Question

match the graphs to their equations.

$$ \frac { ( x - 2 ) ^ { 2 } } { 9 } + \frac { ( y + 1 ) ^ { 2 } } { 16 } = 1 $$

$$ \frac { ( x - 2 ) ^ { 2 } } { 16 } + \frac { ( y - 1 ) ^ { 2 } } { 9 } = 1 $$

$$ \frac { ( x + 2 ) ^ { 2 } } { 9 } + \frac { ( y - 1 ) ^ { 2 } } { 16 } = 1 $$

$$ \frac { ( x + 2 ) ^ { 2 } } { 16 } + \frac { ( y + 1 ) ^ { 2 } } { 9 } = 1 $$

Explanation:

Step1: Recall the standard form of an ellipse

The standard form of an ellipse is \(\frac{(x - h)^{2}}{a^{2}}+\frac{(y - k)^{2}}{b^{2}} = 1\). If \(a>b\), the major axis is horizontal. If \(b > a\), the major axis is vertical. The center of the ellipse is \((h,k)\).

Step2: Analyze each equation

• For equation \(c\): \(\frac{(x - 2)^{2}}{9}+\frac{(y + 1)^{2}}{16}=1\), \(h = 2\), \(k=-1\), \(a = 3\), \(b = 4\). The center is \((2,-1)\) and major axis is vertical.
• For equation \(d\): \(\frac{(x - 2)^{2}}{16}+\frac{(y - 1)^{2}}{9}=1\), \(h = 2\), \(k = 1\), \(a=4\), \(b = 3\). The center is \((2,1)\) and major axis is horizontal.
• For equation \(e\): \(\frac{(x + 2)^{2}}{9}+\frac{(y - 1)^{2}}{16}=1\), \(h=-2\), \(k = 1\), \(a = 3\), \(b = 4\). The center is \((-2,1)\) and major axis is vertical.
• For equation \(f\): \(\frac{(x + 2)^{2}}{16}+\frac{(y + 1)^{2}}{9}=1\), \(h=-2\), \(k=-1\), \(a = 4\), \(b = 3\). The center is \((-2,-1)\) and major axis is horizontal.

Step3: Match the center and major - axis orientation

By observing the centers and the orientation (horizontal/vertical) of the major axes of the ellipses in the graphs:

• Graph \(a\) has a vertical major axis and center \((-2,1)\) which matches equation \(e\).
• Graph \(b\) has a horizontal major axis and center \((-2,-1)\) which matches equation \(f\).
• Graph \(c\) has a vertical major axis and center \((2,-1)\) which matches equation \(c\).
• Graph \(d\) has a horizontal major axis and center \((2,1)\) which matches equation \(d\).

Answer:

\(\frac{(x - 2)^{2}}{16}+\frac{(y - 1)^{2}}{9}=1\) matches graph \(d\)