Chapter 3 Page 1 of 23. Lecture Guide. Math College Algebra Chapter 3. to accompany. College Algebra by Julie Miller

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1 Chapter 3 Page 1 of 23 Lecture Guide Math College Algebra Chapter 3 to accompany College Algebra by Julie Miller Corresponding Lecture Videos can be found at Prepared by Stephen Toner & Nichole DuBal Victor Valley College Last updated: 2/16/13

2 Chapter 3 Page 2 of Quadratic Functions and Applications Quadratic functions are of the form. It is easiest to graph quadratic functions when they are in the form using transformations. Here, the parabola has the vertex at. f. Determine the axis of symmetry. g. Determine the minimum or maximum value of the function. h. Determine the domain and range. 3.1 #10 Use. a. Determine whether the graph of the prabola opens upward or downward. b. Identify the vertex. c. Determine the -intercept(s). Completing the Square 1. Write your equation in the form: d. Determine the -intercept. e. Sketch the function. 2. If there's a leading coefficient, factor it out of the first two terms on the right. 3. Cut the number in front of in half; write this new value on the line below. 4. Square this new value and write the product in the blank on the line above; add/subtract this product at the right to keep the equation balanced. 5. Insert an x, parentheses and exponent on the left to complete the square. 6. Add together the values at the right 7. Write the function in vertex form.

3 Chapter 3 Page 3 of #18 Use. a. Write the function in vertex form. 3.1 #20 Use. a. Write the function in vertex form. b. Identify the vertex. b. Identify the vertex. c. Identify the -intercept(s). c. Identify the -intercept(s). d. Identify the -intercept(s). d. Identify the -intercept(s). e. Sketch the function. e. Sketch the function. f. Determine the axis of symmetry. f. Determine the axis of symmetry. g. Determine the minimum and maximum values of the function. g. Determine the minimum and maximum values of the function. h. State the domain and range. h. State the domain and range.

4 Chapter 3 Page 4 of #30 Use. a. State whether the graph of the parabola opens upward or downward. b. Determine the vertex of the parabola. 3.1 #36 A long jumper leaves the ground at an angle of above the horizontal, at a speed of 11 m/sec. The height of the jumper can be modeled by, where is the jumper s height in meters and is the horizontal distance from the point of launch. a. At what horizontal distance from the point of launch does the maximum height occur? Round to 2 decimal places. b. What is the maximum height of the long jumper? Round to 2 decimal places. c. Determine the -intercept(s). c. What is the length of the jump? Round to 1 decimal place. d. Identify the -intercept(s). e. Sketch the graph. f. Determine the axis of symmetry. 3.1 #42 Two chicken coops are to be built adjacent to one another from 120 ft of fencing. a. What dimensions should be used to maximize the area of an individual coop? g. Determine the minimum and maximum values of the function. h. State the domain and range. b. What is the maximum area of an individual coop?

5 3.2 Introduction to Polynomial Functions Chapter 3 Page 5 of 23 Key Ideas: The domain of a polynomial function is. Informally, a polynomial function can be drawn.

6 Chapter 3 Page 6 of #30 Determine the end behavior of the graph of. 3.2 #44 Find the zeros of the function and state the multiplicities. 3.2 #34 Determine the end behavior of the graph of. The Intermediate Value Theorem: If is a polynomial and, if and have opposite signs, then has at least one zero in the interval. picture: 3.2 #38 Find the zeros of the function and state the multiplicities. 3.2 #50 Determine whether the intermediate value theorem guarantees that the function has a zero on the given interval. 3.2 #40 Find the zeros of the function and state the multiplicities. a. b. c. d.

