Translation

Definition

Translation in analytic geometry involves shifting a graph horizontally, vertically, or both without changing its shape, size, or orientation. This is achieved by adding constants to the $x$ and $y$ coordinates of each point on the graph.

5 Must Know Facts For Your Next Test

1. A horizontal translation shifts the ellipse left or right by adding a constant to the $x$-coordinate.
2. A vertical translation shifts the ellipse up or down by adding a constant to the $y$-coordinate.
3. The standard form of an ellipse $(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1)$ includes translations, with $(h,k)$ being the center of the translated ellipse.
4. Translations do not alter the lengths of the major and minor axes of an ellipse.
5. When translating an ellipse, its foci also shift accordingly along with all other points.

Review Questions

• How does translating an ellipse horizontally affect its equation?
• What are the new coordinates of an ellipse's center if it is translated up by 3 units and left by 4 units?
• Explain how to translate an ellipse given in standard form.

Related terms

Ellipse: $\text{An oval shape that is defined as all points where the sum of distances from two fixed points (foci) is constant.}$

Major Axis: $\text{The longest diameter of an ellipse that passes through its foci.}$

Minor Axis: $\text{The shortest diameter of an ellipse that is perpendicular to its major axis at the center.}$