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 ${\displaystyle y=a(x-h)}$2 ${\displaystyle +k}$. The vertex is found by taking the opposite of the "h" and taking the "k": ${\displaystyle (h,k)}$. The axis of symmetry is ${\displaystyle x=h}$ (opposite of h).

For example:

• ${\displaystyle y=4(x-3)}$2 ${\displaystyle -7}$

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:

• ${\displaystyle 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

• ${\displaystyle 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

• ${\displaystyle 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]

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

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