Section 3.1 Quadratic Functions
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1 Chapter 3 Lecture Notes Page 1 of 72 Section 3.1 Quadratic Functions Objectives: Compare two different forms of writing a quadratic function Find the equation of a quadratic function (given points) Application problems involving quadratic functions Two Different Forms of a Quadratic Function 1. f(x) = ax 2 + bx + c 2. f(x) = a(x h) 2 + k Called: Called: Uses: Uses: Finding vertex: Finding vertex: Page 1 of 72
2 Chapter 3 Lecture Notes Page 2 of 72 Transforming from one form to another Example: f(x) = (x + 9) 2 7 Vertex is: Transform to general form: Find vertex from general form: Transform from general form back to vertex form: Method: Page 2 of 72
3 Chapter 3 Lecture Notes Page 3 of 72 Finding a Quadratic Function Given the vertex and another point. Example: Find the standard (vertex) form of the quadratic function whose graph has vertex (1, -5) and passes through the point (3, 7). Start with the vertex form: f(x) = Substitute in the vertex: f(x) = Still need to solve for a, so use the second point: substitute in the coordinates for x and y: Solve for a: Write equation: Page 3 of 72
4 Chapter 3 Lecture Notes Page 4 of Find both forms of the quadratic function whose graph has vertex 2, 0 and passes through the point (1, 3). Also, sketch the graph of the function and list the information below. Vertex form: General form: Line of symmetry: x = Minimum or Maximum value: y = (indicate which it is, min or max) Domain: Range: Page 4 of 72
5 Chapter 3 Lecture Notes Page 5 of 72 Application Problems Maximizing/Minimizing Example: On July 1, 2004, the Cassini probe became the first spacecraft to orbit the planet Saturn. Although Saturn is about 764 times the size of Earth, it has a very similar gravitational force. The height s (in feet) of an object thrown upward from Saturn s surface with an initial velocity of 100 feet per second is given as a function of time t (in seconds) by: s(t) = 17.28t t Coordinates of maximum (to 2 decimal places): Find the maximum analytically: Height, feet x-coordinate: y-coordinate: Time, seconds How long until the object hit the ground? Estimate from the graph: Find the time analytically: Page 5 of 72
6 Chapter 3 Lecture Notes Page 6 of A dog breeder has 260 feet of fencing material to make a kennel for the dogs along the side of a garage as shown in the figure (therefore, fencing is only needed on three sides of the kennel). (i) Find the dimensions of the kennel which will give the maximum area for the kennel. (ii) What is the maximum area? Let x = width of kennel Write an expression in terms of x for the length of the kennel. Write an equation for the area of the kennel: Area = A(x) = Page 6 of 72
7 Chapter 3 Lecture Notes Page 7 of 72 Section 3.2 Polynomial Functions Objectives: Define polynomial functions and power functions Properties of graphs of polynomial functions, including: o End behavior of graphs o Relationship between degrees, real zeros and turning points Multiplicity of zeros Finding the real zeros of a polynomial function by factoring Sketching the graph of a polynomial function In general in Chapter 3, talking about: Characteristics of polynomial functions, and Finding the zeros of polynomial functions: o o A zero is: any number x that makes f(x) =. A zero is also called: If it is a real number solution, it is an. Already know how to find zeros for two types of functions. Set f(x) = 0, and solve for x. 1. Linear: y = mx + b solve for x 2. Quadratic: y = ax 2 + bx + c solve for x Also can find zeros of higher degree polynomial by factoring and using zero product property. Page 7 of 72
8 Chapter 3 Lecture Notes Page 8 of 72 What if the higher degree polynomial (3 rd degree or more) is not factorable? Can be very difficult to solve! There are formulas for the zeros of 3 rd and 4 th degree polynomials, but they are complicated. NO general formula exists for the zeros of a 5 th degree or higher polynomial. Often end up finding a numerical approximation, rather than an exact solution. Example: a cubic function, f(x) = x 3 x + 1 Exact solution: x = Approximate solution: x Define Polynomial Functions A polynomial function f(x) is: a of terms each term has a different on the variable Notation: f(x) = anx n + an-1x n a2x 2 + a1x + a0 n th degree where, n = an s = Page 8 of 72
