Upwards and Downwards Parabolas
When graphing quadratic relationships, we can describe the curve on the coordinate plane as a parabola. Parabolas open either upwards or downwards, creating a U-shape curve (in the case of an upward parabola) or an upside down U-shape curve (in the case of a downwards parabola).
One form of a quadratic equation is y=ax^{2} + bx + c (this is called standard form). The value of "a" dictates whether the parabola will open up or down. If a is a positive value, the parabola will open upwards. If a is a negative value, the parabola will open downwards.
The Vertex of a Parabola
Given the shape of a parabola, there will always be either a low point or a high point to the curve. This point is referred to as the parabola's vertex. An interesting characteristic of the vertex is that it lies on an invisible line of symmetry. This means that parabolas are symmetrical - we can imagine reflecting one side of the parabola about the line of reflection, and it will match up with points on the other side. This line of reflection is known as the axis of symmetry to the parabola.
Vertex (of a parabola): the minimum or maximum point of a parabola located on the axis of symmetry
Axis of Symmetry: a line of reflection passing through the vertex of a parabola; in up and down parabolas, it is a vertical line.
There are a few ways to find the vertex of a parabola from its equation. When the equation of a parabola is given in standard form, we use the values a and b (the coefficients of the x-squared and x-terms) to calculate the x–coordinate of the vertex. This also describes the equation to the axis of symmetry. Here is the formula we use:
Once we find the value for x, we plug that into the equation and find the associated y-value. This gives us the coordinates of the parabola's vertex.
Now we can use the value of 3/4, or 0.75, for x and solve for y to find the y–coordinate.
The vertex to is located at the point
Vertex Form of a Quadratic Equation
Finding the coordinates to a parabola's vertex is much easier if the equation in given in a different format, specifically the vertex form.
We don't call it vertex form for nothing. The variables h and k represent the x– and y–coordinates of the vertex. So there is virtually no calculations needed in order to identify the vertex. The only tricky thing is to remember that in general, there is a minus sign between x and h, so we need to be mindful of the sign.
It is easy to mistake the sign of h when finding the x–coordinate of the vertex. Because the general formula subtracts h from x, if we see a plus sign in the specific equation, our h value is actually negative.
a line of reflection passing through the vertex of a parabola; in up and down parabolas, it is a vertical line.
the maximum or minimum point of a parabola located on the axis of symmetry