Math 148 Lab 1 Polynomial Graphs Due: Monday Wednesday, April April 10 5 Directions: Work out each problem on a separate sheet of paper, and write your answers on the answer sheet provided. Submit the answer sheet to your recitation instructor. Material from this lab will appear on the first midterm. Purpose: This lab explores how the leading term of a polynomial affects the shape of its graph. In this class we use the graphing calculator to help draw graphs; however, the calculator can be misleading. In section 1.1 a graph of an equation is described as a picture that illustrates the behavior of the equation. With different windows, the calculator can give different pictures. For example, using two different windows, the calculator gives two different views of the graph of y = 0.01x 2 (x " 50) as illustrated below. Window Xmin = "10 Xmax = 10 Ymin = "10 Ymax = 10 Xmin = "10 Xmax = 60 Ymin = "190 Ymax = 190 Graph Which graph better illustrates the equation y = 0.01x 2 (x " 50)? The picture that captures the behavior and important features of the equation is the best choice. We call this picture the complete graph. For y = 0.01x 2 (x " 50), the graph on the right is a better choice; it shows both x-intercepts, x = 0 and x = 50, and the behavior far away from the origin. (This behavior is known as end behavior.) Review of Vocabulary: Before we begin, let s review some math vocabulary. The terms of a polynomial are separated by plus or minus signs; for instance, the terms of the polynomial "14x 3 + 27x 2 +1 are "14x 3, 27x 2, and 1. Because "14x 3 is the term with the highest power on x, we call "14x 3 the leading term of "14 x 3 + 27x 2 +1. We call "14 the leading coefficient of the polynomial "14x 3 + 27x 2 +1 because "14 is the coefficient of the leading term. The leading term of a polynomial and the polynomial itself share the same degree. You may recall that the degree of a polynomial written in terms of x is the highest power of x occurring in the polynomial. For example, the degree of "14 x 3 + 27x 2 +1 is 3, and x 4 + 6 is a polynomial of degree 4. Notice that "14 x 3 + 27x 2 +1 is an odd-degree polynomial, and x 4 + 6 is an even-degree polynomial. 1
Part I: End Behavior By the end behavior of a graph, we mean the shape the graph takes on for very large positive x-values and very large negative x-values. End behavior can be seen on the calculator by zooming out enough until the behavior of the graph far away from the origin is evident. For non-constant polynomials, we can describe the end behavior at the right end and at left end of the graph as either pointing up or pointing down. 1. (a) Complete the tables by filling in the FOUR missing graphs. EVEN DEGEE POLYNOMIALS ODD DEGREE POLYNOMIALS i. y = x 2 " 5 ii. y = "x 2 + 5 iii. y = 0.01x 3 iv. y = "0.01x 3 y = x 4 " 6x 2 + 5 y = "x 4 + 6x 2 " 5 y = x 3 " 4 x + 1 y = "x 3 + 4 x " 1 (b) Conclusion: Look over the tables in part (a). Then complete the charts on the END BEHAVIOR of a polynomial that follow by filling in points up or points down and an accompanying arrow. The first two entries have been completed for you. END BEHAVIOR of EVEN DEGREE POLYNOMIALS Points up! Points up " END BEHAVIOR of ODD DEGREE POLYNOMIALS 2
2. Which graph could be a complete graph of a polynomial whose leading term is "2x 6? i. ii. iii. iv. 3. For each of the polynomials i. Sketch the picture that you see in the standard viewing window, ii. Sketch a complete graph, and iii. Give a good viewing window for a compete graph of the polynomial. (a) 0.02x 4 + 0.24x 3 " 0.18x 2 " 4.4x (b) 2x 3 " 2x 2 " 4x A. Roots & Degree Part II: Roots 4. (a) For each polynomial equation find its x-intercepts either algebraically or graphically. i. y = 2x "1 v. y = x 2 " 30x + 225 ix. y = x 3 " 2x 2 " 5x + 6 ii. y = 2x 2 " 4 x " 6 vi. y = x 2 "11x + 28 x. y = x 3 " x 2 + x "1 iii. y = "17x " 32 vii. y = x 2 " 2x "1 xi. y = x 3 +1.5x 2 " 4 x " 6 iv. y = x 2 + 5x +12 viii. y = x 2 " 2 xii. y = x 3 +18x 2 + 96x +160 (b) For each of the 12 polynomials in (a), state its degree and its number of roots. (c) Conclusion: Guess the relationship between the degree of a polynomial and the number of roots of the polynomial. B. Roots & Linear Factors A polynomial of degree 1 is a linear polynomial. Linear factors of a polynomial are important. For example, the polynomial x 2 + x " 2 factors as (x "1)(x + 2). Therefore (x "1) and (x + 2) are linear factors of x 2 + x " 2. The factorization x 2 + x " 2 = (x "1)(x + 2) makes it easy to solve x 2 + x " 2 = 0; we see that x =1 and x = "2 are zeroes or roots of the polynomial. 5. Consider the polynomial x 2 + 2x "120. (a) Factor the polynomial. (b) What are the linear factors of the polynomial? (c) What are the roots the polynomial? 6. Conclusion: Fill in the blank to make the following statement true: r is a root of the polynomial exactly when is a linear factor of the polynomial. 3
