LESSON PLAN         Day 1              10-1

 

Essential Question:  How are the distance and midpoint formulas used to solve problems?

 

Distance formula:  just like Pythagorean theorem

 

Ex)  find the distance between (4, -3) and (6, 2)

            Draw a triangle and show Pythagorean theorem

 

Show formula:             

CLASSIFYING TRIANGLES

Ex)  determine if the triangle with the vertices (4, 2), (0, 5), (7, -2) is scalene, isosceles, or equilateral

            Find the distances between each pair of points

           

           

so the triangle is isosceles

 

Midpoint formula:  (average of the x's, average of the y's)

 

Ex)  find the midpoint of the segment joining (4, -3) and (6, 2)

 

            Midpoint =

 

Ex)  find the midpoint of the segment joining (1, 7) and (5, 9)                 (3, 8)

 

Ex)  find the equation for the perpendicular bisector of the line segment joining (0, 3) and (-6, 7).

            a)  find the midpoint      (-3, 5)

            b)  find the slope           4/-6 = -2/3

            c)  find the perpendicular slope = 3/2

            d)  write the equation of a line with slope 3/2 through (-3, 5):  y 5 = 3/2(x + 3)

                        y 5 = 3/2x + 9/2        y = 3/2x + 19/2

 

ex)  solve for x: say the distance between the points (6, x) and (2, 3) is 4

           

           

            square both sides:         32 = 16 + (x 3)2

            16 = (x 3)2

            x 3 = 4                    x = 3 4 = 7, -1

LESSON PLAN         Day 2              10-2

 

Essential Question:  What are the features of parabolas and how are they expressed in algebraic and graphical form?

 

Draw a parabola, show the focus, directrix, and axis of symmetry

 

4 types:

 

Direction it opens

Equation

Focus

directrix

Up

x2 = 4py, p > 0

(0, p)

y = -p

Down

x2 = 4py, p < 0

(0, p)

y = -p

Right

y2 = 4px, p > 0

(p, 0)

x = -p

Left

y2 = 4px, p < 0

(p, 0)

x = -p

 

For each of the following, determine if the parabola opens up, down, left, or right.  Then identify the focus and directrix.  Then graph it.

 

Ex)  2y2 = -24x

Y2 = -12x, so the parabola opens to the left

-12x = 4px, -12 = 4p, p = -3

so the focus is at (-3, 0) and the directrix is at x = 3

 

 

Ex)  x2 = 20y

The parabola opens up

20y = 4py, 20 = 4p, p = 5

so the focus is at (0, 5) and the directrix is at y = -5

 

 

Ex)  Write the standard form of the equation of the parabola with a focus at (-3, 0) and the vertex at (0, 0).

The parabola opens left.  (just draw the points if you are unsure)

P = -3

Y2 = 4px, so the equation is y2 = -12x

 

 

Ex)  Write the standard form of the equation of the parabola with a directrix of y = 4 and the vertex at (0, 0).

The parabola opens down.

P = -4

X2 = 4py, so the equation is x2 = -16y

 

 

 

LESSON PLAN         Day 3              10-3

 

Circles

 

Essential Question:  What are the feature of a circle and how equations of circles written?

 

Standard form of the equation of a circle:  x2 + y2 = r2

 

Ex)  name the radius of the circle with equation x2 + y2 = 16.  then graph the circle

 

Ex)  write the standard form of the equation of a circle with a radius of 5 whose center is at the origin.

 

Ex)  ditto but the radius is 32.

 

Ex)  write the standard form of the equation of a circle whose center is at the origin that passes through (2, 5)

 

 


 

LESSON PLAN         Day 4              10-4                 Ellipses

Essential Question:  What are the features of an ellipse and how are equations of ellipses written?

Draw an ellipse both of the following ways:

 

Text Box: Major axis
Text Box: Minor axis
Text Box: vertices
Text Box: Co-vertices
Text Box: foci
Text Box: foci
 

 

 

 

 

 

 

 

 

 

 


 

Equation

Major axis

Vertices

Co-vertices

Foci

 where a > b

Horizontal

(a, 0), (-a, 0)

(0, b), (0, -b)

C2 = a2 b2

(c, 0), (-c, 0)

 where a < b

Vertical

(0, b), (0, -b)

(a, 0), (-a, 0)

C2 = b2 a2

(0, c), (0, -c)

Ex)  Write the equation 9x2 + 100y2 = 900 in standard form.  Then identify the vertices, co-vertices, and foci of the ellipse.  Then graph it.

Equation is .  Vertices are (10, 0) and (-10, 0). 

Co-vertices are (0, 3) and (0, -3).  Since c2 = 100 9, c2 = 91, so c = 91.

So the foci are at (91, 0) and (-91, 0)

Ex)  Write the equation  in standard form.  Then identify the vertices, co-vertices, and foci of the ellipse.  Then graph it.

Equation is .  Vertices are (0, 6) and (0, -6). 

Co-vertices are (12, 0) and (-12, 0).  Since c2 = 36 12, c2 = 24, so c = 24 = 26.

