Quadratic Functions

Definition :

A Quadratic Function may be recognized as a function of x is given by:

f(x) = ax² + b x + c

Where a, the leading coefficient does not = 0 and a,b, and c are real numbers. Term "c" represents the constant of the function.

When graphed the function is a parabola. When discussing parabolas there are a few things to keep in mind.

• All parabolas are symmetric with respect to a line called the axis of symmetry, or the axis of the parabola.
• The point where the axis intersects the parabola is the vertex of the parabola. The vertex is also the highest or lowest point of the functin.
• If the leading coeffiicent is greater than 0, then we can describe the parabola as, concave up because it is rising from left to right. The vertex may be refered to the minimum in this case.
• If the leading coeffiicent is less than 0, then we can describe the parabola as, concave down because it is falling from left to right. The vertex may be refered to the Maximum in this case.
• A Quadratic function's first degree is to the second power.

Sketching Quadratic Functions:

Sketch:

ƒ(x)= 3x2	g(x)= 1/2(x2)

Since these are simple polynomials, all that is required in order to sketch them is a table of worked out values. This can be acheieved by working out the values (substituting) or by using your graphing calculator.

• For a more in depth graphing procedure, you may wish to find the domain and range.

• When Graphed, you will notice that the output of ƒ(x)= 3x² stretches by a factor of 3 and is more narrow.
• Also, when Graphed, you will notice that the output of g(x)= 1/2(x²) shrinks by a factor of (1/2) and is more wide.

Finally shifts may be identified by:

The Standard Form of a Quadratic Function:

When a quadratic function is given as :

f(x) = a(x -h)² + k , a must not equal 0

This function is in standard form. Standard form gives us the vertex of a parabola. Its vertex is given as (h,k). If the leading coefficient is greater than 0, then it opens upward, if it is less than zero, it is downward.

Here is an example of how to solve a quadratic function into its Standard Form:

ƒ ( x) = 2x² + 8x + 7

The Quadratic Equation:

When finding the value of x when the quadratic function is equal to zero you can use the quadratic formula to solve for x :

You can use this formula if you can factor out for F.O.I.L. or want to obtain the value quickly.

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