Graphing Quadratic Equations
where and are all real numbers and .
In this equation, is the -intercept of the parabola.
The sign of determines whether the parabola opens up or down: if is positive, the parabola opens up, and if is negative, it opens down.
If has a high absolute value, the parabola is "skinny"; if it has a low absolute value, the parabola is wide.
Note that the equation for blue parabola has , a positive number greater than ; so it is skinny and opens upward. It also has , so the -intercept is .
The equation for the red parabola has , a negative number close to ; so it is wide, and opens downward. It also has , so the -intercept is .
Even if you know the -intercept, it's not always easy to sketch the graph of a parabola written in standard form. You can use a table of values, OR you convert the equation to another form, such as:
Vertex Form :
This form of the equation for a quadratic function is called vertex form , because we can easily read the vertex of the parabola: the point . The value of is the same as in standard form, and has the same effect on the graph.
Graph the function
Here, the equation is in vertex form. The vertex of the parabola is . Since , the parabola opens downwards, and is a bit wide.
When a quadratic equation can be easily written in factored form , you can use this to draw the graph quickly.
Graph the function .
This equation can be factored and written as
Here, we can see immediately that when equals or , equals . So and are the -intercepts. Since is positive in this case (the coefficient of is ), the graph opens upward.