1. The document discusses parabolas and their key characteristics including focus, directrix, and standard equation forms.
2. A parabola is defined as the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix).
3. The standard equation forms for parabolas are provided depending on the orientation and location of the vertex.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Graphs of the Sine and Cosine Functions LectureFroyd Wess
More: www.PinoyBIX.org
Lesson Objectives
Able to plot the different Trigonometric Graphs
Graph of Sine Function (y = f(x) = sinx)
Graph of Cosine Function (y = f(x) = cosx)
Define the Maximum and Minimum value in a graph
Generalized Trigonometric Functions
Graphs of y = sinbx
Graphs of y = sin(bx + c)
Could find the Period of Trigonometric Functions
Could find the Amplitude of Trigonometric Functions
Variations in the Trigonometric Functions
Graphs of the Sine and Cosine Functions LectureFroyd Wess
More: www.PinoyBIX.org
Lesson Objectives
Able to plot the different Trigonometric Graphs
Graph of Sine Function (y = f(x) = sinx)
Graph of Cosine Function (y = f(x) = cosx)
Define the Maximum and Minimum value in a graph
Generalized Trigonometric Functions
Graphs of y = sinbx
Graphs of y = sin(bx + c)
Could find the Period of Trigonometric Functions
Could find the Amplitude of Trigonometric Functions
Variations in the Trigonometric Functions
9.2 - parabolas 1.ppt discussion about parabolassuser0af920
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. One description of a parabola involves a point and a line. The focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from 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. The graph of a quadratic function y=ax²+bx+c is a parabola if a≠0, and, conversely, a parabola is the graph of a quadratic function if its axis is parallel to the y-axis. The line perpendicular to the 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. One description of a parabola involves a point and a line. The focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from 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. The graph of a quadratic function y=ax²+bx+c is a parabola if a≠0, and, conversely, a parabola is the graph of a quadratic function if its axis is parallel to the y-axis. The line perpendicular to the 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. One description of a parabola involves a point and a line. The focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from 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. The graph of a quadratic function y=ax²+bx+c is a parabola if a≠0, and, conversely, a parabola is the graph of a quadratic function if its axis is parallel to the y-axis. The line perpendicular to the 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. One description of a parabola involves a point and a line. The focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from the directrix and the focus. Another description of a parabola is as a conic s
Finding the opening of the parabola, vertex, axis of symmetry, y-intercept, x- intercept, domain, range, and the minimum/maximum value including the illustration of the graph
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
A parabola is the locus of a point which moves in such a way that its distance from a fixed point is equal to its perpendicular distance from a fixed straight line.
1.1 Focus : The fixed point is called the focus of the Parabola.
1.2 Directrix : The fixed line is called the directrix of the Parabola.
(focus)
2.3 Vertex : The point of intersection of a parabola and its axis is called the vertex of the Parabola.
NOTE: The vertex is the middle point of the focus and the point of intersection of axis and directrix
2.4 Focal Length (Focal distance) : The distance of any point P (x, y) on the parabola from the focus is called the focal length. i.e.
The focal distance of P = the perpendicular distance of the point P from the directrix.
2.5 Double ordinate : The chord which is perpendicular to the axis of Parabola or parallel to Directrix is called double ordinate of the Parabola.
2.6 Focal chord : Any chord of the parabola passing through the focus is called Focal chord.
2.7 Latus Rectum : If a double ordinate passes through the focus of parabola then it is called as latus rectum.
2.7.1 Length of latus rectum :
The length of the latus rectum = 2 x perpendicular distance of focus from the directrix.
2.1 Eccentricity : If P be a point on the parabola and PM and PS are the distances from the directrix and focus S respectively then the ratio PS/PM is called the eccentricity of the Parabola which is denoted by e.
Note: By the definition for the parabola e = 1.
If e > 1 Hyperbola, e = 0 circle, e < 1
ellipse
2.2 Axis : A straight line passes through the focus and perpendicular to the directrix is called the axis of parabola.
If we take vertex as the origin, axis as x- axis and distance between vertex and focus as 'a' then equation of the parabola in the simplest form will be-
y2 = 4ax
3.1 Parameters of the Parabola y2 = 4ax
(i) Vertex A (0, 0)
(ii) Focus S (a, 0)
(iii) Directrix x + a = 0
(iv) Axis y = 0 or x– axis
(v) Equation of Latus Rectum x = a
(vi) Length of L.R. 4a
(vii) Ends of L.R. (a, 2a), (a, – 2a)
(viii) The focal distance sum of abscissa of the point and distance between vertex and L.R.
