Algebra and Trigonometry 5

zeros

in a given function, the values of xx at which y=0y=0, also called roots

global minimum

lowest turning point on a graph; f(a)f(a) where f(a)≤f(x)f(a)≤f(x) for all x.

inverse variation

the relationship between two variables in which the product of the variables is a constant

direct variation

the relationship between two variables that are a constant multiple of each other; as one quantity increases, so does the other

leading term

the term containing the highest power of the variable

power function

a function that can be represented in the form f(x)=kxpf(x)=kxp where kk is a constant, the base is a variable, and the exponent, pp , is a constant

global maximum

highest turning point on a graph; f(a)f(a) where f(a)≥f(x)f(a)≥f(x) for all x.

Factor Theorem kk

kk is a zero of polynomial function f(x)f(x) if and only if (x−k)(x−k) is a factor of f(x)

end behavior

the behavior of the graph of a function as the input decreases without bound and increases without bound

leading coefficient

the coefficient of the leading term

standard form of a quadratic function

the function that describes a parabola, written in the form f(x)=a(x−h)2+kf(x)=a(x−h)2+k, where (h,k)(h,k) is the vertex

general form of a quadratic function

the function that describes a parabola, written in the form f(x)=ax2+bx+cf(x)=ax2+bx+c, where a,b,a,b, and cc are real numbers and a≠0.

degree

the highest power of the variable that occurs in a polynomial

turning point

the location at which the graph of a function changes direction

constant of variation

the non-zero value kk that helps define the relationship between variables in direct or inverse variation

multiplicity

the number of times a given factor appears in the factored form of the equation of a polynomial; if a polynomial contains a factor of the form (x−h)p(x−h)p , x=hx=h is a zero of multiplicity p.

vertex

the point at which a parabola changes direction, corresponding to the minimum or maximum value of the quadratic function

Rational Zero Theorem

the possible rational zeros of a polynomial function have the form pqpq where pp is a factor of the constant term and qq is a factor of the leading coefficient.

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.

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.

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

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

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.

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.

Linear Factorization Theorem

allowing for multiplicities, a polynomial function will have the same number of factors as its degree, and each factor will be in the form (x−c)(x−c) , where cc is a complex number

vertex form of a quadratic function

another name for the standard form of a quadratic function

term of a polynomial function

any aixiaixi of a polynomial function in the form f(x)=anxn+...+a2x2+a1x+a0

invertible function

any function that has an inverse function

Intermediate Value Theorem

for two numbers aa and bb in the domain of f,f, if a<ba<b and f(a)≠f(b),f(a)≠f(b), then the function ff takes on every value between f(a)f(a) and f(b)f(b) ; specifically, when a polynomial function changes from a negative value to a positive value, the function must cross the x-x- axis

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 nn has at most n−1n−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−1n−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)f(a)andf(b) have opposite signs, then there exists at least one value cc between aa and bb for which f(c)=0.f(c)=0. See Example 9.

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 nn has at most n−1n−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−1n−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)f(a)andf(b) have opposite signs, then there exists at least one value cc between aa and bb for which f(c)=0.f(c)=0. See Example 9.

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.

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.

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

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

Division Algorithm

given a polynomial dividend f(x)f(x) and a non-zero polynomial divisor d(x)d(x) where the degree of d(x)d(x) is less than or equal to the degree of f(x)f(x) , there exist unique polynomials q(x)q(x) and r(x)r(x) such that f(x)=d(x)q(x)+r(x)f(x)=d(x)q(x)+r(x) where q(x)q(x) is the quotient and r(x)r(x) is the remainder. The remainder is either equal to zero or has degree strictly less than d(x).

Remainder Theorem

if a polynomial f(x)f(x) is divided by x−kx−k , then the remainder is equal to the value f(k)

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, kk 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 cc 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.

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, kk 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 cc 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.

We can use arrow notation to describe local behavior and end behavior of the toolkit functions f(x)=1xf(x)=1x and f(x)=1x2.f(x)=1x2. 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=x1,x2,...,xn,x=x1,x2,...,xn, vertical asymptotes at x=v1,v2,...,vm,x=v1,v2,...,vm, and no xi=any vj,xi=any vj, then the function can be written in the form f(x)=a(x−x1)p1(x−x2)p2⋯(x−xn)pn(x−v1)q1(x−v2)q2⋯(x−vm)qnf(x)=a(x−x1)p1(x−x2)p2⋯(x−xn)pn(x−v1)q1(x−v2)q2⋯(x−vm)qn

We can use arrow notation to describe local behavior and end behavior of the toolkit functions f(x)=1xf(x)=1x and f(x)=1x2.f(x)=1x2. 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=x1,x2,...,xn,x=x1,x2,...,xn, vertical asymptotes at x=v1,v2,...,vm,x=v1,v2,...,vm, and no xi=any vj,xi=any vj, then the function can be written in the form f(x)=a(x−x1)p1(x−x2)p2⋯(x−xn)pn(x−v1)q1(x−v2)q2⋯(x−vm)qnf(x)=a(x−x1)p1(x−x2)p2⋯(x−xn)pn(x−v1)q1(x−v2)q2⋯(x−vm)qn

rational function

a function that can be written as the ratio of two polynomials

polynomial function

a function that consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power.

continuous function

a function whose graph can be drawn without lifting the pen from the paper because there are no breaks in the graph

smooth curve

a graph with no sharp corners

horizontal asymptote

a horizontal line y=by=b where the graph approaches the line as the inputs increase or decrease without bound.

coefficient

a nonzero real number multiplied by a variable raised to an exponent

Fundamental Theorem of Algebra

a polynomial function with degree greater than 0 has at least one complex zero

joint variation

a relationship where a variable varies directly or inversely with multiple variables

varies inversely

a relationship where one quantity is a constant divided by the other quantity

inversely proportional

a relationship where one quantity is a constant divided by the other quantity; as one quantity increases, the other decreases

varies directly

a relationship where one quantity is a constant multiplied by the other quantity

Descartes' Rule of Signs

a rule that determines the maximum possible numbers of positive and negative real zeros based on the number of sign changes of f(x)f(x) and f(−x)

synthetic division

a shortcut method that can be used to divide a polynomial by a binomial of the form x−k

removable discontinuity

a single point at which a function is undefined that, if filled in, would make the function continuous; it appears as a hole on the graph of a function

axis of symmetry

a vertical line drawn through the vertex of a parabola, that opens up or down, around which the parabola is symmetric; it is defined by x=−b2a.

vertical asymptote

a vertical line x=ax=a where the graph tends toward positive or negative infinity as the inputs approach a

arrow notation

a way to represent symbolically the local and end behavior of a function by using arrows to indicate that an input or output approaches a value

roots

in a given function, the values of xx at which y=0y=0, also called zeros