大学精品课件：parabolas
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Done 2. Decide which way the parabola opens.
Look at the right side. If y: + c → opens up If y: - c → opens down If x: + c → opens right If x: - c → opens left
Colleen Beaudoin For FCIT
Geometric definition: a cone has a plane intersecting it, crossing its vertical axis
Algebraic definition: All points that are equidistant from a given line (the directrix) and a fixed point not on the directrix (the focus)
y = 3: found by moving down 3 from the vertex
To graph:
6. Plot the endpoints of the latus rectum/focal chord (width at the focus). The width is the │c│ at the focus.
Up because y is on the right and 12 is positive
To graph:
3. Plot thBaidu Nhomakorabea vertex (h,k) Note what happens to the signs.
(5,6)
4. Plot the focus: move │¼ c │ from the vertex in the direction that the parabola opens. Mark with an f.
Look at the right side. If y: + c → opens up If y: - c → opens down If x: + c → opens right If x: - c → opens left
To graph:
3. Plot the vertex (h,k) Note what happens to the signs.
To graph:
6. Plot the endpoints of the latus rectum/focal chord (width at the focus). The width is the │c│ at the focus.
7. Sketch the parabola by going through the vertex and the endpoints of the latus rectum. (Be sure to extend the curve and put arrows.)
Any point on the parabola is equidistant to the focus and the directrix.
Example: Point A: d1=d1 Point B: d2=d2
y
B A
Focus Vertex
x
Directrix Axis of Symmetry
8. Identify the axis of symmetry. (The line that goes through the vertex dividing the parabola in half.)
To graph: 1. Put in standard form– squared term on left
4. Plot the focus: move │¼ c │ from the vertex in the direction that the parabola opens. Mark with an f.
5. Draw the directrix: │¼ c │ from the vertex in the opposite direction of the focus (Remember that the directrix is a line.)
y
One variable is squared and one is not.
(How does this differ from linear equations?)
There are many ways the equation of a parabola can be written. We will get the quadratic part (variable that is squared) on the left of the equal sign and the linear part (variable is to the first power) on the right of the equal sign.
L.R. = 12 with endpoints at (-1,9) & (11,9)
7. Sketch the parabola by going through the vertex and the endpoints of the latus rectum. (Be sure to extend the curve and put arrows.)
Equation:
(x - h)2 = c(y – k) OR (y - k)2 = c(x – h)
To graph: 1. Put in standard form (above) – squared term
on left 2. Decide which way the parabola opens.
(5,9): found by moving up 3 from the vertex
5. Draw the directrix: │¼ c │ from the vertex in the opposite direction of the focus (Remember that the directrix is a line.)
Look at the right side. If y: + c → opens up If y: - c → opens down If x: + c → opens right If x: - c → opens left
Colleen Beaudoin For FCIT
Geometric definition: a cone has a plane intersecting it, crossing its vertical axis
Algebraic definition: All points that are equidistant from a given line (the directrix) and a fixed point not on the directrix (the focus)
y = 3: found by moving down 3 from the vertex
To graph:
6. Plot the endpoints of the latus rectum/focal chord (width at the focus). The width is the │c│ at the focus.
Up because y is on the right and 12 is positive
To graph:
3. Plot thBaidu Nhomakorabea vertex (h,k) Note what happens to the signs.
(5,6)
4. Plot the focus: move │¼ c │ from the vertex in the direction that the parabola opens. Mark with an f.
Look at the right side. If y: + c → opens up If y: - c → opens down If x: + c → opens right If x: - c → opens left
To graph:
3. Plot the vertex (h,k) Note what happens to the signs.
To graph:
6. Plot the endpoints of the latus rectum/focal chord (width at the focus). The width is the │c│ at the focus.
7. Sketch the parabola by going through the vertex and the endpoints of the latus rectum. (Be sure to extend the curve and put arrows.)
Any point on the parabola is equidistant to the focus and the directrix.
Example: Point A: d1=d1 Point B: d2=d2
y
B A
Focus Vertex
x
Directrix Axis of Symmetry
8. Identify the axis of symmetry. (The line that goes through the vertex dividing the parabola in half.)
To graph: 1. Put in standard form– squared term on left
4. Plot the focus: move │¼ c │ from the vertex in the direction that the parabola opens. Mark with an f.
5. Draw the directrix: │¼ c │ from the vertex in the opposite direction of the focus (Remember that the directrix is a line.)
y
One variable is squared and one is not.
(How does this differ from linear equations?)
There are many ways the equation of a parabola can be written. We will get the quadratic part (variable that is squared) on the left of the equal sign and the linear part (variable is to the first power) on the right of the equal sign.
L.R. = 12 with endpoints at (-1,9) & (11,9)
7. Sketch the parabola by going through the vertex and the endpoints of the latus rectum. (Be sure to extend the curve and put arrows.)
Equation:
(x - h)2 = c(y – k) OR (y - k)2 = c(x – h)
To graph: 1. Put in standard form (above) – squared term
on left 2. Decide which way the parabola opens.
(5,9): found by moving up 3 from the vertex
5. Draw the directrix: │¼ c │ from the vertex in the opposite direction of the focus (Remember that the directrix is a line.)