9 Chapter 3 Lecture Notes Page 9 of 72 nth degree polynomial 1 st degree linear is called: Example: Leading term & leading coefficient Constant term 2 nd degree quadratic 3 rd degree cubic 4 th degree quartic 5 th degree quintic The leading term is the The leading coefficient is term, the term of highest degree., the number multiplying the leading term. The constant term is the constant (number) at the end of the polynomial. Special cases: No degree polynomial, the zero function, 0 degree polynomial, constant function, Some Properties of ALL Polynomial Functions: Domain is. Graph is. Graph is. End behavior is the function value (y) becomes very positive or very negative as x or x Page 9 of 72
10 Chapter 3 Lecture Notes Page 10 of 72 Which of the parent functions are NOT polynomial functions, and why? (Look back at your graphs from Chapter 2 Handouts) 1. Constant function, f(x) = c 2. Linear (Identify function, f(x) = x 3. Absolute Value function, f(x) = x 4. Quadratic function, f(x) = x 2 5. Square Root Function, f(x) = x 6. Cube Function, f(x) = x 3 7. Cube Root Function, f(x) = 3 x 8. 1 Reciprocal Function, f(x) = x 9. 1 Reciprocal Square Function, f(x) = 2 x Shape/Characteristics of Polynomial Functions Subset of polynomial functions that: JUST have the which is the term, term, and don t have any other terms. Could think of these as being Parent polynomial functions, or as the book calls them: Power Functions f(x) = where, Page 10 of 72
11 Chapter 3 Lecture Notes Page 11 of 72 Page 11 of 72
12 Chapter 3 Lecture Notes Page 12 of 72 From these plots, can see some General Characteristics of Polynomial Power Functions: 1. The plots are continuous and smooth, 2. Domain = 3. The even degree power functions are 4. The odd degree power functions are 5. The left and right end behavior of the degree polynomials is the same on both ends of the graph. 6. The left and right end behavior of the degree polynomials is the opposite at the two ends of the graph. End Behavior means: What the function does for large : When x gets very positive, read: as x approaches infinity When x gets very negative, read: as x approaches neg. inf. Page 12 of 72
13 Chapter 3 Lecture Notes Page 13 of 72 End Behavior of EVEN degree polynomials Example: f(x) = x 2 As x continues to get more positive, y continues to get more positive. We say that: The function goes to infinity as x goes to infinity. Write it as: As x continues to get more negative, y continues to get more positive. We say that: The function goes to infinity as x goes to negative infinity. Write it as: Note that the graph is doing the on both ends of the graph. Negative leading coefficient means: Example: f(x) = x 2 End behavior: Note that the graph is still doing the same thing on both ends of the graph. This is the case for ALL even degree polynomials (not just the power functions) left and right end behavior is the. Page 13 of 72
14 Chapter 3 Lecture Notes Page 14 of 72 End Behavior of ODD degree polynomials Example: f(x) = x 3 End behavior: Note that the graph is doing the on both ends of the graph. Negative leading coefficient: Example: f(x) = x 3 End behavior: Note that the graph is still doing the opposite thing on both ends of the graph. This is the case for ALL odd degree polynomials (not just the power functions) left and right end behavior is the. Page 14 of 72
15 Chapter 3 Lecture Notes Page 15 of 72 To summarize, our book refers to this as the Leading-Term Test: Now, start looking at polynomials with additional terms, so not just plain old power functions with only the leading term. Page 15 of 72
