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: Analyze the first equation \(\frac{(x - 3)^2}{1}-\frac{(y - 2)^2}{4}=1\)

This is a hyperbola in the form \(\frac{(x - h)^2}{a^2}-\frac{(y - k)^2}{b^2}=1\), which opens horizontally (since the \(x\)-term is positive). The center is \((h,k)=(3,2)\). Looking at the graphs, graph c is a horizontal - opening hyperbola (two separate curves opening left and right), so this equation matches graph c.

Step2: Analyze the second equation \(\frac{(y + 2)^2}{1}=\frac{(x - 3)^2}{4}+1\)

Rewrite it as \(\frac{(y + 2)^2}{1}-\frac{(x - 3)^2}{4}=1\). This is a hyperbola in the form \(\frac{(y - k)^2}{a^2}-\frac{(x - h)^2}{b^2}=1\), which opens vertically (since the \(y\)-term is positive). The center is \((h,k)=(3,- 2)\). Looking at the graphs, graph b has a lower hyperbola (opening up and down) with center related to \(y=-2\) (the lower curve is around \(y = - 3\) etc.), so this equation matches graph b.

Step3: Analyze the third equation \(\frac{(y - 2)^2}{1}-\frac{(x - 3)^2}{4}=1\)

This is a hyperbola in the form \(\frac{(y - k)^2}{a^2}-\frac{(x - h)^2}{b^2}=1\), which opens vertically. The center is \((h,k)=(3,2)\). Looking at the graphs, graph a has an upper hyperbola (opening up and down) with center around \(y = 2\) (the upper curve is around \(y = 3\) and the lower curve is around \(y=1\) etc.), so this equation matches graph a.

Answer:

1. \(\boldsymbol{\frac{(x - 3)^2}{1}-\frac{(y - 2)^2}{4}=1}\) matches graph c.
2. \(\boldsymbol{\frac{(y + 2)^2}{1}=\frac{(x - 3)^2}{4}+1}\) (or \(\frac{(y + 2)^2}{1}-\frac{(x - 3)^2}{4}=1\)) matches graph b.
3. \(\boldsymbol{\frac{(y - 2)^2}{1}-\frac{(x - 3)^2}{4}=1}\) matches graph a.