Question

graph the equation: $\frac{(x - 4)^2}{4}+\frac{(y + 1)^2}{1}=1$

Explanation:

Step1: Identify the center of the ellipse

The standard form of an ellipse is \(\frac{(x - h)^{2}}{a^{2}}+\frac{(y - k)^{2}}{b^{2}} = 1\), where \((h,k)\) is the center.
For the equation \(\frac{(x - 4)^{2}}{4}+\frac{(y+1)^{2}}{1}=1\), \(h = 4\), \(k=-1\). So the center is \((4,-1)\).

Step2: Find the values of \(a\) and \(b\)

Since \(a^{2}=4\), then \(a = 2\); since \(b^{2}=1\), then \(b = 1\).

Step3: Determine the vertices

The major axis is parallel to the \(x\) - axis (because \(a^{2}>b^{2}\)).
The vertices are \((h\pm a,k)\). Substituting \(h = 4\), \(a = 2\), \(k=-1\), we get \((4 + 2,-1)=(6,-1)\) and \((4-2,-1)=(2,-1)\).

Step4: Determine the co - vertices

The co - vertices are \((h,k\pm b)\). Substituting \(h = 4\), \(b = 1\), \(k=-1\), we get \((4,-1 + 1)=(4,0)\) and \((4,-1-1)=(4,-2)\).

Answer:

Plot the center \((4,-1)\), vertices \((2,-1)\) and \((6,-1)\), co - vertices \((4,0)\) and \((4,-2)\), and then draw the ellipse passing through these four points.