16 Chapter 3 Lecture Notes Page 16 of 72 Turning Points Turning point A local, where the function turns around These graphs display the maximum no. of turning points possible for the degree of the polynomial. The power function graphs (on p. 13) display the minimum no. possible Degree (n) Max. # Min. # turning turning points points 1 (line) 2 (parabola) 3 (cubic) 4 (quartic) 5 (quintic) 6 For an nth degree polynomial, the graph has AT MOST turning points Odd degree polynomials don t have to Even degree polynomials MUST turn Page 16 of 72
17 Chapter 3 Lecture Notes Page 17 of 72 Number of x-intercepts Degree (n) 1 (line) 2 (parabola) 3 (cubic) 4 (quartic) 5 (quintic) 6 Max. # x-intercepts Min. # x-intercepts An n th degree polynomial has a maximum of x-intercepts Odd degree polynomials must have at least x-intercept Even degree polynomials may have an x-intercept Page 17 of 72
18 Chapter 3 Lecture Notes Page 18 of 72 End behavior (polynomials with additional terms) Already looked at left and right behavior for the power functions: those that just have a leading term. For a polynomial with additional terms, the big picture, or long-term left and right behavior of the polynomial is. p(x) = x 5 h(x) = x 5 6x 3 + 8x + 1 Notice that: The difference is very The leading term (x 5 ) dominates, or controls the function. The end behavior of a polynomial function is completely determined by the end behavior of its. For large absolute values of x, the graph of h(x) looks like the graph of the power function p(x). Page 18 of 72
19 Chapter 3 Lecture Notes Page 19 of 72 To sum up the general characteristics portion: Some questions to consider: 1. How can you tell the difference between the graph of a polynomial function of even degree and the graph of a polynomial function of odd degree? 2. Why does this difference exist? 3. What is the effect of a negative leading coefficient? 4. Can you tell by the graph of a polynomial which degree it is? Page 19 of 72
20 Chapter 3 Lecture Notes Page 20 of 72 Page 20 of 72
21 Chapter 3 Lecture Notes Page 21 of 72 Identifying Zeros of a Polynomial f(x) = x 4 + 6x 3 32x f(x) = x(x 2)(x + 4) 2 From the graph: what are the zeros of f(x)? x = From the factored f(x): what are the zeros of f(x)? f(x) = x(x 2)(x + 4) 2 = 0 Summarize: if c is a zero of f(x), where c just stands for a particular value of x then f(c) = the point is an x-intercept AND is a factor of f(x) For this function: 1. c = 2 is a zero 2. c = 0 is a zero 3. c = 4 is a zero Page 21 of 72
22 Chapter 3 Lecture Notes Page 22 of 72 Multiplicity of Zeros Multiplicity means: Take the factored form of f(x) and write it completely out: Example: f(x) = x(x 2)(x + 4) 2 = 0 Zero Zero Occurs Multiplicity: From the graph, what is different about the zero at 4 vs. the zeros at 0 or 2? At 4: At 0 or 2: Key result: If zero c is EVEN multiplicity: graph x-axis at x = c If zero c is ODD multiplicity: graph x-axis at x = c Page 22 of 72
23 Chapter 3 Lecture Notes Page 23 of 72 Finding the Degree of a Polynomial Example: f(x) = x(x 2)(x + 4) 2 = 0 Write out all the factors: Count up all the x s: Same as adding the of the zeros. Factors multiply out to: Example: f(x) = x 2 (x 1) 2 (x + 5) Write the degree of the polynomial: List the zeros and their multiplicity, and what the graph does at each zero (cross x-axis or just touch x-axis): Zeros: x =? Multiplicity x-intercept Graph does what? Describe the end behavior of the graph: List the maximum number of turning points the graph could have: Page 23 of 72
24 Chapter 3 Lecture Notes Page 24 of 72 Graphing a Polynomial Starting from a polynomial function given in factored form Rough sketch only!! General Method: 1. Determine the. 2. From degree, know the turning points on graph. 3. Find x-intercepts (zeros) from the factors AND their multiplicities will determine if graph or x-axis 4. Find y-intercept, just because it s easy to set x = 0 and solve. 5. Determine the end behavior: i. f(x)?? as x ii. f(x)?? as x - iii. Remember this will be determined entirely by the leading (highest) term of the polynomial 6. Draw a rough graph: i. Mark the x-intercepts and y-intercept ii. Indicate the end behavior iii. Know where it crosses or touches the x-axis iv. Draw smooth curve Do not need to do anything more than this! Page 24 of 72
