conics
DESCRIPTION
Conics. Chapter 7. Parabolas. Definition - a parabola is the set of all points equal distance from a point (called the focus) and a line (called the directrix). Parabolas are shaped like a U or C. Parabolas. Equations - y = a(x - h) 2 + k opens up if a > 0, opens down if a < 0. - PowerPoint PPT PresentationTRANSCRIPT
42510011 0010 1010 1101 0001 0100 1011
Conics
Chapter 7
4251
0011 0010 1010 1101 0001 0100 1011
Parabolas• Definition - a parabola is the set of all points equal distance from a point (called the focus) and a line (called the directrix).
• Parabolas are shaped like a U or C
4251
0011 0010 1010 1101 0001 0100 1011
Parabolas• Equations -
• y = a(x - h)2 + k–opens up if a > 0, opens down if a < 0.
• x = a(y - k)2 + h–opens right if a > 0, opens left if a < 0.
4251
0011 0010 1010 1101 0001 0100 1011
Parabolas• y = a(x - h)2 + k
• x = a(y - k)2 + h
• Vertex - the bottom of the curve that makes up a parabola. Represented by the point (h, k).
4251
0011 0010 1010 1101 0001 0100 1011
Parabolas• Given the following equations
for a parabola, give the direction of opening and the vertex.
• y = (x - 6)2 - 4
• opens up
• vertex is at (6, -4)
4251
0011 0010 1010 1101 0001 0100 1011
Parabolas• x = (y + 5)2 + 4
• opens right.
• vertex = (4, -5)
• y = -5(x + 2)2
• opens down
• vertex = (-2, 0)
4251
0011 0010 1010 1101 0001 0100 1011
Parabolas• x = -y2 - 1
• opens left
• vertex = (-1, 0)
4251
0011 0010 1010 1101 0001 0100 1011
Parabolas• How are we going to graph these?
• Calculator of course!!!
• We will be using the conics menu (#9).
• Typing it will be KEY!!!!!
4251
0011 0010 1010 1101 0001 0100 1011
Parabolas• Notice that you have four choices for parabolas. Two for x = and two for the y = types.
• How would we graph y = (x - 6)2 - 4?
4251
0011 0010 1010 1101 0001 0100 1011
Parabolas• y = (x - 6)2 - 4
• Which form would we use?
• The third one.
• A = 1
• H = 6
• K = -4
4251
0011 0010 1010 1101 0001 0100 1011
Parabolas• y = (x - 6)2 - 4
• We already know that the vertex is at (6, 4), but the calculator will tell us if we hit G-Solv and then VTX (F5, then F4).
4251
0011 0010 1010 1101 0001 0100 1011
Parabolas• Steps to graph a parabola (cause you gotta put in on graph paper for me to see).
• 1) choose the general equation that you will be working with.
4251
0011 0010 1010 1101 0001 0100 1011
Parabolas• 2) Enter your variables.
• 3) Draw (F6)
• 4) Find the vertex (G-solve, then VRX => F5 then F4).
• 5) Plot the vertex on your graph paper.
4251
0011 0010 1010 1101 0001 0100 1011
Parabolas• Now we need to plot a point on each side of the vertex.
• 6) if it is a y = equation, use the x value of the vertex as your reference. Plug in a value larger and smaller into the equation to get your y.
4251
0011 0010 1010 1101 0001 0100 1011
Parabolas• 6) if it is a x = equation, use the y value of the vertex as your reference. Plug in a value larger and smaller into the equation to get your x.
• 7) Plot these two points on your graph paper.
4251
0011 0010 1010 1101 0001 0100 1011
Parabolas• 8) connect your three points in a C or U shape.
• You’re done!!!
4251
0011 0010 1010 1101 0001 0100 1011
Parabolas• Let’s try to graph some together.
• x = (y + 5)2 + 4
• y = -5(x + 2)2
• x = -y2 - 1
4251
0011 0010 1010 1101 0001 0100 1011
Parabolas• Assignment:
• wkst 58
• pg. 420
• #’s 21 - 25
4251
0011 0010 1010 1101 0001 0100 1011
Circles• Definition: the set of all points that are equidistant from a given point (the center). The distance between the center and any point is called the radius.
4251
0011 0010 1010 1101 0001 0100 1011
Circles• Equation -
(x - h)2 + (y - k)2 = r2
• the center is at (h, k)
• the radius is r (notice that in the equation r is squared)
4251
0011 0010 1010 1101 0001 0100 1011
Circles• Give the center and the radius of each equation.
• (x - 1)2 + (y + 3)2 = 9
• center = (2, -3)
• radius = 3
4251
0011 0010 1010 1101 0001 0100 1011
Circles• (x - 2)2 + (y + 4)2 = 16
• center = (2, 4) radius = 4
• (x - 3)2 + y2 = 9
• center = (3, 0) radius = 3
• x2 + (y + 5)2 = 4
• center = (0, -5) radius = 2
4251
0011 0010 1010 1101 0001 0100 1011
Circles• Of course the calculator will do this for us. Let’s look at the circles in the conics menu.
