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.

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• $y=2x$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".