25 Chapter 3 Lecture Notes Page 25 of 72 Example: f(x) = (x + 1) 2 (x 3)(x 1) 1. n = degree 2. max turning points = 3. Zeros: x =? Multiplicity x-intercept Graph does what? 4. y-int: f(0) = 5. End behavior: will behave like: f(x) = f(x) as x f(x) as x Page 25 of 72
26 Chapter 3 Lecture Notes Page 26 of 72 How would this graph change with a negative sign out in front? Example: f(x) = (x + 1) 2 (x 3)(x 1) Changes to the graph: Transformation: zeros are y-intercept becomes: end behavior is, it now behaves like: f(x) = Page 26 of 72
27 Chapter 3 Lecture Notes Page 27 of 72 Intermediate Value Theorem Helps to locate zeros of a polynomial, especially if the polynomial is: Example: Show that f(x) = 2x 3 5x 2 10x + 5 has a root somewhere in the interval [-1,2]. Can approximate the zero using a plotting utility: Page 27 of 72
28 Chapter 3 Lecture Notes Page 28 of For the polynomial function: f(x) = x 2 (x 3)(x + 4) (a) Determine the degree of the polynomial: n = (b) Determine the maximum number of turning points: (c) Find the zeros of f, and their multiplicity. List the x-intercepts. Determine whether the graph of f crosses or just touches the x-axis at each zero. Zeros: x =? Multiplicity x-intercept Graph does what? (d) Find the y-intercept: (e) Determine the end behavior of f: f(x) as x f(x) as x (f) Sketch the graph of f, labeling all of the intercepts. Page 28 of 72
29 Chapter 3 Lecture Notes Page 29 of Write a polynomial function (in factored form) that has the following characteristics: crosses the x-axis at 1 and 4, touches the x-axis at 0 and 2, and is above the x-axis between 0 and 2. Sketch the graph of f, labeling all of the intercepts. To help you get started, fill out the following information: Zeros: x =? Multiplicity x-intercept Graph does what? Factor of f(x): (x c) Write the polynomial in factored form: f(x) = Page 29 of 72
30 Chapter 3 Lecture Notes Page 30 of Note: read that as: the statements MUST be true. (a) It intersects the y-axis in one and only one point. (b) It intersects the x-axis in at most three points. (c) It intersects the x-axis at least once. (d) For x very large, it behaves like the graph of y = x 3. (e) It is symmetric with respect to the origin. (f) It passes through the origin. Page 30 of 72
31 Chapter 3 Lecture Notes Page 31 of 72 Section 3.3 Dividing Polynomials Objectives: Divide polynomials using long division Divide polynomials using synthetic division Use the Factor and Remainder Theorems Overview of Sections 3.3, 3.4 and 3.5 These three sections together are all about: Finding the zeros of a polynomial. Example: f(x) = 4x x 4 x 3 First question is: how many zeros are there to find? Notice that: if a zero has multiplicity > one, then count it more than one time as a zero. Example: a zero of multiplicity 2 counts as two zeros. Remember: each zero corresponds to a factor of the polynomial. Example: f(x) = x(x 2)(x + 4) 2 Therefore: n zeros leads to in the polynomial: In summary: degree n polynomial Page 31 of 72
32 Chapter 3 Lecture Notes Page 32 of 72 Types of Zeros Section 3.4: All complex zeros Section 3.5: What s the relationship between the types of zeros shown here and the x- intercepts of the graph of f(x)? f(x) = x How many total zeros? How many REAL zeros? How many non-real complex zeros? From the plot: what can you guess about any odd-degree polynomial, as far as how many real zeros it must have? A polynomial of ODD degree must have. What about even degree polynomials, how many real zeros must they have? Page 32 of 72
33 Chapter 3 Lecture Notes Page 33 of 72 Overall Goal: 1. Factor the polynomials. 2. This allows us to find the zeros! Just like the example from Section 3.2 In form: f(x) = x 4 + 6x 3 32x In form: f(x) = x(x 2)(x + 4) 2 How to Find Factors of Polynomials The first steps in finding factors of polynomials are: knowing how to divide one polynomial by another, and understanding what The Division Algorithm means. Long Division of a Polynomial Tips for long division: Always put the terms in order If any terms are missing, include a 0 as a placeholder Page 33 of 72
