The standard form of a Quadratic Function is ax² +bx+ c
The graphed form of a quadratic equation is always a parabola
To find the vertex of a parabola use the formula:
f(x) = a (x-h)²+k, where h and k are the coordinates for the vertex point
h or x of the vertex in the graphed parabola is also the line of symmetry, as shown in red in the image above
To find points on the parabola simply make an x and y chart by substituting points into the standard form of the quadratic function
Synthetic Division:
An alternate way to divide polynomials without using long division is synthetic division
To use synthetic division take the opposite form of the divisor's real number (cannot be a variable)
Then use every number in front of x from the quotient (in this case 1, 2, -5, & -1), place them as shown in the picture above
Drag the first number (1) down, then multiply it by the divisor, then add the answer with the next number from the quotient polynomial
Again, multiply the answer by the divisor and add it to the next number in the quotient line (in this case -5+8)
Repeat this step until the last number on the quotient line, if the final answer is 0 there are no remainders to the answer
If the number is anything other than 0, it's the remainder of the answer, subtract one x from the dividend and range the numbers from the synthetic division with descending x variables
In this case the final answer would be:
x²+ 4x+3 and 5/x-2
Rational Functions:
Rational Functions are the quotients of polynomial functions
The Domain of a Rational Function are all possible x numbers for the graph of the function
The Range of a rational function are all possible y numbers, range and domain are usually written like coordinates
Another friend of rational functions are asymptotes, meaning the parts of a graph which are not touched by the function, in the image above the asymptotes are 3 & -3
Polynomial Functions (Monotonic, One-to-One, & Horizontal Line Test):
A monotonic line is a linear equation
The one-to-one linear test is to prove the graph is a function and has an inverse
The Horizontal Line test is to prove the graph is a function
Horizontal Line Test & One-to-one:
A one to one line test proves every domain (x) has exactly one range (y), proving it to be a function
A horizontal line test is to prove the inverse of a function a function
If the line touches more than once, the inverse is not a function
Real Zeros Complex Numbers & Fundamental Theorem of Algebra:
States that every polynomial to the nth power has n solutions
A real number is a number which can be found somewhere along the line of a graph, they can either be positive or negative
An imaginary number is any number which is not a rational number, for example a squared root
The real zeros of a polynomial are non-complex numbers which touch the x-axis
To find the real zeros of a polynomial start by finding the factors of the last coefficient
For Example: f(x)= 2x^4- 7x^3- 8x^2+ 14x +8 The factors are: 8/2= ±1,±2,±4,8/ ±1±2
Then divide each numerator by each denominator
For Example: ±1/±1=±1, ±2/±1=±2, ±8/±1=±8 & ±1/±2=±1/2, ±2/±2=±1, ±4/±2=±1/2, ±8/±2=±4 Excluding the repeating numbers the potential zeros are: ±4, ±1/2, ±1, ±8, ±2
Now synthetically divide the polynomial by the potential zero: 4| 2 -7 -8 14 8