1. Analyzing the Parabola
Week 16

2. Distance and Midpoint Formulas
To find the distance between any two points (a, b) and (c, d), use the distance formula:
Distance = (c – a)2 + (d – b)2
The midpoint of a line is halfway between the two endpoints of a line
To find the midpoint between (a, b) and (c, d), use the midpoint formula:
Midpoint = (a + c) , (b + d)

3. Distance = [(-8) – (-4)]2 + (4 – 2)2
Worked Example
Find the distance between (-4, 2) and (-8, 4). Then find the midpoint between the points.
Distance = [(-8) – (-4)]2 + (4 – 2)2
Distance = (-8 + 4)2 + (2)2
Distance = (-4)2 + (2)2
Distance =
Distance= 20

4. Worked Example
Find the distance between (-4, 2) and (-8, 4). Then find the midpoint between the points.
Midpoint = (-4) + (-8) , 2 + 4
Midpoint = , 6
Midpoint = ( -6, 3)

5. Further Problems Find the distance between (0, 1) and (1, 5).
Find the midpoint between (6, -5) and (-2, -7).
Find the value for x if the Distance = 53 and the endpoints are (-3, 2) and (-10, x).
If you are given an endpoint (3, 2) and midpoint (-1, 5), what are the coordinates of the other endpoint?
1) ) (2, -6) ) x = 0 or x = ) (-5, 8)

6. Conics: Circles
A circle is a set of points equidistant from a center point
The radius is a line between the center and any point on the circle
The equation of a circle is (x – h)2 + (y – k)2 = r2 where the radius is r and the vertex is (h, k)
Sometimes you need to complete the square twice to get the equation in this form (once for x and once for y)
Radius (r)
Vertex (k, h)

7. Radius = 7 and Center is (-2, 6)
Worked Example 1
Find the center and radius of x2 + y2 + 4x – 12y – 9 = 0 and then graph the circle.
x2 + 4x + o + y2 – 12y + o = 9 + o + o
x2 + 4x y2 – 12y + 36 =
(x + 2)2 + (y – 6)2 = 49
Radius = 7 and Center is (-2, 6)

8. Worked Example 2
If a circle has a center (3, -2) and a point on the circle (7, 1), write the equation of the circle.
Find the radius by the distance formula.
Radius = (7 – 3)2 + (1 – (-2))2
r = (4)2 + (3)2
r =
r = 25
r = 5
The equation of the circle will be (x – 3)2 + (y + 2)2 = 25

9. 1) x2 + (y + 2)2 = 4 Center (0, -2) and Radius 2 2) x2 + y2 = 20
Further Problems
Find the center and radius of x2 + y2 + 4y = 0. Then graph the circle.
If a circle has a center (0, 0) and a point on the circle (-2, -4) write the equation of the circle.
1) x2 + (y + 2)2 = 4 Center (0, -2) and Radius ) x2 + y2 = 20 #1

10. THE PARABOLA Parabola Basics
The parabola is a conic section whose eccentricity is unity ie e=1
It is the locus of a point which moves so that its distance from a fixed point (the focus) is always equal to its distance from a fixed straight line (the directrix)
(-a, 0)
(a, 0)

11. ESSENTIAL PROPERTIES
Axis of Symmetry
Parabola
A parabola is a set of points on a plane that are the same distance from a given point called the focus and a given line called the directrix
The axis of symmetry is perpendicular to the directrix and passes through the parabola at a point called the vertex
The latus rectum goes through the focus and is perpendicular to the axis of symmetry
Focus
Latus Rectum
Vertex
Directrix

12. A DIFFERENT PARABOLA The focus is the point (a, 0)
S (a, 0)
(-a, 0) L
The focus is the point (a, 0)
The straight line L is the directrix.
O lies on the parabola and is a units from the focus, S (a, 0)
Like any other point, its distance from the directrix is also a units

13. EQUATION OF THE PARABOLA
Parabola Basics
Let P be any point on the parabola
Distance from the focus = distance from directrix
PA = PS (not too scale)
(x-a)2+(y-0)2=(x+a)2
y2=4ax

14. THE PARABOLA Equation of Parabola
If a parabola has its vertex at the origin and its focus is at S (a, 0), then its equation is given by y2=4ax

15. THE PARABOLA Shape of the Parabola
If a parabola has the equation y2=4ax,
It is symmetrical about the x-axis
y=0 is the axis of the parabola
If a>0, it is not defined for –ve values of x
If a<0, it is not defined for +ve values of x

16. Different Shapes
a>0
a<0
y2=4ax
y2=4ax

17. Think about this A parabola has the following properties
Its vertex is at (0, 0)
Its axis is x=0
Its focus is the point (0, a)
Its directrix is the line y=-a
Sketch the shape of the parabola here

18. Equation of parabola: Std form
y = ax2 + bx + c
If a> 0, the parabola opens upwards
If a< 0, it opens downwards.
The axis of symmetry is the line x = -b/2a
y=ax2+bx+c, a>0

19. Equation of parabola: Vertex form
y = a(x-h)2 + k
(h, k) is the vertex
If a is positive then the parabola opens upwards like a regular "U".
If a is negative, then the graph opens downwards
If |a| > 1, the graph of the parabola widens. This just means that the "U" shape of parabola stretches out sideways.
x=h, the axis of symmetry
(h, k), the vertex

20. Worked Example 1
Write y = x2 + 4x + 1 in the form y = a (x – h)2 + k and name the vertex, axis of symmetry, and the direction the parabola opens.
You can always check your answers by graphing.
y = x2 + 4x + 1
y = (x2 + 4x + o ) + 1 – o
y = (x2 + 4x + 4) + 1 – 4
y = (x + 2)2 – 3
Vertex: (-2, -3)
Axis of Symmetry: x = -2
The parabola opens up because a = 1 so a > 0.

21. Conclusion
The standard form of the equation can be changed into the vertex form by completing the square
It is easier to sketch a parabola when the vertex form of the equation is known

22. Worked Example 2 Graph the equation x2 = 8y.
For the parabola y2 = -16x name the vertex, focus, length of latus rectum, and direction of opening. Also, give the equations of the directrix and axis of symmetry.
Given the vertex (4, 1) and a point on the parabola (8, 3), find the equation of the parabola.
Graph for #2
Graph for #1
2) Vertex: (0,0) Focus: (-4,0) Latus rectum: 16 Direction: left Directrix: x = 4 Axis of symmetry: y = ) y = (1/8)(x – 4)2 + 1

23. Sketching the parabola
What is the graph of the following parabola y = (x–1)² + 1?

24. Analyzing the Equation
Change the standard forms of the following equations into the vertex form by completing the square and sketch the parabola
y = 3x2 + 24x + 50 (h, k) = (–4, 2)
y = – 2x2 + 8x – 5 (h, k) = (2, 3)
4x – y2 = 2y + 13 (h, k) = (3, –1)

25. Homework Sketch the parabolas y2 – 8y = 8x x2 – 4x = 4y

26. IMPORTANT INFORMATION
Form of the equation
y = a (x – h)2 + k
x = a (y – k)2 + h
Axis of Symmetry
x = h
y = k
Vertex
(h, k)
Focus
(h, k + 1/4a)
(h + 1/4a, k)
Directrix
y = k – (1/4)a
x = h – (1/4)a
Direction of Opening
Opens upward when a > 0 and downward when a < 0
Opens to the right when a > 0 and to the left when a < 0
Length of Latus Rectum
1/a units