# Quadratics

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A quadratic equation is a second degree ploynomial. examples are y = x^{2}(This is one of what we call parent functions)

y = 4x^{2} +3x + 4 (4 is the y intercept)

- The generalized statement for quadratics is ax
^{2} + bx + c = y a ≠ 0

- Quadratics open upward if a > 0 and opens downward if a < 0

- The graph of a quadratic is a parabola and it has a vertex.

- If the parabola opens upward the vertex is called the minimum.

- If the parabola opens downward the vertex is called the maximum.

- The vertex form of a quadratic is y = a(x - h)
^{2} + k

- (h, k) are the coordinates of the quadratic's vertex.

- The graph of a quadratic viewed left to right increases on one side of the vertex and decreases on the other side.

- The h coordinate is the x value and is parallel to the vertical axis. This x value is called the axis of symmetry,

Most quadratics are in the form f(x) = ax^{2} + bx + c For example y =3x^{2} + 6x - 4

This is not in the vertex formate and it must be altered to change it to that format.

This is done by using the process called completing the square in the following way:
- y =3x
^{2} + 6x - 4
- y =3(x
^{2} + 2x ) - 4
- y =3(x
^{2} + 2x + 1^{2} ) - 4 - 3
- y =3(x
^{2} + 1) - 7
Therefore the vertex is (-1,-7)

Will this graph open upward or downward?

What is the axis of symmetry?

What is the y intercept?

answer

Try this example:

y = x^{2} + 4x + 5

Will this graph open upward or downward?

What is the axis of symmetry?

What is the y intercept?

What is the vertex?

answer

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