Unit 9 Circles and Other Conics
Find the equation of a circle.
In this lesson, we derived the equation of a circle. We learned that the equation of a circle describes all the points a given distance from the center. Like the distance formula, it is based on the Pythagorean theorem.
Write and graph the equation of a circle.
Find the center and radius of a circle in general form.
In this lesson, we learned to write equations of circles in both standard form and general form. We used the process of completing the square to change an equation from standard form to general form.
Apply understanding of circles and their equations to new situations.
In this lesson, we solved problems about circles that required us to use graphs and formulas such as the Pythagorean theorem, the distance formula, and the midpoint formula. We found it useful to use the equation of the circle to find points on the circle or to determine that a point is not on a circle. Sometimes it was useful to change forms of the equation to find more information about the circle from the equation.
Develop a geometric definition for a familiar shape.
In this lesson, we learned the geometric definition of a parabola. Much like circles, parabolas are a geometric shape that can be constructed from a definition and as a set of points generated from an equation. In the same way that the defining features of a circle are the center and radius, the defining features of a parabola are the focus and directrix.
Write and graph parabolas.
Compare the geometric definition of parabolas with quadratic functions.
In this lesson, we learned that the graph of a quadratic function meets the definition of a parabola. We learned to write equations given the focus and directrix and to find the focus and directrix of the parabola when given the equation of a quadratic function.
Write equations for parabolas with vertical directrices.
Determine the direction of opening for any parabola.
In this lesson, we learned to work with parabolas that have a vertical directrix. We found how to determine if they opened left or right, and how to write an equation of the parabola given a focus and directrix.
Understand the definition of an ellipse.
Understand relationships between parts of an ellipse.
Write the equation of an ellipse.
In this lesson, we learned to understand the definition of an ellipse. We identified many of the features of an ellipse, including the foci, center, and major and minor axes. We found the equation of an ellipse based on the definition and learned to write the equation in standard form with any center.
Compare the equation of an ellipse to the equation of other geometric figures.
Write the equation of a hyperbola.
In this lesson, we learned about hyperbolas, the last of the conic sections. We learned that the graphs of hyperbolas can open up and down or left and right. The definition of a hyperbola is much like the definition of an ellipse, except that a point on a hyperbola is the difference between the distances to the foci and the graph of an ellipse is the sum of the distances to the foci. This makes the equation of a hyperbola like the equation of an ellipse, except the terms are subtracted rather than added.