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Graphing Quadratic Equations

A quadratic equation is a polynomial equation of degree 2 .  The standard form of a quadratic equation is

0 = a x 2 + b x + c

where   a , b and c are all real numbers and a 0 .

In this equation, ( 0 , c ) is the y -intercept of the parabola.

The sign of a determines whether the parabola opens up or down: if a is positive, the parabola opens up, and if a is negative, it opens down.

If a has a high absolute value, the parabola is "skinny"; if it has a low absolute value, the parabola is wide.

Math diagram

Note that the equation for blue parabola has a = 3 , a positive number greater than 1 ; so it is skinny and opens upward. It also has c = 2 , so the y -intercept is 2 .

The equation for the red parabola has a = 1 8 , a negative number close to 0 ; so it is wide, and opens downward. It also has c = 6 , so the y -intercept is 6 .

Even if you know the y -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 : y = a ( x h ) 2 + k

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 ( h , k ) . The value of a is the same as in standard form, and has the same effect on the graph.

Example 1:

Graph the function f ( x ) = 1 2 ( x 1 ) 2 + 2

Here, the equation is in vertex form. The vertex of the parabola is ( 1 , 2 ) . Since a = 1 2 , the parabola opens downwards, and is a bit wide.

Math diagram

When a quadratic equation can be easily written in factored form , you can use this to draw the graph quickly.

Example 2:

Graph the function y = x 2 + x 6 .

This equation can be factored and written as

y = ( x + 3 ) ( x 2 )

Here, we can see immediately that when x equals 3 or 2 , y equals 0 . So ( 3 , 0 ) and ( 2 , 0 ) are the x -intercepts. Since a is positive in this case (the coefficient of x 2 is 1 ), the graph opens upward.

Math diagram

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