Question

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

Step1: Identify vertex of $f(x)$

The vertex of the solid graph $f(x)$ is at $(-2, 1)$.

Step2: Identify vertex of dotted graph

The vertex of the dotted graph is at $(-1, -4)$.

Step3: Calculate horizontal shift

Find change in $x$-coordinate: $-1 - (-2) = 1$. This means a shift right by 1, so replace $x$ with $x-1$.

Step4: Calculate vertical shift

Find change in $y$-coordinate: $-4 - 1 = -5$. This means a shift down by 5, so subtract 5 from $f(x-1)$.

Step5: Form the transformed function

Combine the shifts: $y = f(x-1) - 5$.

Answer:

$y = f(x - 1) - 5$