Question

match the graphs to their equations. f
\\( \frac { ( x + 1 ) ^ { 2 } } { 16 } + \frac { ( y - 3 ) ^ { 2 } } { 1 } = 1 \\)
\\( \frac { ( x - 1 ) ^ { 2 } } { 16 } + \frac { ( y + 3 ) ^ { 2 } } { 1 } = 1 \\)
\\( \frac { ( x + 1 ) ^ { 2 } } { 1 } + \frac { ( y + 3 ) ^ { 2 } } { 16 } = 1 \\)
\\( \frac { ( x - 1 ) ^ { 2 } } { 1 } + \frac { ( y - 3 ) ^ { 2 } } { 16 } = 1 \\)

Explanation:

To match the ellipse equations to their graphs, we use the standard form of an ellipse:

• Horizontal major axis: \(\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\) (\(a > b\)), center \((h, k)\), width \(2a\), height \(2b\).
• Vertical major axis: \(\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1\) (\(a > b\)), center \((h, k)\), width \(2b\), height \(2a\).

Equation 1: \(\frac{(x + 1)^2}{16} + \frac{(y - 3)^2}{1} = 1\)

• Center: \((h, k) = (-1, 3)\) (since \(x + 1 = x - (-1)\), \(y - 3\)).
• \(a^2 = 16 \implies a = 4\), \(b^2 = 1 \implies b = 1\).
• Horizontal major axis ( \(a > b\) ), width \(2a = 8\), height \(2b = 2\).
• Matches graph a (center near \((-1, 3)\), horizontal ellipse).

Equation 2: \(\frac{(x - 1)^2}{16} + \frac{(y + 3)^2}{1} = 1\)

• Center: \((h, k) = (1, -3)\) ( \(x - 1\), \(y + 3 = y - (-3)\) ).
• \(a = 4\), \(b = 1\) (horizontal major axis).
• Matches graph b (center near \((1, -3)\), horizontal ellipse).

Equation 3: \(\frac{(x + 1)^2}{1} + \frac{(y + 3)^2}{16} = 1\)

• Center: \((h, k) = (-1, -3)\) ( \(x + 1\), \(y + 3\) ).
• \(a^2 = 16 \implies a = 4\), \(b^2 = 1 \implies b = 1\).
• Vertical major axis ( \(a > b\) ), width \(2b = 2\), height \(2a = 8\).
• Matches graph with center \((-1, -3)\) and vertical stretch (not shown, but process holds).

Equation 4: \(\frac{(x - 1)^2}{1} + \frac{(y - 3)^2}{16} = 1\)

• Center: \((h, k) = (1, 3)\) ( \(x - 1\), \(y - 3\) ).
• \(a^2 = 16 \implies a = 4\), \(b^2 = 1 \implies b = 1\).
• Vertical major axis, width \(2b = 2\), height \(2a = 8\).
• Matches graph c (center near \((1, 3)\), vertical ellipse).

Graph Matching:

1. \(\frac{(x + 1)^2}{16} + \frac{(y - 3)^2}{1} = 1\) → a
2. \(\frac{(x - 1)^2}{16} + \frac{(y + 3)^2}{1} = 1\) → b
3. \(\frac{(x + 1)^2}{1} + \frac{(y + 3)^2}{16} = 1\) → (vertical ellipse, center \((-1, -3)\))
4. \(\frac{(x - 1)^2}{1} + \frac{(y - 3)^2}{16} = 1\) → c

Answer:

1. \(\boldsymbol{\frac{(x + 1)^2}{16} + \frac{(y - 3)^2}{1} = 1}\) → a
2. \(\boldsymbol{\frac{(x - 1)^2}{16} + \frac{(y + 3)^2}{1} = 1}\) → b
3. \(\boldsymbol{\frac{(x + 1)^2}{1} + \frac{(y + 3)^2}{16} = 1}\) → (vertical ellipse, center \((-1, -3)\))
4. \(\boldsymbol{\frac{(x - 1)^2}{1} + \frac{(y - 3)^2}{16} = 1}\) → c

(Note: For the third equation, if a third graph exists, it would have center \((-1, -3)\) and vertical stretch. The given graphs a, b, c suggest c is vertical with center \((1, 3)\), so the third equation likely matches an implicit graph with center \((-1, -3)\).)