Question

match the graphs to their equations

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

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

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

Explanation:

Step1: Identify the center and orientation of the first equation.

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

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

Step2: Match the first equation to the correct graph.

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

Step3: Rewrite and analyze the second equation.

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

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

Step4: Match the second equation to the correct graph.

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

Step5: Analyze the third equation.

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

This is a vertical hyperbola centered at \((3, 2)\), which is not shown.

Answer:

The matching of the equations to their corresponding graphs is:

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