So the foci are at (0, 26) and (0, -26)

 

Ex)  write the equation of the ellipse with the center at (0, 0), a vertex at (-5, 0), and a focus at (-3, 0).

A2 = 25, c2 = a2 b2, 9 = 25 b2, b2 = 16       equation is

Ex)  write the equation of the ellipse with the center at (0, 0), a vertex at (0, -30), and a focus at (0, 20).

B2 = 900, 900 a2 = 400, a2 = 500     

LESSON PLAN         Day 5              10-5                 Hyperbolas

 

Essential Question:  What are the features of a hyperbola and how are equations of hyperbolas written?

 

Equation

Transverse Axis

Vertices

Asymptotes

Foci

Horizontal

(a, 0), (-a, 0)

C2 = a2 + b2

(c, 0), (-c, 0)

 

Vertical

(0, b), (0, -b)

C2 = a2 + b2

(0, c), (0, -c)

 

Ex)  Write the equation 9x2 16y2 = 144 in standard form.  Identify the vertices, foci, and asymptotes.  Then graph it.

Divide through by 144 to get x2 / 16 y2 / 9 = 1

Vertices:  (4, 0), (-4, 0)

Asymptotes:  +- 3/4x

Foci:  (5, 0), (-5, 0)

 

Ex)  write the equation of a hyperbola with foci (0, 5) and (0, -5) and vertices (0, -3) and (0, 3).  Then graph it

B = 3, so b2 = 9

C = 5 so c2 = 25

C2 = a2 + b2 so 25 = a2 + 9, a2 = 16

Equation y2 / 9 x2 / 16 = 1

Asymptotes:  +- 9/16x = +- 3/4x

 

Ex)  write the equation of a hyperbola with foci (-4, 0) and (4, 0) and vertices (-1, 0) and (1, 0).  Then graph it.

C = 4 so c2 = 16

A = 1 so a2 = 1

C2 = a2 + b2 so 16 = 1 + b2, b2 = 15

Equation x2 / 1 y2 / 15 = 1

Asymptotes:  +- 15x

 


 

LESSON PLAN         Day 6              10-6a

 

Essential Question:  How can conic sections be classified based on a given equation?

 

Classifying conics

 

Parabola:  only 1 of the variables is squared

Circle:  x2 and y2 are added and coefficients are both the same

Ellipse:  x2 and y2 are being added and coefficients are different

Hyperbola:  either x2 or y2 is being subtracted

 

classify the following:

ex)  4x2 9y2 + 32x 144y 548 = 0

ex)  2x2 + 2y2 12x + 4y + 2 = 0

ex)  -12x2 85y + 285 = 0

ex)  2x2 + y2 4x 4 = 0

 

 

Equations with the center at (h, k) (or vertex (h, k) for parabolas)

 

Circle:              (x h)2 + (y k)2 = r2

                        Center = (h, k), radius is r

 

Parabola:          (y k)2 = 4p(x h)                              (x h)2 = 4p(y k)

                        Vertex = (h, k), up or right if p is positive, down or left if p is negative

 

Ellipse:  , a > b             , a < b

                        Center = (h, k)

                        Vertices:  (h + a, k), (h a, k)               vertices:  (h, k + b), (h, k b)

                        Co-vertices:  (h, k + b), (h, k b)         co-vertices:  (h + a, k), (h a, k)

 

Hyperbola:                             

                        Center = (h, k)

                        Vertices:  (h + a, k), (h a, k)               vertices:  (h, k + b), (h, k b)

                        Asymptotes:  still y = +- a/b, but use the center as your center and go from

there with the slope

 

HW:     10-6a   pg 628 629   #21 49 odds

 

 


 

LESSON PLAN         Day 7              10-6b

 

Writing conic equations in standard form (completing the square)

 

Essential Question:  how do you write the equation of a conic section in standard form if it is not centered at the origin?

 

Classify the conic section and write its equation in standard form

 

Parabolas only:  Get all the y's on one side and all the x's on the other side        

 

Ex)  y2 2x 20y + 94 = 0      parabola           form (y k)2 = 4p(x h)

Y2 20y = 2x 94

Complete the square

Y2 20y + 100 = 2x 94 + 100

(y 10)2 = 2x + 4

factor out whatever number is in front of the x

(y 10)2 = 2(x + 2)

 

circles, ellipses, hyperbolas:       put all the x's together and all the y's together

 

ex)  x2 + y2 6x 8y + 24 = 0

x2 6x   + y2 8y = -24

complete the square on both of them

x2 6x + 9 + y2 8y + 16 = -24 + 9 + 16

(x 3)2 + (y 4)2 = 1

 

ex)  4x2 + y2 48x 4y + 48 = 0

4x2 48x + y2 4y = -48

4(x2 12x + 26) + y2 4y + 4 = -48 + 104 + 4

4(x 6)2 + (y 2)2 = 60

divide through by the constant

 

ex)  -9x2 + 4y2 36x 16y 164 = 0

put the negative variable second            4y2 16y 9x2 36x = 164

4(y2 4y + 4) 9(x2 + 4x + 4) = 164 + 16 36

4(y 2)2 9(x + 2)2 = 144

 

 

HW:     10-6     pg 629 #51 61 odds