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College Algebra

Key Concepts

College AlgebraKey Concepts

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- 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.

5.2 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 n will have at most n n x-intercepts and at most n1 n1 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- 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 n1 n1 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 n1 n1 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) and f(b) 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.

5.4 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 xk. xk. 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), f(k), determine the remainder of the polynomial f(x) f(x) when it is divided by xk. xk. 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 (xk) (xk) 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 (xc), (xc), 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.

5.6 Rational 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 x i =any  v j , x i =any  v j , 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.

5.7 Inverses and Radical Functions

  • 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.

5.8 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.
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