Distance The distance between any two points P and Q is written PQ. Find PQ if P is (9, 1) and Q is (2, -1)

Distance The distance between any two points P and Q is written PQ. Find PQ if P is (x 1, y 1 ) and Q is (x 2, y 2 )

The Distance Formula The distance between the points P 1 (x 1, y 1 ) and P 2 (x 2, y 2 ) is:

Midpoint The midpoint of a segment is a point that is equidistant from each endpoint of the segment. Find the midpoint of the segment joining (-3, 2) and (5, 4).

The Distance Formula The midpoint of the segment that joins the points P 1 (x 1, y 1 ) and P 2 (x 2, y 2 ) is:

Midpoint Find an equation of the perpendicular bisector of AB for the points A (-2, 1) and B (1, -3).

Learning Log Summary LT 1 – I can find the distance between any two points and the midpoint of the line segment joining them. To find the distance between two points… To find the midpoint of a segment joining two points…

Closure Homework pg. 404 ~ 1-25 (O)

Conic Sections Several plane curves have second degree equations. We call them conic sections (or conics) because they are obtained by slicing a double cone with a plane. Circle Ellipse Parabola Hyperbola

Definition of a Circle A circle is the set of all points in a plane that are a fixed distance, called the radius, from a fixed point, called the center.

The Equation of a Circle The circle with center (h,k) and radius r has the equation:

The Equation of a Circle Ex) Find an equation of the circle with center (-2,5) and radius 3.

The Equation of a Circle Graph:

Learning Log Summary LT 2 – I can derive the formula for a circle using the center and radius. The equation of a circle is… To graph a circle given its equation…

Closure Homework pg. 410 ~ 1-13 (O)

A Second Degree Equation A second degree equation, written in Standard Form, is: Re-writing an equation in this form will tell us what type of conic section it is describing.

The Equation of a Circle Determine if the graph of the given equation is a circle. If so, find its center and radius.

The Equation of a Circle Determine if the graph of the given equation is a circle. If so, find its center and radius.

Learning Log Summary LT 3 – I can complete the square to rewrite the standard form a circle to identify the center and radius. To re-write a 2 nd degree equation in standard form… To complete the square…

Closure Homework pg. 410 ~ (O), 31, 33

Conic Sections Several plane curves have second degree equations. We call them conic sections (or conics) because they are obtained by slicing a double cone with a plane. Circle Ellipse Parabola Hyperbola

Definition of a Parabola A parabola is the set of all points equidistant from a fixed line, called the directrix, and a fixed point not on the line, called the focus. Directrix Focus Vertex

Definition of a Parabola Ex) If the directrix of a parabola is y=2 and its vertex is (2,4), find the focus. Ex) If the focus of a parabola is (-3,-1) and its vertex and (1,-1), find the directrix.

The Equation of a Parabola Find the equation of a parabola having the point F(3,0) as focus and the line y=4 as directrix. Then graph the parabola.

The Equation of a Parabola Find the equation of a parabola having the point F(0,-2) as focus and the line x=3 as directrix. Then graph the parabola.

The Equation of a Parabola Directrix Focus The parabola whose equation is: Opens upwards if a>0 and downward if a<0. Vertex: (h,k) Axis of Symmetry: x=h The parabola whose equation is: Opens upwards if a>0 and downward if a<0. Vertex: (h,k) Axis of Symmetry: y=k

Learning Log Summary LT 4 – I can derive the formula for a parabola using the focus and directrix. A parabola is defined as… To write the equation of a parabola…

Closure Homework pg. 415 ~ 1-9 (O), 13, 15

Re-Writing Standard Form of a Parabola Find the vertex, focus, directrix and axis of symmetry of the parabola. Then graph it.

Re-Writing Standard Form of a Parabola Find the vertex, focus, directrix and axis of symmetry of the parabola. Then graph it.

Learning Log Summary LT 5 – I can rewrite the standard form of a parabola to identify the vertex, focus, directrix and axis of symmetry. To re-write the standard form of a parabola… To identify the vertex, focus and directrix…

Closure Homework pg. 416 ~ (O)

Definition of an Ellipse An ellipse is the set of all points M in the plane such that the sum of the distances from M to two fixed points (foci) is a given constant.

Definition of an Ellipse The center of an ellipse is the midpoint of the segment joining its foci. The major axis is the chord passing through its foci. The minor axis is the chors perpendicular to the major axis and passing through the center.

Equation of an Ellipse An ellipse having center (0,0) and foci (-c,0) and (c,0), and sum of focal radii 2a has the equation: a 2 > b 2 so… major axis is horizontal

Equation of an Ellipse An ellipse having center (0,0) and foci (0,-c) and (0,c), and sum of focal radii 2b has the equation: b 2 > a 2 so… major axis is vertical

Equation of an Ellipse Ex: Graph the ellipse:

Equation of an Ellipse Ex: Graph and write the equation of an ellipse with vertices (4,0), (-4,0), (0,7) and (0,-7)

Equation of an Ellipse Ex: Graph the ellipse:

Learning Log Summary LT 6 – I can rewrite the standard form of an ellipse to identify the center and length of major/minor axes. To re-write the standard form of an ellipse… To identify the major and minor axes…

Definition of a Hyperbola A h yperbola is the set of all points P in the plane such that the difference between the distances from P to two fixed points is a given constant.

Definition of a Hyperbola The center of a hyperbola is the midpoint of the line segment joining its foci.

Equation of a Hyperbola A hyperbola having center (0,0) and foci (-c,0) and (c,0), and difference of focal radii 2a has the equation: a 2 is positive so… Intercepts are on x-axis.

Equation of a Hyperbola A hyperbola having center (0,0) and foci (0,-c) and (0,c), and difference of focal radii 2b has the equation: b 2 is positive so… Intercepts are on y-axis

Equation of a Hyperbola Ex: Graph the hyperbola:

Equation of a Hyperbola Ex: Graph and write the equation of a hyperbola with foci (-5,0) and (5,0) and asymptotes and.

Equation of a Hyperbola Ex: Graph the hyperbola:

Learning Log Summary LT 7 – I can rewrite the standard form of a hyperbola to identify the foci and asymptotes. To re-write the standard form of a hyperbola… To identify the asymptotes of a hyperbola…

Closure Homework pg. 421 ~ 1-17 (O) – (1-13 graph only!) pg. 430 ~ 1-11 (O) – (graph only)