34 Chapter 3 Lecture Notes Page 34 of 72 Example: F(x) = x 4 3x 2 4 D(x) x 3 Quotient is: Remainder is: From the Division Algorithm, write this as: F(x) = D(x) Q(x) + R(x) Evaluate F(x) for x = 3: F(3) = Dividing by x 3: a = remainder = Dividing by x + 3: a = remainder = So, the remainder equals the function value evaluated at. Page 34 of 72
35 Chapter 3 Lecture Notes Page 35 of 72 Example: x x x 4 2 Quotient is: Remainder is: From the Division Algorithm, write this as: F(x) = D(x) Q(x) + R(x) Evaluate F(x) for x = 2: F( 2) = Same result as before: If F(x) is divided by x a, the remainder = F(a) If F(a) = 0, then is a factor of F(x) If x a is a factor of F(x), then F(a) = Page 35 of 72
36 Chapter 3 Lecture Notes Page 36 of 72 Just saw two ways to determine if a particular value of x = a is a zero of f(x): 1. long division 2. Evaluate Third method is: Synthetic Division Accomplishes EXACTLY the same thing as Easier and more concise Divides a polynomial F(x) by a binomial Both synthetic and long division give the quotient polynomial in addition to telling us if x = a is a zero Evaluating F(a) ONLY tells us if x = a is a zero Example: x x x Write all of the coefficients of F(x) in a row. If there are any missing terms you must include a 0! 2. Leave a blank row, and in the next row down and to the left, put a. The minus sign is built into the (x a), so for (x + 2): a = 3. ALWAYS start by bringing down the 1 st term from the F(x) row. 4. Across the bottom row, always multiplying each number by the 2 out front, so multiply 2 * 1 = 2, and I put it in the middle row under the Down each column, always adding the numbers, and putting the sum in the bottom row. Page 36 of 72
37 Chapter 3 Lecture Notes Page 37 of 72 How to read the answer: the last column is always the Q(x) is the rest of the terms under the box The leading term of Q(x) is than F(x), the polynomial we started with. Each column is the coefficient for the descending terms Quotient: Q(x) = Remainder: R = An even more efficient way to do this is to NOT write down the middle row just do the multiplying and adding in your head, and write down the result. Example: x x x 4 3 Key points: Using synthetic division to divide F(x) by (x a) If last column = 0 a is a of F(x), and is a factor if last column 0 a is of F(x), and is NOT a factor Page 37 of 72
38 Chapter 3 Lecture Notes Page 38 of 72 Example: f(x) = x 4 x 3 7x 2 + x + 6 How many zeros does f(x) have? Test the following values using synthetic division: x = 3, 2, 1, 0, 1, 2, 3, 4 Once you have found the zeros, use them to write the factors, i.e. factor the polynomial Results: Zero: x = Corresponding Factor Fully factored polynomial is: f(x) = Page 38 of 72
39 Chapter 3 Lecture Notes Page 39 of Use long division to find the quotient and the remainder. 4 2 x 2x 1 2 x 2x 1 Quotient is: Quotient is: Remainder is: Remainder is: 2. Use synthetic division to find the quotient and the remainder. 3x 4 2 6x 3x 7 x 2 Quotient is: Remainder is: 3. Determine (using the Factor Theorem) if the linear polynomial is a factor of the second polynomial. x + 3; 3x 4 + 9x 3 4x 2 9x + 9 Page 39 of 72
40 Chapter 3 Lecture Notes Page 40 of 72 Section 3.4 The Real Zeros of a Polynomial Function Objectives: Use the Rational Zeros Theorem to identify possible zeros Learn a procedure to find the real zeros of a polynomial function Techniques to Help Find Real Zeros 1. Descartes Rule of Signs 2. Intermediate Value Theorem 3. Upper Bounds 4. Lower Bounds Descartes Rule of Signs Narrows down the number of: positive real zeros, and negative real zeros Example: f(x) = x 4 x 3 7x 2 + x + 6 Number of sign changes for f(x): Means: To find f( x): Change the sign in front of effect on degree terms or constant term. degree terms, no Number of sign changes for f( x): Means: Page 40 of 72
41 Chapter 3 Lecture Notes Page 41 of 72 Intermediate Value Theorem Upper Bounds There are no real zeros this value. Lower Bounds There are no real zeros this value. Page 41 of 72
