![Page 1: Definition & Construction of a Parabola (Part 1)](https://reader036.vdocuments.mx/reader036/viewer/2022062314/56813bc6550346895da4f1e7/html5/thumbnails/1.jpg)
Let f be the function defined by : (a≠0)
I - Algebraic properties :
1°) Even function : for any x R, f(-x) = f(x).
2°) Rate of growth not constant :
3°) Sign of T = Sign of a on [0 ; +∞[ , and Sign of T = Sign of (-a) on ]- ∞ ; 0]
4°) Chart of the Variations of f :
Definition & Construction of a Parabola (Part 1)
€
T[f , (x1;x2 )] =f (x2 ) − f (x1)
x2 − x1
= a(x2 + x1)
€
f : x a ax 2
![Page 2: Definition & Construction of a Parabola (Part 1)](https://reader036.vdocuments.mx/reader036/viewer/2022062314/56813bc6550346895da4f1e7/html5/thumbnails/2.jpg)
Definition & Construction of a Parabola (Part 1)II- Geometric Properties :
1°) The curve has (Oy) as an axis of symmetry. For that reason the curve is called a Parabola.2°) The Parabola is tangent to the (Ox) Axis in O.3°) The Parabola passes through the point A(1 ; a).4°) If a > 0 the Parabola concavity is directed towars the positive y :
(as one can say the « the bowl can hold water »)If a < 0 the Parabola concavity is directed towards the negative y :(as one can say the « the bowl cannot hold water »)
5°) The Parabola intercepts the 1st bisector line (y = x) at point B(1/a ; 1/a)6°) On B the Parabola is tangent to the line joining B to the middle of the segment located under the
tangent, which is the point of abscissa 1/2a7°) By symmetry with respect to Oy we get the points A’(-1 ;a) and B’(-1/a ; 1/a)8°) If a is very small compared to the unity (a << 1) , the parabola is widely opened, inversely
if a >>1 the parabola is very narrow around the axis of symmetry.9°) The parabola contains absolutely no piece of a straight line.10°) The branches spread indefinitely in the direction of the Oy axis.
![Page 3: Definition & Construction of a Parabola (Part 1)](https://reader036.vdocuments.mx/reader036/viewer/2022062314/56813bc6550346895da4f1e7/html5/thumbnails/3.jpg)
1°) The Parabola is symmetrical through the (Oy ) axis.It was named so because of that property.
Definition & Construction of a Parabola (Part 1)II- Geometric Properties :
![Page 4: Definition & Construction of a Parabola (Part 1)](https://reader036.vdocuments.mx/reader036/viewer/2022062314/56813bc6550346895da4f1e7/html5/thumbnails/4.jpg)
2°) The Parabola is tangent to the (Ox) Axis in O (0;0).
Definition & Construction of a Parabola (Part 1)II- Geometric Properties :
![Page 5: Definition & Construction of a Parabola (Part 1)](https://reader036.vdocuments.mx/reader036/viewer/2022062314/56813bc6550346895da4f1e7/html5/thumbnails/5.jpg)
3°) The Parabola passes through the point A(1 ; a).
Definition & Construction of a Parabola (Part 1)II- Geometric Properties :
![Page 6: Definition & Construction of a Parabola (Part 1)](https://reader036.vdocuments.mx/reader036/viewer/2022062314/56813bc6550346895da4f1e7/html5/thumbnails/6.jpg)
Definition & Construction of a Parabola (Part 1)II- Geometric Properties :
4°) If a > 0 the Parabola concavity is directed towars the positive y :
(as one can say the « the bowl can hold water »)If a < 0 the Parabola concavity is directed towards the négative y :
(as one can say the « the bowl cannot hold water »)
![Page 7: Definition & Construction of a Parabola (Part 1)](https://reader036.vdocuments.mx/reader036/viewer/2022062314/56813bc6550346895da4f1e7/html5/thumbnails/7.jpg)
Definition & Construction of a Parabola (Part 1)II- Geometric Properties :
5°) The Parabola intercepts the 1st bisector line (y = x) at point B(1/a ; 1/a).
