Algebra II/Parabola

A parabola is an approximate u-shaped curve in which any point is equidistant from the focus (fixed point) and the directrix (fixed straight line). The standard form of the parabola is $$y = a(x - h)$$2 $$+ k$$. The vertex is found by taking the opposite of the "h" and taking the "k": $$(h, k)$$. The axis of symmetry is $$x = h$$ (opposite of h).

For example:


 * $$y = 4(x - 3)$$2 $$- 7$$

Vertex: (3, -7) Axis of Symmetry: x = 3 Positive or negative?: The "4" in this equation represents whether the graph is going up (positive) or is negative (down). In this case, since we have a positive "4", it is going up (and therefore, positive).

Here are a few tricky ones:


 * $$y = -3(x + 1)$$2

Vertex: (-1, 0) [no presence of a "k", so therefore, a zero takes its place] Axis of Symmetry: x = -1 Positive or negative?: Negative


 * $$y = -x$$2 - 7

Vertex: (0, -7) [no presence of a "h", so therefore, a zero takes its place] Axis of Symmetry: x = 0 Positive or negative?: Negative


 * $$y = x$$2

Vertex: (0, 0) Axis of Symmetry: x = 0 [no presence of a "h", so therefore, a zero takes its place] Positive or negative?: Positive

Quadratic Function → Standard Form [Parabola Equation]
--
 * $$y = x$$2 $$- 4x + 5$$
 * Bring the "5" to the other side, or the "c" (constant term).
 * $$y - 5= x$$2 $$- 4x$$
 * Divide the "4", or the "bx" (linear term), by "2". Then square it and add it to both sides.
 * $$y - 1= (x$$ $$- 2)$$2
 * Bring the constant term to the other side.
 * $$y = (x$$ $$- 2)$$2$$+ 1$$
 * You're finished. This is your answer--now you can figure out the vertex and the AOS. The vertex for this problem is (2, 1) and the AOS is x = 2.
 * $$y = 2x$$2 $$+ 16x - 5$$
 * Bring the constant term to the other side
 * $$y + 5 = 2x$$2 $$+ 16x$$
 * Break down "$$2x$$2 $$+ 16x$$"
 * $$y + 5 = 2(x$$2$$ + 8x + [?])$$
 * Divide the linear team by 2, then square that number, multiply the number by "2" (the 2 infront of the paranthesis) and add it on both sides.
 * $$y + 37 = 2(x$$2$$ + 8x + 16)$$
 * Break down "$$2(x$$2$$ + 8x + 16)$$".
 * $$y + 37 = 2(x + 4)$$2
 * Move the "37" to the other side. Your problem is finished!
 * $$y = 2(x + 4)$$2 $$- 37$$
 * The vertex is (-4, -37), the AOS is x = -4 and the parabola here is positive due to the positive "2".