2.1 Quadratic Functions -let a, b, and c be real numbers with a not equal to 0
- When a>0, the parabola opens up and the vertex is the low point
When a<0, the parabola opens down and the vertex is the high point
Transforming Graphs
- Example 1:
- When compared to its parent graph, every output shrinks by a factor of 1/4
Finding a polynomial function with given zeros: -2,-2,1/2,10
2.3 Real zeros
Review Synthetic Division***************
Rational Zero Test
Possible Rational zeros= Factors of constant term/Factors of leading coefficient
Possible Rational zeros= +-3, +-1/+-2, +-1
Possible Rational Zeros= +-1, +-3, +-1/2, +-3/2
2.4 Complex numbers
In the Standard form a+bi, the real number a is called the real part of the complex number and the number bi is called the imaginary part of the complex number.
sum: (a+bi)+(c+di)= (a+c)+(b+d)i
difference: (a+bi)-(c+di)= (a-c)+(b-d)i
Conjugates of complex numbers
Conjugate of a+bi is a-bi
2.5 The fundamental theorm of Algebra
If f(x) is a polynomial of degree n, where n>0, f has at least one zero in the complex number system.
ex: The first-degree polynomial f(x)=x+8 has exactly one zero. This zero is x=-8
ex: The second-degree polynomial function f(x)=(x+9)(x-2) has exactly two zeros. These zeros are x=2 and x=-9
Complex zeros occur in conjugate pairs:
If a+bi, where b cannot equal 0, is a zero of a function, the conjugate a-bi is also a zero of the function.
ex: If -10i is a zero, you know that 10i is also a zero of the polynomial function.
Finding the zeros of a polynomial function:
step 1: Use the Rational Zero Test to determine possible zeros
step 2: Use Synthetic Division to find the real zeros
step 3: Use the Quadratic Formula to determine the remaining zeros
Resources for additional help:
1. Khanacademy.org
2. Purplemath.com
1.