42 Chapter 3 Lecture Notes Page 42 of 72 Last piece: how do you know which numbers to start testing to see if they are zeros? For a rational number q p which is a zero of the polynomial F(x): F(x) = a n x n + an-1x n a1x + a 0 gives values of gives values of Therefore, start the problems by listing ALL possible rational zeros: Page 42 of 72
43 Chapter 3 Lecture Notes Page 43 of 72 Example: f(x) = 2x 4 11x Find all zeros. a0 = factors of a0 = p: an = factors of an = q: p : q These are ALL the possible rational zeros. Could also have: irrational (real) zeros, or complex non-real zeros This theorem tells us NOTHING about those. Apply Descartes Rule of signs: f(x) = 2x 4 11x has sign changes: f(-x) = has sign changes: Some tips on testing possible rational zeros: Start at +1 and go up Do the integers first Remember to check for UB or LB Take the first factor out: f(x) = Page 43 of 72
44 Chapter 3 Lecture Notes Page 44 of 72 Two ways to continue solving this: 1. Factor Q(x) using factoring by grouping 2. Continue synthetic division with Q(x) 1. Factor by grouping: 2. Continue with synthetic division Start with Q(x), the depressed equation : Q(x) = a0 = p = factors of a0: an = q = factors of an: p : q Page 44 of 72
45 Chapter 3 Lecture Notes Page 45 of 72 Take out the second factor: f(x) = When you get down to a quadratic: Do not need to continue with synthetic division Finish by solving quadratic: factor, square root property, or QF Solution: Page 45 of 72
46 Chapter 3 Lecture Notes Page 46 of Find all rational zeros of the given polynomial function. (Note: ALL of the polynomial s zeros are rational, so basically, find all of them). Then write the function in factored form. f(x) = x 4 5x x 16 Start by listing all possible rational zeros: Page 46 of 72
47 Chapter 3 Lecture Notes Page 47 of 72 Section 3.5 The Complex Zeros of a Polynomial Function Objectives: Find ALL zeros of a polynomial function Conjugate Pairs Theorem Polynomial Factors as: Can be factored over: x 2 4 x 2 3 x KEY: remember that non-real complex zeros come in CONJUGATE PAIRS!! If a bi is a zero so is Notation is: if z = a + bi is a complex number, then the notation for its complex conjugate is: z Page 47 of 72
48 Chapter 3 Lecture Notes Page 48 of 72 Example: x 4 7x x 2 38x 60 zero = 1 + 3i Find the remaining zeros. Second zero: Write factors: Multiply them together: Note: this quadratic is an irreducible quadratic factor of the polynomial, it can t be factored over the. Use long division to divide out the quadratic from the polynomial, to find the other two zeros. Page 48 of 72
49 Chapter 3 Lecture Notes Page 49 of 72 Construct a polynomial P(x) Example: Find the polynomial P(x) with real coefficients having degree 4, a leading coefficient of 3 and zeros 2, 1 and 3i. Total number of zeros = Page 49 of 72
50 Chapter 3 Lecture Notes Page 50 of 72 Summary of Method Finding ALL Zeros of a Polynomial Function 1. Identify total number of zeros (equal to of polynomial with real-numbered coefficients). 2. Use Descartes Rule of Signs to identify how many positive/negative real zeros the polynomial may have. This is optional, but may be useful. 3. Use Rational Zeros Theorem to list all possible rational (real) zeros. List p: all factors of List q: all factors of List p : all possible rational zeros q 4. Use synthetic division to start testing possible rational zeros. Remember to include a 0 for any! 5. Remember to look for: Upper bounds: all values in quotient row are Lower bounds: values in quotient row in sign Intermediate Value Theorem: remainder column changes from. 6. When the quotient polynomial is reduced to a cubic: try factoring by grouping, or continue with synthetic division 7. When the quotient polynomial is reduced to a quadratic: try factoring, or square root property, or use Quadratic Formula 8. List all zeros and their multiplicity. Also can write the factored form of the polynomial. Page 50 of 72
51 Chapter 3 Lecture Notes Page 51 of Use the given zero to find the remaining zeros for the function. Write the polynomial in fully factored form. P(x) = x 4 2x 3 + x 2 8x 12 2i is a zero Page 51 of 72
