Algebra and Trigonometry 2e

# Key Concepts

### Key Concepts

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

### 5.2Power 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 $n n$ will have at most $n n$ x-intercepts and at most $n−1 n−1$ turning points. See Example 8, Example 9, Example 10, Example 11, and Example 12.

### 5.3Graphs 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 $x- x-$ intercepts of a polynomial function is to graph the function and identify the points at which the graph crosses the $x- x-$ axis. See Example 5.
• The multiplicity of a zero determines how the graph behaves at the $x- x-$ 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 $n n$ has at most $n−1 n−1$ 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 $n−1 n−1$ 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 have opposite signs, then there exists at least one value $c c$ between $a a$ and $b b$ for which $f( c )=0. f( c )=0.$ See Example 9.

### 5.4Dividing 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 $x−k. x−k.$ See Example 3, Example 4, and Example 5.
• Polynomial division can be used to solve application problems, including area and volume. See Example 6.

### 5.5Zeros of Polynomial Functions

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

### 5.6Rational Functions

• We can use arrow notation to describe local behavior and end behavior of the toolkit functions $f(x)= 1 x f(x)= 1 x$ and $f(x)= 1 x 2 . f(x)= 1 x 2 .$ 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 $x= x 1 , x 2 ,…, x n , x= x 1 , x 2 ,…, x n ,$ vertical asymptotes at $x= v 1 , v 2 ,…, v m , x= v 1 , v 2 ,…, v m ,$ and no then the function can be written in the form
$f(x)=a (x− x 1 ) p 1 (x− x 2 ) p 2 ⋯ (x− x n ) p n (x− v 1 ) q 1 (x− v 2 ) q 2 ⋯ (x− v m ) q n f(x)=a (x− x 1 ) p 1 (x− x 2 ) p 2 ⋯ (x− x n ) p n (x− v 1 ) q 1 (x− v 2 ) q 2 ⋯ (x− v m ) q n$

See Example 12.

• The inverse of a quadratic function is a square root function.
• If $f −1 f −1$ is the inverse of a function $f, f,$ then $f f$ is the inverse of the function $f −1 . f −1 .$ 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.

### 5.8Modeling 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.
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