Chapter 1 Functions and Graphs Copyright © 2014, 2010, 2007 Pearson Education, Inc Distance and Midpoint Formulas; Circles

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 Find the distance between two points. Find the midpoint of a line segment. Write the standard form of a circle’s equation. Give the center and radius of a circle whose equation is in standard form. Convert the general form of a circle’s equation to standard form. Objectives:

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 The Distance Formula The distance, d, between the points (x 1, y 1 ) and (x 2, y 2 ) in the rectangular coordinate system is To compute the distance between two points, find the square of the difference between the x-coordinates plus the square of the difference between the y-coordinates. The principal square root of this sum is the distance.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 Example: Using the Distance Formula Find the distance between (–1, –3) and (2, 3). Express the answer in simplified radical form and then round to two decimal places.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 Find the distance. Express the answer in simplified radical form and then round to two decimal places. 1)(-2,-6) & (3,-4) 2)(0,-2,) & (4,3) 3)( &

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 The Midpoint Formula Consider a line segment whose endpoints are (x 1, y 1 ) and (x 2, y 2 ). The coordinates of the segment’s midpoint are To find the midpoint, take the average of the two x-coordinates and the average of the two y-coordinates.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 Example: Using the Midpoint Formula Find the midpoint of the line segment with endpoints (1, 2) and (7, –3).

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 Find the midpoint of the line segment with endpoints 1)(-3, -4) & (6, –8). 2) 3)

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 Definition of a Circle A circle is the set of all points in a plane that are equidistant from a fixed point, called the center. The fixed distance from the circle’s center to any point on the circle is called the radius.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10 The Standard Form of the Equation of a Circle The standard form of the equation of a circle with center (h, k) and radius r is

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11 Example: Finding the Standard Form of a Circle’s Equation Write the standard form of the equation of the circle with center (0, –6) and radius 10. The standard form of the equation of the circle is

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 Example: Using the Standard Form of a Circle’s Equation to Graph the Circle Find the center and the radius of the circle whose equation is The center is (–3, 1). The radius is

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13 Example: Using the Standard Form of a Circle’s Equation to Graph the Circle Find the center and the radius of the circle whose equation is Graph the equation. The center is (–3, 1) The radius is

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14 Example: Using the Standard Form of a Circle’s Equation to Graph the Circle Find the center and the radius of the circle whose equation is Use the graph of the equation to identify the relation’s domain and range. The center is (–3, 1). The radius is The domain is [–5, –1]. The range is [–1, 3].

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15 The General Form of the Equation of a Circle The general form of the equation of a circle is where D, E, and F are real numbers.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16 Example: Converting the General Form of a Circle’s Equation to Standard Form and Graphing the Circle Write in standard form: The center of the circle is (–2, 2) The radius of the circle is

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 17 Example: Write equation of a circle Write equation of a circle whose center at (3,-4) and passing through the point (-1,2) The center of the circle is (3, -4) The radius of the circle is

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 18 Example: Write equation of a circle Write equation of a circle with diameter between (-2,3) and (4,-5). The center of the circle is (1,-1) The radius of the circle is 5