Note that x 2 " 2 can be factored. In problem 4 (a), we see that the roots are x = ± 2. So the factors are (x " 2) and (x " (" 2)) = (x + 2). Hence written as the product of its linear factors, x 2 " 2 = (x " 2)(x + 2). 7. Write each of the following polynomials from # 4 as a product of its linear factors: (a) x 2 "11x + 28 (b) 2x 2 " 4x " 6 (Note: Don t forget the leading coefficient of 2.) (c) x 2 " 2x "1 (Hint: Use the quadratic formula.) (d) x 3 +1.5x 2 " 4 x " 6 (Hint: Use your calculator to find the roots.) C. Roots & Multiplicity The only roots of (x "1)(x + 2) are x =1 and x = "2. In section 1.2, we learned that this is equivalent to saying that x =1 and x = "2 are the only x-intercepts of the graph of y = (x "1)(x + 2). Likewise, x =1 and x = "2 are the only roots and hence only x-intercepts of (x "1) 2 (x + 2). However (x "1)(x + 2) and (x "1) 2 (x + 2) are not the same polynomial. In fact, one has degree 2 and the other has degree 3. (You can see this by expanding each polynomial: (x "1)(x + 2) = x 2 + x " 2 and (x "1) 2 (x + 2) = x 3 " 3x + 2.) Moreover, they have different graphs! So, if they have the same x-intercepts, what makes their graphs different? An obvious difference between the two polynomials is the number of times the factor (x "1) appears in each: In (x "1)(x + 2), the factor (x "1) appears once, and in (x "1) 2 (x + 2) = (x "1)(x "1)(x + 2), the factor (x "1) appears twice. For this reason, we say that 1 is a root of multiplicity one of (x "1)(x + 2), whereas 1 is a root of multiplicity 2 of (x "1) 2 (x + 2). 8. The multiplicity of a root determines whether the graph touches or crosses the x-axis at the respective x- intercept. Look over the table given below. Then answer the question following the table. y = 2(x "1)(x + 2) y = "(x "1) 2 (x + 2) y = 3(x "1) 3 (x + 2) y = 0.8(x "1) 4 (x + 2) Graph Roots x =1, once x = "2, once x =1, twice x = "2, once x =1, three times x = "2, once x =1, four times x = "2, once Conclusion: How does the multiplicity of a root determine whether the graph touches or crosses the x-axis at the x-intercept? 4
9. There are many polynomials that have exactly the roots 2, 6, and "11. (a) If possible, produce a polynomial of degree 3 with this property. (b) Find another polynomial of degree 3 with this same property. (c) If possible, produce a polynomial of degree 4 with this property. (d) If possible, produce a polynomial of degree 2 with this property. Part III: Putting It All Together 10. Consider the polynomial "(x + 2)(x " 20) 2. (a) What is its degree? (b) Describe its end behavior. (c) List its roots and their respective multiplicities. (d) Taking the roots into consideration, what would be good choices for Xmin and Xmax? (e) On your graphing calculator, set Xmin and Xmax to your choices from part (d) and press [GRAPH]. You might not have a complete graph. The ZoomFit feature of your calculator can help you find a complete graph. (Press [ZOOM] and choose 0: ZoomFit.) Sketch a picture of the graph. Be sure to label the x- and y-intercepts. (f) Why is this feature called ZoomFit? What does it do? [Hint: Do Xmin and Xmax change? Do Ymin and Ymax change?] 11. In 9 c. there are several possible polynomials. Find one with y-intercept 1089. [Hint: Choose a good leading coefficient.] 5
Name: Recitation Time: 1. (a) Math 148 Lab 1 Answer Sheet Page 1 of 3 i. y = x 2 " 5 ii. y = "x 2 + 5 iii. y = 0.01x 3 iv. y = "0.01x 3 (b) END BEHAVIOR of EVEN DEGREE POLYNOMIALS Points up! Points up " END BEHAVIOR of ODD DEGREE POLYNOMIALS 2. 3. (a) i. ii. iii. Xmin = Xmax = Ymin = Ymax = (b) i ii. iii. Xmin = Xmax = Ymin = Ymax = 6
Name: Page 2 of 3 4. (a) x-intercepts (b) Degree Number of Roots i. i. ii. iii. iv. ii. iii. iv. v. v. vi. vii. viii. ix. vi. vii. viii. ix. x. x. xi. xii. xi. xii. (c) 5. (a) (b) (c) 6. 7
Name: Page 3 of 3 7. (a) (b) (c) (d) 8. 9. (a) (b) (c) (d) 10. (a) (b) (c) (d) Xmin = Xmax = (e) (f) 11. 8
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