3.1 - Quadratic Functions

Definitions

Polynomial function in one variable of degree n: A function with one variable raised to whole number powers (the largest being n) and with real coefficients.; The standard form is f(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₂x² + a₁x + a₀, a_n≠0
Constant function: A polynomial function in one variable of degree 0.; Polynomial form: f(x)=a₀; Standard form: f(x) = c
Linear function: A polynomial function in one variable of degree 1.; Polynomial form: f(x)= a₁x + a₀; Standard form: f(x) = ax + b
Quadratic function: A polynomial function in one variable of 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: A polynomial function in one variable of degree 3.; Polynomial form: f(x)= a₃x³ + a₂x² + a₁x + a₀
Quartic function: A polynomial function in one variable of degree 4.; Polynomial form: f(x)= a₄x⁴ + a₃x³ + a₂x² + a₁x + a₀; For powers higher than 4, they are usually just referred to by their degree - example "A 5^th degree polynomial"
Parabola: The graph of a quadratic function
Axis of symmetry (for a parabola): The line of symmetry through the center of the parabola
Vertex: The intersection of the axis of symmetry and the parabola. It will be the minimum point on the graph if a>0 and the maximum point on the graph if a<0.

A new "standard form"

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

We're going to complete the square and place it into a form where the translations are easily interpreted. This time, instead of dividing through by a, let's factor an a out of the x-terms instead.

f(x) = a [ x² + (b/a) x + ? ] + c

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

f(x) = a [ x + (b/2a) ]² + ?

One thing to be careful of here. When you add the b²/(4a²), you are really multiplying it by the a that you factored out, so it is really just a b²/(4a). This time, instead of adding it to both sides of the equation, add it and subtract it on the same 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 substitutions, this can be written in the new standard form.

f(x) = a ( x - h )² + k

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

Do not worry about what k is, but you might want to memorize the value for h.

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

There are three translations involved here.

The y-coordinates have been multiplied by a. This is the same a that was in the original problem. If a>0, then the parabola opens up and the vertex is at the bottom. If a<0, then the parabola opens down and the vertex is at the top.
There has been a horizontal shift. Instead of the x-coordinate of the vertex being at x=0, it is now at x=h, where h=-b/(2a). Since the axis of symmetry passes through the vertex, that means that the axis of symmetry is now x=-b/(2a).
There has been a vertical shift. The y-coordinate of the vertex is now at y=k. It is not worth your time to memorize the formula for the vertical shift. It isn't that hard, it is -a times the discriminant of the quadratic, but it is easier to find the x-coordinate, and plug that back into the equation to find the y-coordinate.

Unless the coefficients are really nasty (ie, decimals), you may find it quicker to complete the square to find the vertex than to let x=-b/(2a) and then find the y-coordinate.

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

Extrema - Maximum and Minimums

Absolute Minimum: If a>0, then the parabola will open up and the vertex will be the lowest point on the graph. Since it is lower than all other points, not just those around it, it is an absolute minimum instead of a relative minimum. Since the coordinates of the vertex are (h,k), the "absolute minimum of the function is k when x=h."
Absolute Maximum: If a<0, then the parabola will open down and the vertex will be the highest point on the graph. Since it is higher than all other points, not just those around it, it is an absolute maximum instead of a relative maximum. Since the coordinates of the vertex are (h,k), the "absolute maximum of the function is k when x=h."

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