- Horizontal Shift – this moves the entire parabola left or right without changing its basic shape.
- Vertical Shift – this moves the entire parabola up or down without changing its basic shape.
- Width Change – this makes the parabola wider or narrower, changing its basic shape.
What determines if a parabola is narrow or wide?
What determines if a parabola is narrow or wide? The coefficient of the quadratic term, a, determines how wide or narrow the graphs are, and whether the graph turns upward or downward. A positive quadratic coefficient causes the ends of the parabola to point upward. The greater the quadratic coefficient, the narrower the parabola.
How do you shift a parabola left and right?
is the center of the parabola. To shift the parabola left of right, the value of h changes. Since there is a negative sign in the parent function, a positive value moves the parabola to the left and a negative value moves it to the right. that has been translated right 2 spaces and up 4 spaces.
How many points are needed to determine a parabola?
Sketching Parabolas
- Find the vertex. We’ll discuss how to find this shortly. ...
- Find the y y -intercept, (0,f (0)) ( 0, f ( 0)).
- Solve f (x) = 0 f ( x) = 0 to find the x x coordinates of the x x -intercepts if they exist. ...
- Make sure that you’ve got at least one point to either side of the vertex. This is to make sure we get a somewhat accurate sketch. ...
- Sketch the graph. ...
How to shade a parabola?
To draw a parabola in PowerPoint 2013 for Windows, follow these steps:
- Launch PowerPoint 2013 for Windows. ...
- Within the View tab of the Ribbon, select the Guides check-box (highlighted in red within Figure 2 ).
- Figure 2: Guides check-box selected
- This will show the guides on the slide, as shown in Figure 3. ...
- Figure 3: Guides showing on the slide
What value affects the width of the parabola?
a determines the width and the direction of the parabola: The larger |a| becomes, the wider the parabola. If a is positive, the parabola opens upward, and if a is negative, the parabola opens downward. c determines the y-intercept of the parabola; there will always be a point at (0, c).
What causes a parabola to stretch?
When we multiply a function by a positive constant, we get a function whose graph is stretched or compressed vertically in relation to the graph of the original function. If the constant is greater than 1, we get a vertical stretch; if the constant is between 0 and 1, we get a vertical compression.
How do you make a parabola equation wider?
The greater the quadratic coefficient, the narrower the parabola. The lesser the quadratic coefficient, the wider the parabola.
How do you widen a graph?
To stretch or shrink the graph in the y direction, multiply or divide the output by a constant. 2f (x) is stretched in the y direction by a factor of 2, and f (x) is shrunk in the y direction by a factor of 2 (or stretched by a factor of ).
What is a parabola?
The parabola is a member of the family of conic sections. In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U- shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
What happens when a parabola has parallel chords?
A corollary of the above discussion is that if a parabola has several parallel chords, their midpoints all lie on a line parallel to the axis of symmetry. If tangents to the parabola are drawn through the endpoints of any of these chords, the two tangents intersect on this same line parallel to the axis of symmetry (see Axis-direction of a parabola ).
What is the distance between the vertex and the focus?
The distance between the vertex and the focus, measured along the axis of symmetry, is the "focal length". The " latus rectum " is the chord of the parabola that is parallel to the directrix and passes through the focus. Parabolas can open up, down, left, right, or in some other arbitrary direction.
What is the locus of points in that plane that are equidistant from both the directrix
The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and a plane parallel to another plane that is tangential to the conical surface.
What is the line that splits the parabola through the middle called?
The line perpendicular to the directrix and passing through the focus (that is, the line that splits the parabola through the middle) is called the "axis of symmetry". The point where the parabola intersects its axis of symmetry is called the " vertex " and is the point where the parabola is most sharply curved.
What is the area between a parabola and a chord?
The chord itself ends at the points where the line intersects the parabola. The area enclosed between a parabola and a chord (see diagram) is two-thirds of the area of a parallelogram that surrounds it.
How to find the length of arcs of a parabola?
Arc length. If a point X is located on a parabola with focal length f , and if p is the perpendicular distance from X to the axis of symmetry of the parabola, then the lengths of arcs of the parabola that terminate at X can be calculated from f and p as follows, assuming they are all expressed in the same units.
What is the vertex of a parabola?
First, if a a is positive then the parabola will open up and if a a is negative then the parabola will open down. Secondly, the vertex of the parabola is the point (h,k) ( h, k).
What is the axis of symmetry of a parabola?
Every parabola has an axis of symmetry and, as the graph shows, the graph to either side of the axis of symmetry is a mirror image of the other side. This means that if we know a point on one side of the parabola we will also know a point on the other side based on the axis of symmetry.
What is the line with each parabola called?
The dashed line with each of these parabolas is called the axis of symmetry.
Overview
As a graph of a function
The previous section shows that any parabola with the origin as vertex and the y axis as axis of symmetry can be considered as the graph of a function
For the parabolas are opening to the top, and for are opening to the bottom (see picture). From the section above one obtains:
• The focus is ,
History
The earliest known work on conic sections was by Menaechmus in the 4th century BC. He discovered a way to solve the problem of doubling the cube using parabolas. (The solution, however, does not meet the requirements of compass-and-straightedge construction.) The area enclosed by a parabola and a line segment, the so-called "parabola segment", was computed by Archimedes by the method …
In a cartesian coordinate system
If one introduces Cartesian coordinates, such that and the directrix has the equation , one obtains for a point from the equation . Solving for yields
This parabola is U-shaped (opening to the top).
The horizontal chord through the focus (see picture in opening section) is called the latus rectum; one half of it is the semi-latus rectum. The latus rectum is par…
Similarity to the unit parabola
Two objects in the Euclidean plane are similar if one can be transformed to the other by a similarity, that is, an arbitrary composition of rigid motions (translations and rotations) and uniform scalings.
A parabola with vertex can be transformed by the translation to one with the origin as vertex. A suitable rotation around the origin can then transform the p…
Conic section and quadratic form
The diagram represents a cone with its axis AV. The point A is its apex. An inclined cross-section of the cone, shown in pink, is inclined from the axis by the same angle θ, as the side of the cone. According to the definition of a parabola as a conic section, the boundary of this pink cross-section EPD is a parabola.
A cross-section perpendicular to the axis of the cone passes through the verte…
Proof of the reflective property
The reflective property states that if a parabola can reflect light, then light that enters it travelling parallel to the axis of symmetry is reflected toward the focus. This is derived from geometrical optics, based on the assumption that light travels in rays.
Consider the parabola y = x . Since all parabolas are similar, this simple case r…
Properties related to Pascal's theorem
A parabola can be considered as the affine part of a non-degenerated projective conic with a point on the line of infinity , which is the tangent at . The 5-, 4- and 3- point degenerations of Pascal's theorem are properties of a conic dealing with at least one tangent. If one considers this tangent as the line at infinity and its point of contact as the point at infinity of the y axis, one obtains three stateme…