### Key Concepts

## 5.1 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 $x\text{-}$ intercepts, are the points at which the parabola crosses the $x\text{-}$ axis. The $y\text{-}$ intercept is the point at which the parabola crosses the $y\text{-}$ 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 $y\text{-}$ 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.
- The vertex and the intercepts can be identified and interpreted to solve real-world problems. See Example 9.

## 5.2 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.

## 5.3 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 $x\text{-}$ intercepts of a polynomial function is to graph the function and identify the points at which the graph crosses the $x\text{-}$ axis. See Example 5
**.** - The multiplicity of a zero determines how the graph behaves at the $x\text{-}$ 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 $f(a)\text{and}f(b)$
have opposite signs, then there exists at least one value $c$
between $a$
and $b$
for which $f\left(c\right)=0.$
See Example 9
**.**

## 5.4 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
**.**

## 5.5 Zeros of Polynomial Functions

- To find $f(k),$ determine the remainder of the polynomial $f(x)$ when it is divided by $x-k.$ This is known as the Remainder Theorem. See Example 1.
- According to the Factor Theorem, $k$ is a zero of $f(x)$ if and only if $(x-k)$ is a factor of $f(x).$ See Example 2.
- According to the Rational Zero Theorem, 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.

## 5.6 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 ${x}_{i}=\text{any}{v}_{j},$ then the function can be written in the form

$$\begin{array}{l}\begin{array}{l}\hfill \\ f(x)=a\frac{{(x-{x}_{1})}^{{p}_{1}}{(x-{x}_{2})}^{{p}_{2}}\cdots {(x-{x}_{n})}^{{p}_{n}}}{{(x-{v}_{1})}^{{q}_{1}}{(x-{v}_{2})}^{{q}_{2}}\cdots {(x-{v}_{m})}^{{q}_{n}}}\hfill \end{array}\hfill \end{array}$$

See Example 12.

## 5.7 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.

## 5.8 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.