(ix) If length of any double ordinate of parabola
y2 = 4ax is 2 𝑙 then coordinates of end points of this Double ordinate are
𝑙2 𝑙2
, 𝑙
and
, 𝑙 .
4a
4a
3.2 Other standard Parabola :
Equation of Parabola Vertex Axis Focus Directrix Equation of Latus rectum Length of Latus rectum
y2 = 4ax (0, 0) y = 0 (a, 0) x = –a x = a 4a
y2 = – 4ax (0, 0) y = 0 (–a, 0) x = a x = –a 4a
x2 = 4ay (0, 0) x = 0 (0, a) y = a y = a 4a
x2 = – 4ay (0, 0) x = 0 (0, –a) y = a y = –a 4a
Standard form of an equation of Parabola
Ex.1 If focus of a parabola is (3,–4) and directrix is x + y – 2 = 0, then its vertex is (A) (4/15, – 4/13)
(B) (–13/4, –15/4)
(C) (15/2, – 13/2)
(D) (15/4, – 13/4)
Sol. First we find the equation of axis of parabola
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Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
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2. If a plane intersects the cone when it is slanted the same as the side of the cone, (formally, when it is parallel to the slant height), the conic section is a parabola. This is shown below:
3. Parabolas Parabola: the set of points in a plane that are the same distance from a given point called the focus and a given line called the directrix . Directrix The light source is the Focus The cross section of a headlight is an example of a parabola...
5. Directrix Focus d 1 d 1 d 2 d 2 d 3 d 3 Also, notice that the distance from the focus to any point on the parabola is equal to the distance from that point to the directrix... We can determine the coordinates of the focus, and the equation of the directrix, given the equation of the parabola.... Vertex Notice that the vertex is located at the midpoint between the focus and the directrix...
6. Standard Equation of a Parabola: (Vertex at the origin) Equation Focus Directrix x 2 = 4py (0, p) y = –p Equation Focus Directrix y 2 = 4px (p, 0) x = –p (If the x term is squared, the parabola is up or down) (If the y term is squared, the parabola is left or right)
7. Examples: Determine the focus and directrix of the parabola y = 4x 2 : Since x is squared, the parabola goes up or down… Solve for x 2 x 2 = 4py y = 4x 2 4 4 x 2 = 1/4y Solve for p 4p = 1/4 p = 1/16 Focus: (0, p) Directrix: y = –p Focus: (0, 1/16) Directrix: y = –1/16 Let’s see what this parabola looks like...
8. Examples: Determine the focus and directrix of the parabola – 3y 2 – 12x = 0 : Since y is squared, the parabola goes left or right… Solve for y 2 y 2 = 4px – 3y 2 = 12x – 3y 2 = 12x –3 –3 y 2 = –4x Solve for p 4p = –4 p = –1 Focus: (p, 0) Directrix: x = –p Focus: (–1, 0) Directrix: x = 1 Let’s see what this parabola looks like...
9. Examples: Write the standard form of the equation of the parabola with focus at (0, 3) and vertex at the origin. Since the focus is on the y axis,(and vertex at the origin) the parabola goes up or down… x 2 = 4py Since p = 3 , the standard form of the equation is x 2 = 12y Example: Write the standard form of the equation of the parabola with directrix x = –1 and vertex at the origin. Since the directrix is parallel to the y axis,(and vertex at the origin) the parabola goes left or right… y 2 = 4px Since p = 1 , the standard form of the equation is y 2 = 4x
10. Equation Parabola at (h,k) If the vertex of the parabola is at (h,k), the standard equation for the parabola are as summarised below.
11. Opens downwards y=k+p (h,k-p) (x-h) 2 =-4p(y-k) Opens upwards y=k-p (h,k+p) (x-h) 2 =4p(y-k) Opens to the left x=h+p (h-p,k) (y-k) 2 =-4p(x-h) Opens to the right x=h-p (h+p,k) (y-k) 2 =4p(x-h) Shape Directrix Focus Parabola
17. Example 6 A necklace hanging between two fixed points A and B at the same level. The length of the necklace between the two point is 100 cm. The mid point of the necklaceis 8 cm below A and B. Assume that the necklace hangs in the form of parabolic curve, find the equation of the curve.
