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]

$y=x$ $y=x$ ² $-4x+5$ $-4x+5$
- Bring the "5" to the other side, or the "c" (constant term).
$y-5=x$ $y-5=x$ ² $-4x$ $-4x$
- Divide the "4", or the "bx" (linear term), by "2". Then square it and add it to both sides.
$y-1=(x$ $y-1=(x$ $-2)$ $-2)$ ²
- Bring the constant term to the other side.
$y=(x$ $y=(x$ $-2)$ $-2)$ ² $+1$ $+1$
- 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.

$y=2x$ $y=2x$ ² $+16x-5$ $+16x-5$
- Bring the constant term to the other side
$y+5=2x$ $y+5=2x$ ² $+16x$ $+16x$
- Break down " $2x$ ² $+16x$ "
$y+5=2(x$ $y+5=2(x$ ² $+8x+[?])$ $+8x+[?])$
- 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.
$y+37=2(x$ $y+37=2(x$ ² $+8x+16)$ $+8x+16)$
- Break down " $2(x$ ² $+8x+16)$ ".
$y+37=2(x+4)$ $y+37=2(x+4)$ ²
- Move the "37" to the other side. Your problem is finished!
$y=2(x+4)$ $y=2(x+4)$ ² $-37$ $-37$
- The vertex is (-4, -37), the AOS is x = -4 and the parabola here is positive due to the positive "2".

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