### About Graphing Ellipses:

When we graph an ellipse that is centered at the origin, we first plot the x and y intercepts. We then sketch a smooth curve to connect the four points. When our ellipse is not centered at the origin, we first find the center, and then we consider how the points have shifted vs an ellipse that is centered at the origin. We can use this information to obtain four points and sketch the graph.

Test Objectives

- Demonstrate the ability to identify the center of an ellipse
- Demonstrate the ability to apply a horizontal and or vertical shift
- Demonstrate the ability to sketch the graph of an ellipse

#1:

Instructions: Graph each ellipse.

a) $$\frac{x^2}{36} + \frac{y^2}{16} = 1$$

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#2:

Instructions: Graph each ellipse.

a) $$\frac{x^2}{25} + \frac{y^2}{9} = 1$$

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#3:

Instructions: Graph each ellipse.

a) $$\frac{(x-1)^2}{9} + \frac{(y-3)^2}{4} = 1$$

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#4:

Instructions: Graph each ellipse.

a) $$\frac{x^2}{49} + \frac{(y + 2)^2}{9} = 1$$

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#5:

Instructions: Graph each ellipse.

a) $$\frac{\left(x - \frac{1}{2}\right)^2}{36} + \frac{(y - 3)^2}{4} = 1$$

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Written Solutions:

#1:

Solutions:

a)
Center: (0,0)

x-intercepts: (6,0),(-6,0)

y-intercepts: (0,4),(0,-4)

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#2:

Solutions:

a)
Center: (0,0)

x-intercepts: (5,0),(-5,0)

y-intercepts: (0,3),(0,-3)

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#3:

Solutions:

a)
Center: (1,3)

additional points:(4,3),(-2,3),(1,5),(1,1)

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#4:

Solutions:

a)
Center: (0,-2)

additional points:(7,-2),(-7,-2),(0,1),(0,-5)

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#5:

Solutions:

a) $$Center: \left(\frac{1}{2},3\right)$$ $$additional\hspace{.25em}points: $$$$\left(\frac{13}{2},3\right),\left(\frac{-11}{2},3\right)$$ $$\left(\frac{1}{2},1\right),\left(\frac{1}{2},5\right)$$