(no subject)
"Parabolic Reasoning:" a Lesson in Parabolas:
Definition:
A parabola is the locus (set) of points for which the distance from the focus equals the distance to the directrix. This can be more easily explained by a picture.
Translations and Transformations:
There are four basic operations in translating (shifting up, down, or sideways or flipping) or transforming (stretching or compressing) a graph. These can be easily remembered by the use of four parameters A, B, C, and D.
Consider the following polynomial:
y = A(Bx + C)^2 + D
We begin with the graph y = x^2 which also (by "completing the square") equals 1(1x + 0)^2 + 0. We know that the graph of this function has it's vertex (or global minimum) at the origin and opens upward. This graph has the parameters A = 1, B = 1, C = 0, D = 0.
Each of the parameters affects the graph of y = x^2 in a certain way. Let's look at each individually and then work with them in combination.
A determines the direction in which the parabola opens. If A is positive, the graph opens upwards; if it is negative, the graph will open downward. A can be made to be 1 or -1 for simplicity by algebra. If you want to flip a parabola, just change the sign of A.
B determines the "steepness" of the parabola. For example, B = 2 makes the graph of y(x) increase twice as fast. B = 1/3 makes y(x) increase only 1/3 as fast, and so the graph is less steep.
C controls horizontal translation of the graph. For example, y = x^2 + 2x + 2 equals (x + 1)^2 + 1 so in this case C = 1. For values of C, the graph is shifted -C units. In the example, the graph y = x^2 would be shifted one unit to the left by C.
D controls vertical translation of the graph. In our example (x + 1)^2 + 1, the graph of y = x^2 would be shifted up one unit by D.
The easiest way to deal with these parameters is to work with them one at a time. Remember, A is up or down (or left or right), B is steepness, C is right and left, and D is up and down. For graphs of the form x = y^2, the same rules can be used, but some of the directions change. A controls if the parabola opens left or right (to the right if A is positive), B controls the steepness just as before, C shifts the graph up or down. If C is positive, the graph shifts downward. Finally, D shifts the graph left or right - if D is positive, the graph shifts to the right.
Relating the VERTEX, FOCUS, and DIRECTRIX:
The vertex has coordinates (h,k). The vertex can be found from the equation for a parabola by the method of completing the square, and making the appropriate translation.
EXAMPLE:
y^2 + 2y + 2 = x Completing the Square gives: (y + 1)^2 + 1 = x
This gives a horizontal translation of the graph y = x^2 of one to the left and one upward. This corresponds to shifting the vertex of the graph of y = x^2 by the same amount, hence the vertex is (-1,1).
To find the focus, we use the equation: 4p(y-k) = (x-h)^2 where p is the distance to the focus, the vertex is (h,k), and the focus is (x,y).
The directrix is the line perpendicular to the axis of symmetry (the line through the vertex) through the point the same distance from the vertex as the focus, that is -p.
Hopefully that will get you started.
Love,
Nickolas
Definition:
A parabola is the locus (set) of points for which the distance from the focus equals the distance to the directrix. This can be more easily explained by a picture.
Translations and Transformations:
There are four basic operations in translating (shifting up, down, or sideways or flipping) or transforming (stretching or compressing) a graph. These can be easily remembered by the use of four parameters A, B, C, and D.
Consider the following polynomial:
y = A(Bx + C)^2 + D
We begin with the graph y = x^2 which also (by "completing the square") equals 1(1x + 0)^2 + 0. We know that the graph of this function has it's vertex (or global minimum) at the origin and opens upward. This graph has the parameters A = 1, B = 1, C = 0, D = 0.
Each of the parameters affects the graph of y = x^2 in a certain way. Let's look at each individually and then work with them in combination.
A determines the direction in which the parabola opens. If A is positive, the graph opens upwards; if it is negative, the graph will open downward. A can be made to be 1 or -1 for simplicity by algebra. If you want to flip a parabola, just change the sign of A.
B determines the "steepness" of the parabola. For example, B = 2 makes the graph of y(x) increase twice as fast. B = 1/3 makes y(x) increase only 1/3 as fast, and so the graph is less steep.
C controls horizontal translation of the graph. For example, y = x^2 + 2x + 2 equals (x + 1)^2 + 1 so in this case C = 1. For values of C, the graph is shifted -C units. In the example, the graph y = x^2 would be shifted one unit to the left by C.
D controls vertical translation of the graph. In our example (x + 1)^2 + 1, the graph of y = x^2 would be shifted up one unit by D.
The easiest way to deal with these parameters is to work with them one at a time. Remember, A is up or down (or left or right), B is steepness, C is right and left, and D is up and down. For graphs of the form x = y^2, the same rules can be used, but some of the directions change. A controls if the parabola opens left or right (to the right if A is positive), B controls the steepness just as before, C shifts the graph up or down. If C is positive, the graph shifts downward. Finally, D shifts the graph left or right - if D is positive, the graph shifts to the right.
Relating the VERTEX, FOCUS, and DIRECTRIX:
The vertex has coordinates (h,k). The vertex can be found from the equation for a parabola by the method of completing the square, and making the appropriate translation.
EXAMPLE:
y^2 + 2y + 2 = x Completing the Square gives: (y + 1)^2 + 1 = x
This gives a horizontal translation of the graph y = x^2 of one to the left and one upward. This corresponds to shifting the vertex of the graph of y = x^2 by the same amount, hence the vertex is (-1,1).
To find the focus, we use the equation: 4p(y-k) = (x-h)^2 where p is the distance to the focus, the vertex is (h,k), and the focus is (x,y).
The directrix is the line perpendicular to the axis of symmetry (the line through the vertex) through the point the same distance from the vertex as the focus, that is -p.
Hopefully that will get you started.
Love,
Nickolas