Graphs of Quadratic Function
y = x^2
This function's graph is a combination of two parallel curves called a parabola. There is one value of for each positive and negative pair of value of . The graph has a minimum value of 0 at the origin, which is also the turning point.
Due to its nature the graph can be used to find square roots.
y = -x^2
Same as the graph of the above function. However all values for are negative. Also the curves are inverted and moving downwards.
y = ax^2
Same as the first function, however controls the steepness of the curve. The higher its value, the steeper the curve.
y = x^2 +/- a
Same as the first function but the graph is raised or lowered according to the value of
y = (x-a)^2
Displaces the graph units to the right, or to the left if its value is added instead of subtracted.
y = (x-b)^2 + a
A general function for any possible position on the coordinate plane with any possible steepness. Implies the standard form of
ax^2 + bx + c = 0
Describes the points where the curve cuts the x-axis. They can be read from a plotted graph where equals .
Inequalities such as and define all the values of above and below all points where . Therefore they can be solved for all points on the parabola that fall into the regions.
On the other hand general inequalities such as and describe the regions above and below the curve .
Algebraic techniques can be used to solve such equations with any required degree of accuracy.
For example, an equation such as can be put transformed to find the corresponding values of where .
So two valid points of the curve are and .
Solving by Completing the Square
Given the equation
Remove the constant on the right:
Divide throughout by the coefficient of if it is anything but unity
Add a number that produces a complete square on the left side: (Always the square of half the coefficient of )
This factors into:
Solving by Factorization
If the product of two factors is , then at least one of the factors has to be as well.
Therefore factoring an equation can lead to the values of where .
This method can be used to solve equations of higher degrees than .
Given any quadratic
Simultaneous equations of the second degree
A term of is a term of the second degree much like or .
Therefore <?kt xyz ?> is a term of the third degree.
Simultaneous quadratic equations are only solvable in a very narrow range of cases.
One such case is when one of the equations is a linear equation. In this case the approach to solving is to solve the linear equation for one unknown and substitute it into the quadratic equation.
If the resulting equation happens to have a negative value for , it is correct and flipping signs by multiplying by is not legal.
Solving Quadratic Inequalities
Factorization allows to solve these inequalities. Given , results in when factorized.
Since the product of the factors is negative, one factor has to come out negative while the other has to be positive.
In case of they would both have to yield the same sign for the product to be a positive number.
If then On the other hand if then .
For to yield a pair of opposing signs, has to be a value between and .
The solution for a positive product would be either or , that is a value that yields the same sign for both factors.
This method also works for inequalities of higher degrees.
Number lines can come handy when solving these inequalities.