7 Chapter 3 Page 7 of 23 Key Ideas: Let represent a polynomial function of degree. Then the graph of has at most turning points. Let be a polynomial function and let be a zero of. Then the point is an - intercept of the graph of. Furthermore, o If o If is a zero of multiplicity, then the graph crosses the -axis at. The point is called a cross point. is a zero of multiplicity, then the graph touches the -axis at and turns back around (does not cross the -axis). The point is called a touch point. 3.2 #60 a. Determine the minimum degree of the polynomial based on the number of turning points. b. Detemine whether the leading coefficient is positive or negative based on the end behavior and whether the degree of the polynomial is odd or even. c. Approximate the real zeros of the function and determine their multiplicities. For exercises 58 and 60, determine if the graph can represent a polynomial function. If so, assume that the end behavior and all turning points are represented in the graph. 3.2 #58 a. Determine the minimum degree of the polynomial based on the number of turning points. b. Detemine whether the leading coefficient is positive or negative based on the end behavior and whether the degree of the polynomial is odd or even. c. Approximate the real zeros of the function and determine their multiplicities.

8 Chapter 3 Page 8 of #64 Sketch. 3.3 Divison of Polynomials and the Remainder and Factor Theorems Division vocabulary: q( x) r( x) d( x) f ( x) is called the. is called the. is called the. is called the. 3.3 #17 a. Use long division to divide. 3.2 #68 Sketch. b. Identify the dividend, divisor, quotient, and remainder. Dividend: Divisor: Quotient: Remainder: c. Check the result from part (a) with the division algorithm.

9 Chapter 3 Page 9 of #24 Use long division to divide: 3.3 #40 Use synthetic division to divide. 3.3 #44 Use synthetic division to divide. 3.3 #28 Use long division to divide: 3.3 #50 Use the remainder theorem to evaluate for the given values of a. b. c. ( ) d.

10 Chapter 3 Page 10 of #58 Use the remainder theorem to determione if the given numer is a zero of the polynomial. c. Use the quadratic formula to solve the equation. a. b. d. Find the zeros of the polynomial. 3.3 #66 a. Factor, given that is a zero. 3.3 #64 a. Use synthetic division and the factor theorem to determine if is a factor of. b. Solve b. Use synthetic division and the factor theorem to determine if is a factor of. 3.3 #76 Write a degree 2 polynomial with zeros and.

11 Chapter 3 Page 11 of Zeros of Polynomials The Rational Zero Theorem If has integer coefficients and, and if (written in lowest terms) is a rational zero of, then Theorem: If the sum of the coefficients is, then is a zero (and is a factor). If after changing the signs of the coefficients of the odd-degreed terms, the sum of the "new" coefficients is zero, then is a zero. (Note: This theorem is not in your textbook.) is a factor of the constant term is a factor of the leading coefficient 3.4 #30 Find all the zeros. The rational zero theorem does not guarantee the existence of rational zeros. Rather, it indicates that if a rational zero exists for a polynomial, then it must be of the form. 3.4 #18 List the possible rational zeros. 3.4 #26 Find all the zeros.

12 Chapter 3 Page 12 of #34 Find all the zeros. b. Factor as a product of linear factors. c. Solve the equation. 3.4 #50 Write a polynomial of lowest degree with zeros of (mulitplicity 2) and (mulitiplicity 1) and with. 3.4 #40 has as a zero. a. Find all the zeros.

13 Chapter 3 Page 13 of #56 Determine the number of possible positive and negative real zeros for the given function. 3.4 #76 Find the zeros and their multiplicities. Consider using Descartes rule of signs and the upper and lower bound theorem to limit your search for rational zeros. (Hint: see exercise 68.) 3.4 #68 a. Determine if the upper bound theorem identifies as an upper bound for the real zeros of. b. Determine if the lower bound theorem identifies as a lower bound for the real zeros of.

14 Chapter 3 Page 14 of Rational Functions Rational functions are of the form factors in common. where and have no We want to analyze graphs and behaviors of rational functions, but need some notation: If and DO have factors in common, you get removable discontinuities in your graph. For example: Example: Graph and describe the behaviors as approaches the -axis from both sides. Also describe the behavior of as increases and decreases without bound. Identifying Vertical and Horizontal Asymptotes Three Types of Discontinuities: removable discontinuities Consider where and have no factors in common. If is a zero of, then is a vertical asymptote of the graph of. nonremovable (gap) discontinuities nonremovable (asymptotic) discontinuities If is a rational function with numerator of degree and denominator of degree, then 1. If, then has no horizontal asymptote. 2. If, then the line (the -axis) is the horizontal asymptote of. 3. If, then the line is the horizontal asymptote of.

