph, observe the horizontal shift. A horizontal shift of \( h \) units: \( y = f(x - h) \) shifts right by \( h \), \( y = f(x + h) \) shifts left by \( h \). Here, the dotted graph seems to shift left (since it's to the left of the solid graph) or right? Wait, looking at the graph, the solid is at \( x \geq 0 \) (assuming the solid is a curve starting at origin, going down). The dotted is to the left (negative x-direction) and up? Wait, no, let's check vertical shift. Vertical shift: \( y = f(x) + k \) shifts up by \( k \), \( y = f(x) - k \) shifts down. Wait, the dotted graph: let's see the vertex or key point. The solid graph (black) has a key point at (0,0) maybe? The dotted (gray) is at, say, shifting left by 1 (so \( x + 1 \)) and up by 5? Wait no, wait the options: let's recall transformation rules.

Wait, the solid is \( y = f(x) \). Let's check each option:

1. \( y = f(x - 1) - 5 \): shifts right 1, down 5. Not matching.
2. \( y = f(x + 1) + 5 \): shifts left 1, up 5.
3. \( y = f(x + 1) - 5 \): shifts left 1, down 5.
4. \( y = f(x - 1) + 5 \): shifts right 1, up 5.

Looking at the graph, the dotted graph is to the left of the solid (so horizontal shift left: \( x + 1 \)) and up (so +5). So the correct transformation is \( y = f(x + 1) + 5 \)? Wait no, wait the original solid: let's see the shape. Wait, maybe I got the shift wrong. Wait, the solid is a curve starting at (0,0) going down. The dotted is a curve starting at (-1, 5)? Wait, no, maybe the solid is \( y = f(x) \) (like \( y = -\sqrt{x} \), starting at (0,0), decreasing). The dotted would be \( y = -\sqrt{x + 1} + 5 \), which is \( y = f(x + 1) + 5 \), since \( f(x) = -\sqrt{x} \), so \( f(x + 1) = -\sqrt{x + 1} \), then +5 shifts up 5. So the dotted graph is \( y = f(x + 1) + 5 \). Wait, but let's check the options. The options are:

• \( y = f(x - 1) - 5 \)
• \( y = f(x + 1) + 5 \)
• \( y = f(x + 1) - 5 \)
• \( y = f(x - 1) + 5 \)

So the correct one is \( y = f(x + 1) + 5 \), which is the second option? Wait no, the options are:

First option: \( y = f(x - 1) - 5 \)

Second: \( y = f(x + 1) + 5 \)

Third: \( y = f(x + 1) - 5 \)

Fourth: \( y = f(x - 1) + 5 \)

Wait, maybe I made a mistake. Wait, the solid graph (black) is \( y = f(x) \), the dotted (gray) is shifted left by 1 (so \( x + 1 \)) and up by 5 (so +5). So the function is \( y = f(x + 1) + 5 \), which is the second option? Wait no, the first option is \( y = f(x - 1) - 5 \), second is \( y = f(x + 1) + 5 \), third is \( y = f(x + 1) - 5 \), fourth is \( y = f(x - 1) + 5 \). So the correct answer should be \( y = f(x + 1) + 5 \), which is the second option? Wait, no, let's re-express.

Wait, horizontal shift: if the graph moves left by \( h \), the function is \( f(x + h) \). Vertical shift: up by \( k \), \( +k \). So if the dotted graph is left 1 and up 5, then \( y = f(x + 1) + 5 \), which is the second option (the one with \( +1 \) and \( +5 \)).

Step2: Confirm vertical shift

The vertical shift: the dotted graph is above the solid graph, so it's a vertical shift up. So \( +5 \). Horizontal shift: left, so \( x + 1 \). So combining, \( y = f(x + 1) + 5 \).

Answer:

\( y = f(x + 1) + 5 \) (the option with \( y = f(x + 1) + 5 \))