Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page 1 of 3 Topic 5: Quadratic Functions (Part 1) Definition: A quadratic function is a function which can be written as f x ax bx c, where a 0. Properties of Quadratic Functions 1. The graph of a quadratic function is a parabola. If a is positive, the parabola opens up. If a is negative, the parabola opens down. 1. Parabolas have a turning point called a vertex. The general form of a quadratic function provides clues about the vertex, axis of symmetry, and extreme value but requires investigation in order to find them. Thus, a potentially more useful form of a quadratic function might provide direct information about these properties. Compare the graph of a basic squaring function. x f x. Parabolas have an axis of symmetry which is a vertical line passing through the vertex. 4. Quadratic functions have exactly one increasing interval and exactly one decreasing interval. 5. The vertex is an extremum. If the parabola opens up, the function value at the vertex is a minimum value. If it opens down, the function value is a maximum value. To the graph of a transformed squaring function. x f x 4
Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page of 3 Vertex Form of a Quadratic Function (also called Standard Form): f x a x h k If a function is written in vertex form, how can we find the properties of parabola? 1. Direction of Opening: If a > 0, the parabola opens up. If a < 0, the parabola opens down.. Vertex of the Parabola: Vertex is at (h, k). 3. Axis of Symmetry. The axis of symmetry is x = h. 4. Increasing/Decreasing: If a > 0, f decreases over (, h) and increases over (h, ). If a < 0. f increases over (, h) and decreases over (h, ). 5. Extreme function value: If the parabola opens up, f(h) = k is the minimum value of f. If it opens down, f(h) = k is the maximum value of f. By applying the completing the square technique, it is possible to rewrite a quadratic function into vertex form. Steps to put a quadratic function into vertex form: 1. Group the x² and x terms.. As relevant, factor a from the group. 3a. Complete the square inside the parentheses. 3b. Compensate the function outside the parentheses. 4. Factor and simplify.
Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page 3 of 3 Ex. 1 Rewrite the function in vertex form and then identify the properties associated with the function. Ex. Rewrite the function in vertex form and then identify the properties associated with the function. f x x 6x 7 f x x 8x 10 f opens Vertex: (, ) Axis of Symmetry: Increasing: Decreasing: Extremum is a value of Intercepts: f opens Vertex: (, ) Axis of Symmetry: Increasing: Decreasing: Extremum is a value of Intercepts:
Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page 4 of 3 Ex. 3 Rewrite the function in vertex form and then identify the properties associated with the function. Ex. 4 Rewrite the function in vertex form and then identify the properties associated with the function. f x 3x 6x 1 f x x 4x 7 3 f opens Vertex: (, ) Axis of Symmetry: Increasing: Decreasing: Extremum is a value of Intercepts: f opens Vertex: (, ) Axis of Symmetry: Increasing: Decreasing: Extremum is a value of Intercepts:
Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page 5 of 3 Topic 6: Quadratic Functions (Part ) When given the vertex of a quadratic function, it is possible to uniquely find the rule of the function if you know only one other point on the parabola. Ex. Find the function whose graph is a parabola with vertex 1, and passes through the point,4. Ex. 1 Find the function whose graph is a parabola with vertex 3,4 and passes through the point 1, 8.
Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page 6 of 3 Knowing some information about a quadratic function should allow you to sketch an approximate graph of the function. Ex. 3 Given a quadratic function defined by f x a x h k, draw a graph which satisfies the following properties. a. a > 0, h > 0, k < 0 b. a < 0, axis of symmetry is x = 3, k < 0 Ex. 4 Given a quadratic function defined by f x a x h k, draw a graph which satisfies the following properties. a. a < 0, h < 0, k < 0 b. a > 0, h < 0, minimum value is 3
Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page 7 of 3 Given a quadratic function in general form, another way to find the vertex is by the vertex formula. For a quadratic function vertex is at b b f,. a a f x ax bx c, the Ex. 3 An object is launched from a catapult with the height of the object, x seconds after launch, being estimated at f x 0.41x 5.x 0.19 meters. Use the vertex formula to estimate when the object reaches its maximum height and what the maximum height will be. Use the quadratic formula to estimate how long it will take for the object to return to the ground.
Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page 8 of 3 A common problem in economics is to calculate when a maximum profit occurs and what the maximum profit will be. While many problems of this nature require calculus, algebra can be used if the model for the problem is quadratic. Ex. 4 A contractor for installing junction boxes determines that installing x boxes each day will result in profits of P x 0.x 8x 480 dollars. Use the quadratic formula to find how many junction boxes he must install in order to break even each day and then use the vertex formula to calculate the number of boxes he should install to maximize his profits and what that profit will be.
Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page 9 of 3 Topic 7: Polynomial Expressions and Functions More Terminology Recall Recall the definitions of polynomials and terms. Definition: Definition: A polynomial is a sum of terms. A polynomial term is a product of constants and variables with non-negative integer exponents. Which of the following represent polynomial expressions? Of the expressions that are not polynomials, identify the parts that are not properly polynomial terms. 3 x 6x x 5 x 4 x 4 3x 6x 9 x 100 100 3 5x 7x 0.5 3 x x 3 x Definition: Definition: Definition: Definition: Definition: Definition: The degree of a term is equal to the sum of the exponents of variables in the term. The degree of a polynomial is equal to the greatest degree for any term in the polynomial. The leading term of a polynomial is the term with the highest degree in a one variable polynomial. The leading coefficient of a polynomial is the coefficient of the leading term. A constant term is a term whose degree is zero and is usually represented by a number. Descending order is an ordering scheme for polynomials of one variable where the terms are arranged by degree from highest to lowest.
Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page 10 of 3 Polynomial Functions and their parts Definition: A polynomial function is a function defined by a polynomial expression. What is the simplest 8 th degree polynomial function? Ex. Answer the following questions about the polynomial function. 4 3 6 5 f x x x x How can you generalize an 8 th degree monomial function? a. What is the degree of f? b. What is the constant term of f? True or False: Every 8 th degree function has a term of degree 5. c. What is the leading term of f? d. What is the leading coefficient of f? True or False: 8 th degree functions can have a term of degree 9. e. Write f in descending order.
Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page 11 of 3 General Polynomial Function Form How can you generalize an 8 th degree polynomial function? Turning Points Definition: A turning point of a graph is a point where the function changes from increasing to decreasing or vice versa. Turning points correspond with local extrema (plural of extremum). How can you generalize an n th degree polynomial function? Maximum (sing.) Maxima (pl.) Minimum (sing.) Minima (pl.) An n th degree polynomial can have at most n 1 turning points. Polynomial functions with k turning points must be at least degree (k + 1). Functions of odd degree have an even number of turning point and functions of even degree have an odd number of turning points.
Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page 1 of 3 Topic 8: Behavior of a Polynomial Function End behavior is the term used to describe what happens to function values as x gets very large positive (x ) or very large negative (x ). With polynomial functions, end behavior is always either increasing without bound (graphically, going up) or decreasing without bound (graphically, going down). The end behavior of a polynomial function is determined by the leading coefficient and the degree of the function. 1. The leading coefficient affects right end behavior (x ). a. If the leading coefficient is positive, then the right end of the graph will go up. b. If the leading coefficient is negative, then the right end of the graph will go down.. The function s degree affects left end behavior(x ). a. If the degree of the function is even, then the left end of the graph will having matching behavior to the right end of the graph. b. If the degree of the function is odd, then the left end of the graph will have opposite behavior to the right end of the graph.
Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page 13 of 3 Ex. 1 Determine the end behavior of f and justify your answer. Ex. Determine the end behavior of f and justify your answer. 5 f x 4x 7x 1 3 6 5 4 f x x x x
Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page 14 of 3 Zeros of a Function Definition: A zero of a function f is an x-value such that f(x) = 0. Recall the Zero Product Property: If A B = 0, then either A = 0 or B = 0. This means that factors of a function can be solved to find the zeros of the function. Ex. 3 Find the intercepts of each polynomial function and discuss the end behavior. 1 1 P x x x x x-intercepts on the graph of a function correspond to zeros of the function. Thus a function in factored form easily gives up its x-intercepts. Intercepts: Ex. Find the zeros of f x x 6x 5. End behavior:
Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page 15 of 3 Ex. 4 Find the intercepts of each polynomial function and discuss the end behavior. 3 P x x x x Find a polynomial of leading coefficient of 1 with the zeros as given. Write in factored form. Ex. 5a, 1, 4 Degree 3 Intercepts: Ex. 5b 5, 1, 0, 3 Degree 4 End behavior:
Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page 16 of 3 Multiplicity of a Zero Definition: Theorem: The multiplicity of a zero is the number of times a zero of a function occurs in the fully factored form. The sum of the multiplicities of all zeros should be equal to the degree of the function. Effects of multiplicity on the graph of a function Ex. 6 Find the intercepts of each polynomial function, describing the behavior of the graph at each x-intercept, and discuss the end behavior. 1 3 P x x 1 x 4 1. At an x-intercept corresponding to a zero of odd multiplicity the graph of the function will cross the x-axis.. At an x-intercept corresponding to a zero of even multiplicity the graph of the function will turn at the x-axis. 3. The higher the multiplicity of a zero, the greater the localized flatness of the graph at the corresponding x-intercept. Intercepts: End behavior: For consistency we will describe an x-intercept corresponding to a zero of multiplicity of three or higher as having flatness around the intercept.
Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page 17 of 3 Ex. 7 Find the intercepts of each polynomial function, describing the behavior of the graph at each x-intercept, and discuss the end behavior. 3 3 P x x x 3 x 4 400 Find a polynomial with the properties given below. Write in factored form. Ex. 8a Degree 4 Leading coefficient of Zero 1, multiplicity 1 Zero, multiplicity Zero 4, multiplicity 1 Intercepts: Ex. 8b Degree 6 Leading coefficient of 1 8 Zero, multiplicity 3 Zero 0, multiplicity 1 Zero 3, multiplicity End behavior:
Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page 18 of 3 Topic 9: Long Division of Polynomials Recall how to do long division of numbers. 7109 3 Long Division Algorithm for Polynomials 1. Divide the leading term of the divisor into the leading term of the dividend. Write in the quotient.. Multiply the result of step 1 to the divisor and write under the dividend. 3. Subtract. 4. Repeat steps 1-3 by bringing down terms of the dividend after subtraction. The algorithm is completed once the remaining leading term under the dividend is of lower degree than the degree of the divisor. To check your division: quotient divisor remainder numerator
Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page 19 of 3 Ex. 1 Divide. Identify the quotient and remainder. 3 x 5x x 3 x
Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page 0 of 3 Ex. Divide. Identify the quotient and remainder. 4 3 x 6x x 7 x x
Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page 1 of 3 Ex. 3 Divide. Identify the quotient and remainder. 3 4x 8x 6x 5 x 1
Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page of 3 Topic 30: Synthetic Division of Polynomials In addition to the long division, there is another method for dividing polynomials called synthetic division. Drawbacks to Synthetic Division Rarely feasible if the denominator is nonlinear. Requires extra manipulation if the leading coefficient of a linear denominator is not 1. Advantages to Synthetic Division Far easier, quicker, and less messy than long division. Most of the cases where division is useful can be handled by synthetic division. Synthetic Division Algorithm for Polynomials Px ( ) For a division problem in the form, x c c a n a n a a a a set up the problem as 1 0 where a n through a 0 are the coefficients of P. 1. Bring the first remaining coefficient down under the second line.. Multiply the number under the second line to c. Write the product under the next coefficient of P. 3. Add and write the sum under the second line. 4. Repeat steps and 3 until the coefficients of P are exhausted. 5. Draw a box around the last number under the second line. This is the remainder. 6. The preceding numbers under the second line are the coefficients of the quotient which should be exactly one degree less than P.
Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page 3 of 3 Ex. 1 Divide. Identify the quotient and remainder. Ex. Divide. Identify the quotient and remainder. 3 x 5x x 3 x 4 3 x 5x x 9 x 3
Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page 4 of 3 Ex. 3 Divide. Identify the quotient and remainder. Ex. 4 Divide. Identify the quotient and remainder. 3 x 8x x x 1 3 x 9x x 8 x 4
Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page 5 of 3 Remainder and Factor Theorems 1. The remainder found by synthetic division is equal to P(c). (Remainder Theorem). If the remainder found by division is zero, then Q(x) is a factor of P(x). (In other words, Q times something is P). Use previous examples to evaluate: 3 Px ( ) x 5x x 3 at x = 4 3 Px ( ) x 5x x 9 at x = 3 3 Px ( ) x 8x xat x = -1
Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page 6 of 3 Topic 31: Rational Functions (part 1) Definition: A rational function is defined by the quotient of two polynomial functions. Let R represent a rational function and P and Q P x represent polynomial functions, then R x Q x Definition:. An asymptote is a line that a graph approaches but does not intersect as values approach ±. Intercepts and Vertical Asymptotes of Rational Functions The y-intercept occurs at R(0). The x-intercept(s) occur wherever P(x) = 0 but Q(x) 0. (That is, an x-value makes the numerator equal 0 but not the denominator.) Vertical Asymptotes occur wherever Q(x) = 0 but P(x) 0. (That is, an x-value makes the denominator equal 0 but not the numerator.) A vertical asymptote of a rational function is a line x = h where h is a solution to Q(x) = 0 and not P(x) = 0.
Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page 7 of 3 Ex. 1a Find the intercepts and vertical asymptote(s) of R. R x x x 4x 1 3x 4 Horizontal Asymptotes of Rational Functions Horizontal Asymptotes correspond with end behavior. Unlike polynomial functions which either increase or decrease without bound as x, rational functions may approach a fixed #. To determine if a rational function has a horizontal asymptote, identify the degrees of the numerator and denominator, then: a: If the degree of the numerator > the degree of the denominator, then the rational function does not have a horizontal asymptote. b: If the degree of the numerator < the degree of the denominator, then the rational function has a horizontal asymptote of y = 0. c: If the degree of the numerator = the degree of the denominator, then the rational function has a horizontal asymptote of y = k where k is the ratio of the leading coefficients of P and Q.
Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page 8 of 3 Ex. 1b Sketch a graph of the rational function. R x x x 4x 1 3x 4 Ex. Identify the attributes of the rational function and then sketch a graph. R x 6 x x 1 y-intercept: x-intercept(s): vertical asymptote(s): horizontal asymptote:
Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page 9 of 3 Ex. 3 Identify the attributes of the rational function and then sketch a graph. Ex. 4 Identify the attributes of the rational function and then sketch a graph. R x x x 3x 4 R x x x 18 7x 6 y-intercept: y-intercept: x-intercept(s): x-intercept(s): vertical asymptote(s): vertical asymptote(s): horizontal asymptote: horizontal asymptote:
Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page 30 of 3 Topic 3: Rational Functions (part ) Definition: A slant asymptote, also called an oblique asymptote, is an asymptote that is neither horizontal nor vertical. Ex. 1 Identify the attributes of the rational function and then sketch a graph. R x x 4x 6 x Slant asymptotes occur whenever the numerator is exactly one degree greater than the denominator. The quotient found by dividing the rational expression defines the rule of the slope of the slant asymptote (i.e. y = quotient). With rational functions, it is not possible to have both a horizontal asymptote and a slant asymptote. y-intercept: x-intercept(s): vertical asymptote(s): horizontal/slant asymptote:
Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page 31 of 3 Holes of a Rational Function A hole in a function is created when the rule of the rational function can be simplified and the denominator causes a point-wise loss of definition. Ex. Identify the attributes of the rational function and then sketch a graph. R x x 4 x x 8 Holes occur wherever P(x) = 0 and Q(x) = 0. (That is, an x-value makes both the numerator and denominator equal 0.) To find the ordered pair coordinates of a hole, set the common factor of P and Q equal to zero and solve. The solution will be the x-value of the hole. To find the y-value of the hole, simplify the rational expression and evaluate the x-value of the hole. y-intercept: x-intercept(s): vertical asymptote(s): horizontal/slant hole: asymptote:
Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page 3 of 3 Sketching a graph from limited information. Although the graph may not be unique, it is possible to envision a graph of a rational function when given information about its intercepts and asymptotes. Ex. 4 Sketch a rational function subject to the given conditions. Answers may vary. Asymptotes: y =, y = x = 0 Intercepts: ( 1, 0) (0, 1) Ex. 3 Sketch a rational function subject to the given conditions. Answers may vary. Asymptotes: y = 1 x = 1 Intercepts: (, 0) (0, )