15 Chapter 3 Page 15 of #18 Write the domain of in interval notation. 3.5 #28 Determine the vertical asymptote of the graph of. 3.5 #30 Determine the vertical asymptotes of the graph of. 3.5 #24 Refer to the graph of the function and complete the statement. a. As. For exercises 36-40, (a) identify the horizontal asymptote (if any), and (b) if the graph of the function has a horizontal asymptote, determine the point where the graph corsses the horizontal asymptote. 3.5 #36 b. As. c. As. 3.5 #38 d. As. e. The graph in increasing over the interval(s). f. The graph in decreasing over the interval(s). g. The domain is. h. The range is. i. The vertical asymptote is the line. 3.5 #40 j. The horizontal asymptote is the line.

16 Chapter 3 Page 16 of #74 Graph. 3.5 #48 Identify the asymptotes. 3.5 #60 Graph by using a transformation of the graph of. 3.5 #82 Graph.

17 Chapter 3 Page 17 of #86 Graph. 3.6 Polynomial and Rational Inequalities 3.6 #18 The graph of is given. Solve the inequalities. a. b. c. d. 3.6 #26 Solve the equations and inequalities. 3.5 #89 Graph. a. b. c. d. e.

18 Chapter 3 Page 18 of #28 Solve. 3.6 #62 Solve the inequalities. a. b. 3.6 #42 Solve. c. d. 3.6 #74 Solve. 3.6 #56 The graph of is given. Solve the inequalities. a. b. c. d.

19 Chapter 3 Page 19 of 23 The vertical position of an object moving upward or downward under the influence of gravit is given by, where 3.6 #102 Write the domain in interval notation. is the acceleration due to gravity (32 ft/sec 2 or 9.8 m/sec 2 ). is the time of travel. is the initial velocity. is the initial vertical position. 3.6 #86 Suppose that a basketball player jumps straight up for a rebound. a. If his initial velocity is 16 ft/sec, write a function modeling his vertical position (in ft) at a time seconds after leaving the ground. b. Find the times after leaving the ground when the player will be at a height of more than 3 ft in the air. Find the domain graphically 3.6 #94 Write the domain in interval notation.

20 Chapter 3 Page 20 of Variation Direct Variation 3.7 #30 The number of people that a ham can serve varies directly as the weight of the ham. An 8-lb ham feeds 20 people. Inverse Variation a. How many people will a 10-lb ham serve? In exercises 12-20, write a variation model using as the constant of variation. 3.7 #12 Simple interest on a loan or investment varies directly as the amount the loan. of b. How many people will a 15-lb ham serve? 3.7 #14 The time of travel is inversely proportional to the rate of travel. c. How many people will an 18-lb ham serve. d. If a ham feeds 30 people, what is the weight of the ham? 3.7 #18 The variable is directly proportional to the square of and inversely proportional to the square of. 3.7 #20 The variable varies jointly as and and inversely as the cube root of. 3.7 #40 The resistance of a wire varies directly as its length and inversely as the square of its diameter. A 50-ft wire with a 0.2-in. diameter has a resistance of Find the resistance of a 40-ft wire with a diameter of 0.1-in.

21 Chapter 3 Page 21 of 23 Some Chapter 3 Review Problems 4. Find the vertex: 1. Given that is a zero, find the other zeroes of. 2. Sketch: 5. Graph: 3. Solve:

22 6. Graph: Chapter 3 Page 22 of Find all the zeroes of. 8. Solve:

23 Chapter 3 Page 23 of Use the Rational Zero Theorem and Descartes' Rule of Signs to find all the zeroes of. 10. Given, a. Determine the end behavior of the graph of the function. b. List all possible rational zeros. c. Find all the zeros of (and state the multiplicities of). d. Determine the -intercepts. e. Determine the -intercepts. f. Is even, odd, or neither? g. Graph.