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  Theory of Quadratic Functions

Basic Definitions of Quadratic Functions:

Polynomial function in one variable of degree n: It is a function with one variable raised to the whole number powers (that is the biggest being n) and with real coefficients.

The standard form is f(x) = anxn + an-1xn-1 + ... + a2x2 + a1x + a0, an ≠ 0

Constant function: It is a polynomial function in one variable of the degree 0.

Polynomial form: f(x) = a0
Standard form: f(x) = c

Linear function: It is a polynomial function in one variable of the degree 1.

Polynomial form: f(x) = a1x + a0
Standard form: f(x) = ax + b

Quadratic function: It is a polynomial function in one variable of the degree 2.

Polynomial form: f(x)= a2x2 + a1x + a0
Standard form 1: f(x) = ax2 + bx + c
Standard form 2: f(x) = a (x-h)2 + k

Cubic function: It is a polynomial function in one variable of the degree 3.

Polynomial form: f(x)= a3x3 + a2x2 + a1x + a0

Quartic function: It is a polynomial function in one variable of the degree 4.

Polynomial form: f(x) = a4x4 + a3x3 + a2x2 + a1x + a0

For powers more than 4, they are generally referred to by their degree-example ‘A 5th degree polynomial’.

Parabola: It is the graph of a quadratic function

Axis of symmetry (for a parabola): It is the line of symmetry via the center of the parabola.

Vertex: It is the intersection of the axis of symmetry and parabola. This will be the minimum point on the graph when a>0 and the maximum point on the graph when a<0.

A new standard form:

The previous standard form for a parabola was written like any of other polynomial, f(x) = ax2 + bx + c, a ≠ 0.

We are going to complete the square and put it to a form where the translations are simply interpreted. This time, rather than dividing via by a, let us factor an a out of x-terms rather.

f(x) = a [x2 + (b/a) x + ? ] + c

Go ahead and take half of the x-coefficient and place it on the next line.

f(x) = a [x + (b/2a) ]2 + ?

The one thing will be kept in mind. Whenever you add the b2/(4a2), you are really multiplying it by a which you factored out, therefore it is really just a b2/(4a). This time, rather than adding it to the both sides of equation, add it and subtract it on similar side of the equation.

f(x) = a [x2 + (b/a) x + b2/(4a2)] + c - b2/(4a)
f(x) = a [x + (b/2a)]2 + (4ac - b2)/(4a)

With a couple of replacements, this can be written in a new standard form.

f(x) = a ( x - h )2 + k

Here, h = -b/(2a) and k = (4ac - b2)/(4a)

Do not worry regarding what k is, however you might wish to memorize the value for h.

The x-coordinate of vertex is -b/(2a). The y-coordinate is what you get whenever you plug-b/(2a) back to the original function for x.

There are three translations comprised here.

a) The y-coordinates have been multiplied by the a. It is the same a that was in the original problem. When a>0, then the parabola opens up and the vertex is at bottom. When a<0, then the parabola opens down and the vertex is at top.

b) There consists a horizontal shift. Rather than the x-coordinate of vertex being at x = 0, it is now at x = h, where h = -b/(2a). As the axis of symmetry passes via the vertex, that signifies that the axis of symmetry is now x = -b/(2a).

c) There consists a vertical shift. The y-coordinate of vertex is now at y = k. This is not worth your time to memorize the formula for vertical shift. It is not that hard, it is -a times the discriminant of quadratic, however it is simpler to find out the x-coordinate, and plug that back to the equation to find out the y-coordinate.

Unless the coefficients are in fact nasty (that is, decimals), you might find out it quicker to complete the square to find out the vertex than to let x = -b/(2a) and then find out the y-coordinate.

However do note that the vertex is now at (h, k) rather of (0,0).

Extrema - Maximum and Minimums:

Absolute Minimum:

When a>0, then the parabola will open-up and the vertex will be the lowest point on graph. As it is lower than all other points, not just such around it, it is an absolute minimum rather than relative minimum. As the coordinates of the vertex are (h, k), the ‘absolute minimum of the function is k whenever x=h’.

Absolute Maximum:

When a<0, then the parabola will open down and the vertex will be the maximum point on the graph. As it is higher than all other points, not just such around it, it is an absolute maximum rather than a relative maximum. As the coordinates of the vertex are (h, k), the ‘absolute maximum of the function is k whenever x=h’.

Note that the proper format for answering a maximum or minimum question is to give the minimum or maximum value (that is the y-coordinate) and where it takes place (that is the x-coordinate).

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