Question

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match the graphs to their equations

\frac{(y + 2)^2}{4} = \frac{(x - 3)^2}{4} + 1

\frac{(x - 3)^2}{4} - \frac{(y - 2)^2}{4} = 1

\frac{(y - 2)^2}{4} - \frac{(x - 3)^2}{4} = 1

Explanation:

Step1: Analyze the first equation

\[\frac{(y+2)^2}{4} - \frac{(x-3)^2}{4} = 1\]

Step2: Identify its geometric properties

This is a vertical hyperbola centered at \((3, -2)\).

Step3: Match with graph b

Graph b shows a vertical hyperbola centered at \((3, -2)\).

Step4: Analyze the second equation

\[\frac{(x-3)^2}{4} - \frac{(y-2)^2}{4} = 1\]

Step5: Identify its geometric properties

This is a horizontal hyperbola centered at \((3, 2)\).

Step6: Match with graph a

Graph a shows a horizontal hyperbola centered at \((3, 2)\).

Answer:

The matching of the equations to their corresponding graphs is:

1. \(\frac{(y+2)^2}{4} = \frac{(x-3)^2}{4} + 1\) matches with b
2. \(\frac{(x-3)^2}{4} - \frac{(y-2)^2}{4} = 1\) matches with a
3. \(\frac{(y-2)^2}{4} - \frac{(x-3)^2}{4} = 1\) does not match any of the shown graphs.