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Using Coordinates to Prove Geometric Theorems About Circles and ParabolasAdapted from Walch Education

ImportantA 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 = r 2. This is based on the fact that any point (x, y) is on the circle if and only if6.2.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas2

RememberCompleting 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 Parabolas3

Dont forgetThe 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 Parabolas4Key ConceptsThe 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 Parabolas5Vertex of a ParabolaThe axis of symmetry intersects the parabola at the vertex. The x-coordinate of the vertex isThe 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 Parabolas6

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 Parabolas7Practice (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 Parabolas8

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