Jay Abramson

Key Concepts

3.1 Complex Numbers

The square root of any negative number can be written as a multiple of $i .$ See Example 1.
To plot a complex number, we use two number lines, crossed to form the complex plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis. See Example 2.
Complex numbers can be added and subtracted by combining the real parts and combining the imaginary parts. See Example 3.
Complex numbers can be multiplied and divided.
To multiply complex numbers, distribute just as with polynomials. See Example 4, Example 5, and Example 8.
To divide complex numbers, multiply both the numerator and denominator by the complex conjugate of the denominator to eliminate the complex number from the denominator. See Example 6, Example 7, and Example 9.
The powers of $i$ are cyclic, repeating every fourth one. See Example 10.

3.2 Quadratic Functions

A polynomial function of degree two is called a quadratic function.
The graph of a quadratic function is a parabola. A parabola is a U-shaped curve that can open either up or down.
The axis of symmetry is the vertical line passing through the vertex. The zeros, or $x -$ intercepts, are the points at which the parabola crosses the $x -$ axis. The $y -$ intercept is the point at which the parabola crosses the $y -$ axis. See Example 1, Example 7, and Example 8.
Quadratic functions are often written in general form. Standard or vertex form is useful to easily identify the vertex of a parabola. Either form can be written from a graph. See Example 2.
The vertex can be found from an equation representing a quadratic function. See Example 3.
The domain of a quadratic function is all real numbers. The range varies with the function. See Example 4.
A quadratic function’s minimum or maximum value is given by the $y -$ value of the vertex.
The minimum or maximum value of a quadratic function can be used to determine the range of the function and to solve many kinds of real-world problems, including problems involving area and revenue. See Example 5 and Example 6.
Some quadratic equations must be solved by using the quadratic formula. See Example 9.
The vertex and the intercepts can be identified and interpreted to solve real-world problems. See Example 10.

3.3 Power Functions and Polynomial Functions

A power function is a variable base raised to a number power. See Example 1.
The behavior of a graph as the input decreases beyond bound and increases beyond bound is called the end behavior.
The end behavior depends on whether the power is even or odd. See Example 2 and Example 3.
A polynomial function is the sum of terms, each of which consists of a transformed power function with positive whole number power. See Example 4.
The degree of a polynomial function is the highest power of the variable that occurs in a polynomial. The term containing the highest power of the variable is called the leading term. The coefficient of the leading term is called the leading coefficient. See Example 5.
The end behavior of a polynomial function is the same as the end behavior of the power function represented by the leading term of the function. See Example 6 and Example 7.
A polynomial of degree $n$ will have at most $n$ x-intercepts and at most $n - 1$ turning points. See Example 8, Example 9, Example 10, Example 11, and Example 12.

3.4 Graphs of Polynomial Functions

Polynomial functions of degree 2 or more are smooth, continuous functions. See Example 1.
To find the zeros of a polynomial function, if it can be factored, factor the function and set each factor equal to zero. See Example 2, Example 3, and Example 4.
Another way to find the $x -$ intercepts of a polynomial function is to graph the function and identify the points at which the graph crosses the $x -$ axis. See Example 5.
The multiplicity of a zero determines how the graph behaves at the $x -$ intercepts. See Example 6.
The graph of a polynomial will cross the horizontal axis at a zero with odd multiplicity.
The graph of a polynomial will touch the horizontal axis at a zero with even multiplicity.
The end behavior of a polynomial function depends on the leading term.
The graph of a polynomial function changes direction at its turning points.
A polynomial function of degree $n$ has at most $n - 1$ turning points. See Example 7.
To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that the final graph has at most $n - 1$ turning points. See Example 8 and Example 10.
Graphing a polynomial function helps to estimate local and global extremas. See Example 11.
The Intermediate Value Theorem tells us that if $f (a) and f (b)$ have opposite signs, then there exists at least one value $c$ between $a$ and $b$ for which $f (c) = 0.$ See Example 9.

3.5 Dividing Polynomials

Polynomial long division can be used to divide a polynomial by any polynomial with equal or lower degree. See Example 1 and Example 2.
The Division Algorithm tells us that a polynomial dividend can be written as the product of the divisor and the quotient added to the remainder.
Synthetic division is a shortcut that can be used to divide a polynomial by a binomial in the form $x - k .$ See Example 3, Example 4, and Example 5.
Polynomial division can be used to solve application problems, including area and volume. See Example 6.

