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Properties of Quadratic Functions

In this video, we are going to look at properties of quadratic functions. Quadratic functions are written in the form y=ax^2+bx+c. The most basic one is y=x^2. These functions are parabolas and are “U”-shaped. The point where it stops and changes top the other direction is known as the vertex. The axis of symmetry of the parabola passes through the vertex.
The graphing calculator is a helpful tool when it comes to dealing with quadratic functions. After graphing the function, the vertex can sometimes be seen as a point on the table. The axis of symmetry has the equation of the x-coordinate of the vertex. In the function y=x^2+4x-2 the table helped us determine that the vertex lies at (-2,-6).
Without a graphing calculator, we can still find this information.
The formula for the axis of symmetry is written as x=\frac{-b}{2a}
In the function y=x^2+4x-2  a=1, b=4, c=-2 . When these values are substituted into the formula, x=\frac{-(4)}{2(1)} we find that the axis of symmetry is at x=-2.
Since the vertex is a point on the graph, we can use the original function, and the x-value of the axis of symmetry to solve for the point where the vertex lies.
If we substitute -2 for x in this function, we have
which is simplified to
and finally
This means that the vertex lies at the point (-2,-6), which agrees with the same point that we determined from the graphing calculator.

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