![Page 8: Definition & Construction of a Parabola (Part 1)](https://reader036.vdocuments.mx/reader036/viewer/2022062314/56813bc6550346895da4f1e7/html5/thumbnails/8.jpg)
Definition & Construction of a Parabola (Part 1)II- Geometric Properties :
6°) On B (1/a;1/a) the Parabola is tangent tothe line joining B to the middle of the segment located under the tangent,which is the point of abscissa 1/2a
![Page 9: Definition & Construction of a Parabola (Part 1)](https://reader036.vdocuments.mx/reader036/viewer/2022062314/56813bc6550346895da4f1e7/html5/thumbnails/9.jpg)
Definition & Construction of a Parabola (Part 1)II- Geometric Properties :
6°) On B (1/a;1/a) the Parabola is tangent to the line joining B to the middle of the segment located under the tangent, which is the point of abscissa 1/2a
![Page 10: Definition & Construction of a Parabola (Part 1)](https://reader036.vdocuments.mx/reader036/viewer/2022062314/56813bc6550346895da4f1e7/html5/thumbnails/10.jpg)
Definition & Construction of a Parabola (Part 1)II- Geometric Properties :
8°) If a is very small compared to the unity (a << 1) , the parabola is widely opened, Inversely if a >>1 the parabola is very narrow around the axis of symmetry.
![Page 11: Definition & Construction of a Parabola (Part 1)](https://reader036.vdocuments.mx/reader036/viewer/2022062314/56813bc6550346895da4f1e7/html5/thumbnails/11.jpg)
Definition & Construction of a Parabola (Part 1)II- Geometric Properties :
9°) The parabola contains absolutely no piece of a straight line.Because the rate of growth is never constant.
![Page 12: Definition & Construction of a Parabola (Part 1)](https://reader036.vdocuments.mx/reader036/viewer/2022062314/56813bc6550346895da4f1e7/html5/thumbnails/12.jpg)
Definition & Construction of a Parabola (Part 1)II- Geometric Properties :
10°) The branches spread indefinitely in the direction of the Oy axis.
![Page 13: Definition & Construction of a Parabola (Part 1)](https://reader036.vdocuments.mx/reader036/viewer/2022062314/56813bc6550346895da4f1e7/html5/thumbnails/13.jpg)
Definition & Construction of a Parabola (Part 1)II- Geometric Properties : (summary)
1°) The curve has (Oy) as an axis of symmetry. For that reason the curve is called a Parabola.
2°) The Parabola is tangent to the (Ox) Axis in O.
3°) The Parabola passes through the point A(1 ; a).
4°) If a > 0 the Parabola concavity is directed towars the positive y :
(as one can say the « the bowl can hold water »)
If a < 0 the Parabola concavity is directed towards the négative y :
(as one can say the « the bowl cannot hold water »)
5°) The Parabola intercepts the 1st bisector line (y = x) at point B(1/a ; 1/a)
6°) On B the Parabola is tangent to the line joining B to the middle of the segment located under the tangent, which is the point of abscissa 1/2a
7°) By symetry with respect to Oy we get the points A’(-1 ;a) and B’(-1/a ; 1/a)
8°) If a is very small compared to the unity (a << 1) , the parabola is widely opened, inverselyif a >>1 the parabola is very narrow around the axis of symmetry.
9°) The parabola contains absolutely no piece of a straight line.
10°) The branches spread indefinitely in the direction of the Oy axis.
![Page 14: Definition & Construction of a Parabola (Part 1)](https://reader036.vdocuments.mx/reader036/viewer/2022062314/56813bc6550346895da4f1e7/html5/thumbnails/14.jpg)
Y = ½ x2 Y = - ¼ x2
Definition & Construction of a Parabola (Part 1)II- Geometric Properties : Graphs
![Page 15: Definition & Construction of a Parabola (Part 1)](https://reader036.vdocuments.mx/reader036/viewer/2022062314/56813bc6550346895da4f1e7/html5/thumbnails/15.jpg)
Second degree Functions
![Page 16: Definition & Construction of a Parabola (Part 1)](https://reader036.vdocuments.mx/reader036/viewer/2022062314/56813bc6550346895da4f1e7/html5/thumbnails/16.jpg)
Definition & Construction of a Parabola (Part II)
![Page 17: Definition & Construction of a Parabola (Part 1)](https://reader036.vdocuments.mx/reader036/viewer/2022062314/56813bc6550346895da4f1e7/html5/thumbnails/17.jpg)
Definition & Construction of a Parabola (Part II)