Chapter 2.1-2.5 Review

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

Completing the square

2.2 Polynomial Functions of higher degree

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

3.  http://mathbits.com/MathBits/TeacherResources/PreCalculus/PreCalculusT.html

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