52 Chapter 3 Lecture Notes Page 52 of Find all zeros of the polynomial function. Write the polynomial in fully factored form. P(x) = x 3 5x 2 + 7x + 13 Page 52 of 72
53 Chapter 3 Lecture Notes Page 53 of 72 Section 3.6 Rational Functions Objectives: Analyze rational functions Sketch a graph of a rational function Polynomial functions, or P(x), where: P(x) = of terms the domain is the graph is smooth and continuous Rational functions: defined as the of two polynomials. Different from polynomial functions: The domain is by the denominator. The graph is smooth, but often continuous. Page 53 of 72
54 Chapter 3 Lecture Notes Page 54 of 72 Parent rational function: f(x) = x 1 Domain: x y x y As x is approaching 0 from the negative (left) side, f(x) is: As x is approaching 0 from the positive (right) side, f(x) is: In symbols: Page 54 of 72 In symbols:
55 Chapter 3 Lecture Notes Page 55 of 72 Definition of a Vertical Asymptote Page 55 of 72
56 Chapter 3 Lecture Notes Page 56 of 72 Locating Vertical Asymptotes Key Point: vertical asymptotes will occur at the of the denominator. AS LONG AS: the rational function is in which means: N(x) and D(x) do NOT have Example: f(x) = x 2 x 3 4x 5 Factored: Page 56 of 72
57 Chapter 3 Lecture Notes Page 57 of 72 Example: f(x) = x x 2 2 3x 10 8x 15 Factor the denominator: Fully factored: Domain: The graph has a at x = Page 57 of 72
58 Chapter 3 Lecture Notes Page 58 of 72 Horizontal Asymptotes Parent function: f(x) = x 1 Left/right end behavior of the graph: Indicates that the line is a horizontal asymptote Horizontal asymptotes are a little different than vertical asymptotes: Graph touch or cross a horizontal asymptote whereas Graph can touch or cross a vertical asymptote. A rational function has at most horizontal asymptote How to Find the Horizontal Asymptotes Statement: As x, the behavior of a rational function is determined by the of its numerator and denominator. Page 58 of 72
59 Chapter 3 Lecture Notes Page 59 of 72 Case 1: n < m degree of numerator < degree of denominator Example: f(x) = 3x 4 x 2 1 End behavior: Case 1 Result: N( x) For f a rational function = D( x) When degree of N(x) < degree of D(x), is the horizontal asymptote Page 59 of 72
60 Chapter 3 Lecture Notes Page 60 of 72 Case 2: n = m degree of numerator = degree of denominator Example: f(x) = 4x x End behavior: Case 2 Result: N( x) For f a rational function = D( x) When degree of N(x) = degree of D(x), The line y = is a horizontal asymptote. Page 60 of 72
61 Chapter 3 Lecture Notes Page 61 of 72 Case 3: n > m degree of numerator > degree of denominator x Example: f(x) = 3 3x 4 6x 3 End behavior: Case 3 Result: N( x) For f a rational function = D( x) when degree of N(x) > degree of D(x), there horizontal asymptote. f(x) as x Page 61 of 72
62 Chapter 3 Lecture Notes Page 62 of 72 Special Case: n = m + 1 degree of numerator is 1 more than degree of denominator 3x 2 1 Example: f(x) = x 2 End behavior: HOWEVER! The graph has what is called an:. Q(x) = the equation of the asymptote, in form: Special Case Result: N( x) For f(x) a rational function = D( x) when degree of N(x) is one greater than degree of D(x) N( x) there will be an oblique asymptote = quotient of D( x) in form: f(x) Q(x), the line mx + b as x Use long or synthetic division to find the oblique asymptote, Q(x) Note: f(x) is still unbounded, i.e. f(x) as x Page 62 of 72
63 Chapter 3 Lecture Notes Page 63 of 72 Summary of Horizontal Asymptotes For a rational function f(x) in lowest terms (i.e. N(x) and D(x) have no common factors): If: n < m Horizontal Asymptote is: n = m n > m n is 1 greater than m Page 63 of 72
64 Chapter 3 Lecture Notes Page 64 of 72 Analyzing a Rational Function Use the graph of the rational function f(x) to complete each statement. 1. As x 1 +, f(x). 2. As x 1 -, f(x). 3. As x, f(x). 4. As x, f(x). 5. The domain of f(x) is. 6. The equation of the vertical asymptote is. 7. The equation of the horizontal asymptote is. Page 64 of 72