• The 5th and 6th choices are circles. We will be using the 5th choice most often.
4251
0011 0010 1010 1101 0001 0100 1011
Circles• Let’s graph (x - 1)2 + (y + 3)2 = 9 using the calculator.
• Select the correct equation and plug in h, k and r.
• h = 1, k = -3, and r = 3
4251
0011 0010 1010 1101 0001 0100 1011
Circles• Draw it.
• By hitting G-Solv we can get the center and radius.
• Check it with what we found earlier.
4251
0011 0010 1010 1101 0001 0100 1011
Circles• Graphing on paper
• 1) plot the center.
• 2) make 4 points, one up, down, left and right from the center. The distance between the points and the center is the radius.
4251
0011 0010 1010 1101 0001 0100 1011
Circles• 3) Connect the points in a circular fashion. DO NOT create a square. This will take practice.
4251
0011 0010 1010 1101 0001 0100 1011
Circles• (x - 1)2 + (y + 3)2 = 9
• Center = (1, -3) Radius = 3
x
y
4251
0011 0010 1010 1101 0001 0100 1011
Circles - graph these• (x - 2)2 + (y + 4)2 = 16
• center = (2, 4) radius = 4
• (x - 3)2 + y2 = 9
• center = (3, 0) radius = 3
• x2 + (y + 5)2 = 4
• center = (0, -5) radius = 2
4251
0011 0010 1010 1101 0001 0100 1011
Circles
•Assignment:
•Circles wkst 60/61
4251
0011 0010 1010 1101 0001 0100 1011
Ellipses• Do not call ellipses ovals, even though they have the same shape.
• Equation:
(x−h)2
a2 +(y−k)2
b2 =1
4251
0011 0010 1010 1101 0001 0100 1011
Ellipses• The center is at (h, k).
• a is the horizontal distance from the center to the edge of the oval.
• b is the vertical distance from the center to the edge of the oval
4251
0011 0010 1010 1101 0001 0100 1011
Ellipses• Give the center of the ellipse, a, and b.
• center is (0, 2), a = 4, b = 2
x2
16+
(y−2)2
4=1
4251
0011 0010 1010 1101 0001 0100 1011
Ellipses• center is (0, 2), a = 4, b = 2
• to graph we will plot the center, then use a to create points on each side of the center and use b to create points above and below the center.
4251
0011 0010 1010 1101 0001 0100 1011
Ellipses• center is (0, 2), a = 4, b = 2
x
y
4251
0011 0010 1010 1101 0001 0100 1011
Ellipses• The calculator will be useful in confirming your answer, but will not give you the center or any of the distances. We use the next to the last option for ellipses.
4251
0011 0010 1010 1101 0001 0100 1011
Ellipses• Graph - find the center and a & b. Check using the calc.x2
9+y2
25=1
(x+3)2
4+(y+1)2 =1
4251
0011 0010 1010 1101 0001 0100 1011
Ellipses•Assignment:•wkst 63/64
4251
0011 0010 1010 1101 0001 0100 1011
Hyperbola• Hyperbola look like two parabolas facing out from each other.
• I am not going to make you graph them by hand. Just use the calculator.
4251
0011 0010 1010 1101 0001 0100 1011
Hyperbola• the equation is just like that of an ellipse except that the fractions are being subtracted.
(x−h)2
a2 −(y−k)2
b2 =1
4251
0011 0010 1010 1101 0001 0100 1011
Hyperbola• Enter the following equation into the calculator.
• h = -3, k = 5, a = 3, b = 2
(x+3)2
9−
(y−5)2
4=1
4251
0011 0010 1010 1101 0001 0100 1011
Hyperbola• hit G-Solve, then VTX (F4).
• this will give you one of the two vertices, use the arrow keys to get the other.
• graph these points.
4251
0011 0010 1010 1101 0001 0100 1011
Hyperbola• use the x or y intercepts (which will be given to you using G-Solv) to sketch the graph.
4251
0011 0010 1010 1101 0001 0100 1011
Hyperbola•Assignment•wkst 66/67
4251
0011 0010 1010 1101 0001 0100 1011
Conics• Points to use to distinguish between the conics sections.
• the equation of a parabola is the ONLY equation where only one variable is being squared.
4251
0011 0010 1010 1101 0001 0100 1011
Conics• for circles, both x and y are being squared, it is usually not set equal to 1 and there are no fractions.
4251
0011 0010 1010 1101 0001 0100 1011
Conics• for ellipses, both variables are squared, and the equation is the sum of fractions set equal to 1
4251
0011 0010 1010 1101 0001 0100 1011
Conics• for hyperbola, both variables are squared, and the equation is the difference of fractions set equal to 1
4251
0011 0010 1010 1101 0001 0100 1011
Conics•Assignment
•Conics worksheet