3.6 Zeros of Polynomial Functions

To find $f (k),$ determine the remainder of the polynomial $f (x)$ when it is divided by $x - k .$ See Example 1.
$k$ is a zero of $f (x)$ if and only if $(x - k)$ is a factor of $f (x) .$ See Example 2.
Each rational zero of a polynomial function with integer coefficients will be equal to a factor of the constant term divided by a factor of the leading coefficient. See Example 3 and Example 4.
When the leading coefficient is 1, the possible rational zeros are the factors of the constant term.
Synthetic division can be used to find the zeros of a polynomial function. See Example 5.
According to the Fundamental Theorem, every polynomial function has at least one complex zero. See Example 6.
Every polynomial function with degree greater than 0 has at least one complex zero.
Allowing for multiplicities, a polynomial function will have the same number of factors as its degree. Each factor will be in the form $(x - c),$ where $c$ is a complex number. See Example 7.
The number of positive real zeros of a polynomial function is either the number of sign changes of the function or less than the number of sign changes by an even integer.
The number of negative real zeros of a polynomial function is either the number of sign changes of $f (- x)$ or less than the number of sign changes by an even integer. See Example 8.
Polynomial equations model many real-world scenarios. Solving the equations is easiest done by synthetic division. See Example 9.

3.7 Rational Functions

We can use arrow notation to describe local behavior and end behavior of the toolkit functions $f (x) = \frac{1}{x}$ and $f (x) = \frac{1}{x^{2}} .$ See Example 1.
A function that levels off at a horizontal value has a horizontal asymptote. A function can have more than one vertical asymptote. See Example 2.
Application problems involving rates and concentrations often involve rational functions. See Example 3.
The domain of a rational function includes all real numbers except those that cause the denominator to equal zero. See Example 4.
The vertical asymptotes of a rational function will occur where the denominator of the function is equal to zero and the numerator is not zero. See Example 5.
A removable discontinuity might occur in the graph of a rational function if an input causes both numerator and denominator to be zero. See Example 6.
A rational function’s end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions. See Example 7, Example 8, Example 9, and Example 10.
Graph rational functions by finding the intercepts, behavior at the intercepts and asymptotes, and end behavior. See Example 11.
If a rational function has x-intercepts at $x = x_{1}, x_{2}, \dots, x_{n},$ vertical asymptotes at $x = v_{1}, v_{2}, \dots, v_{m},$ and no $x_{i} = any v_{j},$ then the function can be written in the form
$\begin{array}{l} \begin{array}{l} f (x) = a \frac{{(x - x_{1})}^{p_{1}} {(x - x_{2})}^{p_{2}} \dots {(x - x_{n})}^{p_{n}}}{{(x - v_{1})}^{q_{1}} {(x - v_{2})}^{q_{2}} \dots {(x - v_{m})}^{q_{n}}} \end{array} \end{array}$
See Example 12.

3.8 Inverses and Radical Functions

The inverse of a quadratic function is a square root function.
If $f^{- 1}$ is the inverse of a function $f,$ then $f$ is the inverse of the function $f^{- 1} .$ See Example 1.
While it is not possible to find an inverse of most polynomial functions, some basic polynomials are invertible. See Example 2.
To find the inverse of certain functions, we must restrict the function to a domain on which it will be one-to-one. See Example 3 and Example 4.
When finding the inverse of a radical function, we need a restriction on the domain of the answer. See Example 5 and Example 7.
Inverse and radical and functions can be used to solve application problems. See Example 6 and Example 8.

3.9 Modeling Using Variation

A relationship where one quantity is a constant multiplied by another quantity is called direct variation. See Example 1.
Two variables that are directly proportional to one another will have a constant ratio.
A relationship where one quantity is a constant divided by another quantity is called inverse variation. See Example 2.
Two variables that are inversely proportional to one another will have a constant multiple. See Example 3.
In many problems, a variable varies directly or inversely with multiple variables. We call this type of relationship joint variation. See Example 4.