Parabolas: 4 different situations for the coefficient a
a>1, 0<a<1, 0>a>-1, and -1>a. These tell you how wide or narrow the parabola is, and whether it opens in a positive or a negative direction.
Two general forms for a parabola, where (h, k) is the vertex:
y=a(x-h)^2 + k (vertical parabola, up or down) and x=a(y-k)^2 + h, (horizontal parabola, left or right).
The value “a” also tells you about the directrix and the focus, since a=1÷(4p). The directrix and focus are both “p” units away from the vertex. Whether that is “p” units up, down, left, or right depends on the problem. Remember, the focus is always inside the parabola, and the directrix is always outside the parabola, so use that to determine how you use p.
The latus rectum is the line parallel to the directrix, going through the focus, connecting two points on the parabola. Its length is equal to 4p, or 1/a.
Remember: when you complete the square for a parabola, you maintain the balance of the equation by taking whatever you added inside the parenthesis to complete the square, multiplying it by the coefficient if there is one, and subtracting it outside the parenthesis, but on the same side of the equation!
Using Zeroes to Graph:
“Zeroes” is the nickname mathematicians use that stands for “the solutions of an equation when it is set equal to zero.” The term “roots” is another alternative name for “zeroes,” or “the solutions of an equation when it is set equal to zero.” If an equation is factorable, you probably want to use “zeroes.” Once you find the zeroes, plot them on the graph, and know that the line of symmetry for the parabola will go exactly between them, and that the x value of the line of symmetry can also be used to find the vertex (the vertex is always on the line of symmetry).
Hyperbolas: Two general forms, where (h, k) is the center:
For both of these equations, we need a third equation: a^2+b^2=c^2
Notice the similarities and differences with these equations and those of ellipses! Here, a^2 is always below the positive part of the equation. a^2 is not necessarily the larger denominator. It is just the denominator for the positive part of the equation. Once you identify a, b, and c, remember: a is the distance from the center to each vertex, either up and down (vertical hyperbola) or left and right (horizontal hyperbola). c is the distance from the center to each focus, either up and down (vertical hyperbola) or left and right (horizontal hyperbola). Then, you use 2b to complete the width or height of the box with 2a, and you draw your asymptotes passing through the corners of the box and through the center. The asymptotes show you where the graph of the hyperbola approaches, but never actually reaches.
Rearranging equations into standard form:
Look for whether or not both the x’s and y’s have second degree terms. If there is a y but no y^2, solve for y and put it into parabola form. Same for an x but no x^2. If there is an x^2 and a y^2 term, group the x’s and set up the problem so that you are ready to complete the square. Remember, you need to factor out the leading coefficient first, and whatever you add to complete the square inside the parenthesis needs to be multiplied by that coefficient outside the parenthesis, because you actually added the term that many times. For hyperbolas, remember that the equation needs to be set equal to 1, like in ellipses.
To tell the difference between an equation for an ellipse and a hyperbola, look for the minus sign. If both the x^2 and y^2 terms are positive, then you will have an ellipse. If one of them is negative, then you will end up with subtraction, and the general form will look like a hyperbola. Once you decide an equation looks like that of an ellipse, you still have to determine whether or not it’s a circle…