### 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$\text{\xe2\u20ac\u2030}x\text{-}$intercepts, are the points at which the parabola crosses the$\text{\xe2\u20ac\u2030}x\text{-}$axis. The$\text{\xe2\u20ac\u2030}y\text{-}$intercept is the point at which the parabola crosses the$\text{\xe2\u20ac\u2030}y\text{-}$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 functionâ€™s minimum or maximum value is given by the$\text{\xe2\u20ac\u2030}y\text{-}$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$\text{\xe2\u20ac\u2030}n\text{\xe2\u20ac\u2030}$
will have at most$\text{\xe2\u20ac\u2030}n\text{\xe2\u20ac\u2030}$
*x-*intercepts and at most$\text{\xe2\u20ac\u2030}n\xe2\u02c6\u20191\text{\xe2\u20ac\u2030}$ 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$\text{\xe2\u20ac\u2030}x\text{-}$intercepts of a polynomial function is to graph the function and identify the points at which the graph crosses the$\text{\xe2\u20ac\u2030}x\text{-}$axis. See Example 5
**.** - The multiplicity of a zero determines how the graph behaves at the$\text{\xe2\u20ac\u2030}x\text{-}$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$\text{\xe2\u20ac\u2030}n\text{\xe2\u20ac\u2030}$
has at most$\text{\xe2\u20ac\u2030}n\xe2\u02c6\u20191\text{\xe2\u20ac\u2030}$
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$\text{\xe2\u20ac\u2030}n\xe2\u02c6\u20191\text{\xe2\u20ac\u2030}$
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$\text{\xe2\u20ac\u2030}f(a)\text{and}f(b)\text{\xe2\u20ac\u2030}$
have opposite signs, then there exists at least one value$\text{\xe2\u20ac\u2030}c\text{\xe2\u20ac\u2030}$
between$\text{\xe2\u20ac\u2030}a\text{\xe2\u20ac\u2030}$
and$\text{\xe2\u20ac\u2030}b\text{\xe2\u20ac\u2030}$
for which$\text{\xe2\u20ac\u2030}f\left(c\right)=0.\text{\xe2\u20ac\u2030}$
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$\text{\xe2\u20ac\u2030}x\xe2\u02c6\u2019k.\text{\xe2\u20ac\u2030}$
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$\text{\xe2\u20ac\u2030}f(k),\text{\xe2\u20ac\u2030}$determine the remainder of the polynomial$\text{\xe2\u20ac\u2030}f(x)\text{\xe2\u20ac\u2030}$when it is divided by$\text{\xe2\u20ac\u2030}x\xe2\u02c6\u2019k.\text{\xe2\u20ac\u2030}$This is known as the Remainder Theorem. See Example 1.
- According to the Factor Theorem,$\text{\xe2\u20ac\u2030}k\text{\xe2\u20ac\u2030}$is a zero of$\text{\xe2\u20ac\u2030}f(x)\text{\xe2\u20ac\u2030}$ if and only if$\text{\xe2\u20ac\u2030}(x\xe2\u02c6\u2019k)\text{\xe2\u20ac\u2030}$ is a factor of$\text{\xe2\u20ac\u2030}f(x).$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$\text{\xe2\u20ac\u2030}(x\xe2\u02c6\u2019c),\text{\xe2\u20ac\u2030}$where$\text{\xe2\u20ac\u2030}c\text{\xe2\u20ac\u2030}$ 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$\text{\xe2\u20ac\u2030}f(\xe2\u02c6\u2019x)\text{\xe2\u20ac\u2030}$ 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$\text{\xe2\u20ac\u2030}f(x)=\frac{1}{x}\text{\xe2\u20ac\u2030}$and$\text{\xe2\u20ac\u2030}f(x)=\frac{1}{{x}^{2}}.\text{\xe2\u20ac\u2030}$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 functionâ€™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$\text{\xe2\u20ac\u2030}x={x}_{1},{x}_{2},\xe2\u20ac\xa6,{x}_{n},\text{\xe2\u20ac\u2030}$vertical asymptotes at$\text{\xe2\u20ac\u2030}x={v}_{1},{v}_{2},\xe2\u20ac\xa6,{v}_{m},\text{\xe2\u20ac\u2030}$and no$\text{\xe2\u20ac\u2030}{x}_{i}=\text{any}{v}_{j},\text{\xe2\u20ac\u2030}$then the function can be written in the form

$$\begin{array}{l}\begin{array}{l}\hfill \\ f(x)=a\frac{{(x\xe2\u02c6\u2019{x}_{1})}^{{p}_{1}}{(x\xe2\u02c6\u2019{x}_{2})}^{{p}_{2}}\xe2\u2039\xaf{(x\xe2\u02c6\u2019{x}_{n})}^{{p}_{n}}}{{(x\xe2\u02c6\u2019{v}_{1})}^{{q}_{1}}{(x\xe2\u02c6\u2019{v}_{2})}^{{q}_{2}}\xe2\u2039\xaf{(x\xe2\u02c6\u2019{v}_{m})}^{{q}_{n}}}\hfill \end{array}\hfill \end{array}$$

See Example 12.

### 5.7 Inverses and Radical Functions

- The inverse of a quadratic function is a square root function.
- If$\text{\xe2\u20ac\u2030}{f}^{\xe2\u02c6\u20191}\text{\xe2\u20ac\u2030}$ is the inverse of a function$\text{\xe2\u20ac\u2030}f,\text{\xe2\u20ac\u2030}$ then$\text{\xe2\u20ac\u2030}f\text{\xe2\u20ac\u2030}$ is the inverse of the function$\text{\xe2\u20ac\u2030}{f}^{\xe2\u02c6\u20191}.\text{\xe2\u20ac\u2030}$ 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.