Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

1. Adapted from Walch Education Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

2. Important • A theorem is any statement that is proven or can be proved to be true. • The standard form of an equation of a circle with center (h, k) and radius r is (x – h)2 + (y – k)2 = r2. This is based on the fact that any point (x, y) is on the circle if and only if 6.2.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

3. Remember… • Completing the square is the process of determining the value of and adding it to x2+ bx to form the perfect square trinomial • A quadratic function can be represented by an equation of the form f(x)= ax2 + bx + c, where a ≠ 0. 6.2.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

4. Don’t forget… • The graph of any quadratic function is a parabola that opens up or down. • A parabola is the set of all points that are equidistant from a fixed line, called the directrix, and a fixed point not on that line, called the focus. • The parabola, directrix, and focus are all in the same plane. 6.2.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

5. Key Concepts • The distance between the focus and a point on the parabola is the same as the distance from that point to the directrix. • The vertex of the parabola is the point on the parabola that is closest to the directrix. • Every parabola is symmetric about a line called the axis of symmetry. 6.2.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

6. Vertex of a Parabola • The axis of symmetry intersects the parabola at the vertex. • The x-coordinate of the vertex is • The y-coordinate of the vertex is The standard form of an equation of a parabola that opens up or down and has vertex (h, k) is (x – h)2 = 4p(y – k), where p ≠ 0 and p is the distance between the vertex and the focus and between the vertex and the directrix. 6.2.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

7. Parabolas • Parabolas that open up or down represent functions, and their equations can be written in either of the following forms: y = ax2 + bx+ c or (x – h)2 = 4p(y – k). • The standard form of an equation of a parabola that opens right or left and has vertex (h, k) is (y – k)2 = 4p(x – h) 6.2.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

8. Practice (in class) • Given the point A (–6, 0), prove or disprove that point A is on the circle centered at the origin and passing through • Prove or disprove that the quadratic function graph with vertex (–4, 0) and passing through (0, 8) has its focus at (–4, 1). 6.2.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

9. Thanks for Watching! Ms. Dambreville