65 Chapter 3 Lecture Notes Page 65 of 72 Procedure for Graphing a Rational Function Given a rational function: f(x) = N( x) D( x) Note: all of the following steps are on an as applicable basis. 1. Factor numerator and denominator. 2. Find domain Exclude values where denominator 3. Special case: cancel any common factors in numerator and denominator. N( x) Function is now in lowest terms = D( x) (If you do cancel a factor, remember that there will be a hole in the graph, called a ). 4. Solve for and plot the x-intercept(s) (if any): Set N(x) Solve for x. Solutions are x-intercepts. 5. Solve for and plot the y-intercept (if there is one): Evaluate 6. Solve for and draw the vertical asymptote(s) (if any): set D(x) Solve for x Solutions are the vertical asymptotes values that x can never take on because it would make the denominator = 0 Generally the same as values excluded from domain, unless you have a special case of a common factor (see item 3.) 7. Solve for and draw the horizontal or oblique asymptote (if any): Compare degree of N(x) to degree of D(x). Use Summary of Horizontal Asymptotes table to determine asymptote. Page 65 of 72
66 Chapter 3 Lecture Notes Page 66 of Test for symmetry. If f( x) = f(x) symmetric with respect to the If f( x) = f(x) symmetric with respect to the 9. Construct a sign chart to analyze sign of f(x) between all zeros. Use as a guide in plotting Draw a number line indicating each zero/vertical asymptote Separates x-axis into intervals Pick a test number in each interval Calculate sign ( ) of f(x) for test number ALL f(x) values in that interval will have the same sign ( ) 10. Calculate any additional points if needed. Can use the points that you calculated for the sign chart in step 9. Use your knowledge of what the function will do as it approaches a vertical asymptote As x a vertical asymptote, f(x) Use your knowledge of what the function will do as x (the end behaviors). As x, f(x) the horizontal or oblique asymptote 11. Finish plot. Draw smooth lines and add arrows to indicate where the graph continues. Page 66 of 72
67 Chapter 3 Lecture Notes Page 67 of 72 Example: f(x) = 2 x x 12 2 x 4 1. Factor numerator and denominator: 2. Find domain: 3. Special case: cancel any common factors in numerator and denominator. 4. Solve for and plot the x-intercept(s): 5. Solve for and plot the y-intercept: 6. Solve for and draw the vertical asymptote(s): 7. Solve for and draw the horizontal or oblique asymptote (if any): 8. Test for symmetry. 9. Construct a sign chart to analyze sign of f(x) between all zeros. 10. Calculate any additional points if needed. 11. Finish plot. Page 67 of 72
68 Chapter 3 Lecture Notes Page 68 of 72 Example: f(x) = 2 x x 12 2 x 4 Page 68 of 72
69 Chapter 3 Lecture Notes Page 69 of 72 Problem #1: Use the following steps to graph the rational function (plot on next page): x 2 1 f ( x) x 1. Factor numerator and denominator: 2. Find domain: 3. Special case: cancel any common factors in numerator and denominator. 4. Solve for and plot the x-intercept(s): 5. Solve for and plot the y-intercept: 6. Solve for and draw the vertical asymptote(s): 7. Solve for and draw the horizontal or oblique asymptote (if any): 8. Test for symmetry (optional). 9. Construct a sign chart to analyze sign of f(x) between all zeros/vertical asymptotes. 10. Calculate any additional points if needed. Page 69 of 72
70 Chapter 3 Lecture Notes Page 70 of 72 Problem #1: x f ( x) 2 x 1 Page 70 of 72
71 Chapter 3 Lecture Notes Page 71 of 72 Problem #2: Use the following steps to graph the rational function: f ( x) x 2 3x 2 x 1 1. Factor numerator and denominator: 2. Find domain: 3. Special case: cancel any common factors in numerator and denominator. 4. Solve for and plot the x-intercept(s): 5. Solve for and plot the y-intercept: 6. Solve for and draw the vertical asymptote(s): 7. Solve for and draw the horizontal or oblique asymptote (if any): 8. Test for symmetry (optional). 9. Construct a sign chart to analyze sign of f(x) between all zeros/vertical asymptotes. 10. Calculate any additional points if needed. Page 71 of 72
72 Chapter 3 Lecture Notes Page 72 of 72 Problem #2: f ( x) x 2 3x 2 x 1 Page 72 of 72
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