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2 –3 –4 –5 –6 86. 21 3 4 5 6 x –6 –5 –4 –3 –2 y 6 5 4 3 2 1 –1–1 –2 –3 –4 –5 –6 87. 21 3 4 5 6 x –6 –5 –4 –3 –2 y 6 5 4 3 2 1 –1–1 –2 –3 –4 –5 –6 78. h(x) =# #1 __ x + 2 3 81. p(t) = −2 + 3t 84. r(x) = 4 88. 21 3 4 5 6 x –5 –4 –3 –2 y 5 4 3 2 1 –1–1 –2 –3 –4 –5 21 3 4 5 x Download the OpenStax text for free at http://c... |
000 100. Table 3 shows the input, w, and output, k, for a linear function k. a. Fill in the missing values of the table. b. Write the linear function k, round to 3 decimal places. w −10 5.5 67.5 b k 30 −26 a −44 Table 3 101. Table 4 shows the input, p, and output, q, for a linear function q. a. Fill in the missing valu... |
://cnx.org/content/col11759/latest. 30 8 CHAPTER 4 LINEAR FUNCTIONS 111. Find the equation of the line perpendicular to the line g(x) = −0.01x + 2.01 through the point (1, 2). For the following exercises, use the functions f (x) = −0.1x + 200 and g(x) = 20x + 0.1. 112. Find the point of intersection of the lines f and ... |
. In particular, historical data shows that 1,000 shirts can be sold at a price of $30, while 3,000 shirts can be sold at a price of $22. Find a linear equation in the form p(n) = mn + b that gives the price p they can charge for n shirts. 118. A farmer finds there is a linear relationship between the number of bean st... |
linear models to solve problems involving quantities with a constant rate of change, we typically follow the same problem strategies that we would use for any type of function. Let’s briefly review them: 1. Identify changing quantities, and then define descriptive variables to represent those quantities. When appropri... |
t) = mt + b ↑ ↑ −400 3500 M(t) = − 400t + 3500 To find the x-intercept, we set the output to zero, and solve for the input. 0 = −400t + 3500 t = 3500 ____ 400 = 8.75 The x-intercept is 8.75 weeks. Because this represents the input value when the output will be zero, we could say that Emily will have no money left after... |
x) = mx + b = −250x + 1000 Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 4.2 MODELING WITH LINEAR FUNCTIONS 311 0 = −250x + 1000 1000 = 250x 4 = x x = 4 The x-intercept is the number of months it takes her to reach a balance of $0. The x-intercept is 4 months, so it will take Ha... |
equation, we need the initial value and the rate of change, or slope. To determine the rate of change, we will use the change in output per change in input. m = change in output __ change in input The problem gives us two input-output pairs. Converting them to match our defined variables, the year 2004 would correspon... |
the population in 2014. b. Identify the year in which the population will reach 54,000. Using a Diagram to Model a Problem It is useful for many real-world applications to draw a picture to gain a sense of how the variables representing the input and output may be used to answer a question. To draw the picture, first ... |
write formulas for the distance each person has walked. A(t) = 4t E(t) = 3t For this problem, the distances from the starting point are important. To notate these, we can define a coordinate system, identifying the “starting point” at the intersection where they both started. Then we can use the variable, A, which we ... |
.4 hours, or 24 minutes. Q & A… Should I draw diagrams when given information based on a geometric shape? Yes. Sketch the figure and label the quantities and unknowns on the sketch. Example 3 Using a Diagram to Model Distance Between Cities There is a straight road leading from the town of Westborough to Agritown 30 mi... |
to the junction. distance#=# √ —— (x2 − x1)2 + (y2 − y1)2 — (18 − 0)2 + (6 − 0)2 = √ ≈ 18.974 miles Analysis One nice use of linear models is to take advantage of the fact that the graphs of these functions are lines. This means real-world applications discussing maps need linear functions to model the distances betwe... |
define two functions in Table 1. Input Outputs Initial Value Rate of Change d, distance driven in miles K(d): cost, in dollars, for renting from Keep on Trucking M(d) cost, in dollars, for renting from Move It Your Way Up-front fee: K(0) = 20 and M(0) = 16 K(d) = $0.59/mile and P(d) = $0.63/mile Table 1 A linear funct... |
x.org/content/col11759/latest. SECTION 4.2 SECTION EXERCISES 317 4.2 SECTION EXERCISES VERBAL 1. 2. 3. 4. ALGEBRAIC 5. yx=f x=+x f(x 6. x __ f(x=– x f(x 7. y 8. __ f(x=– x f(x xg x=f(x=x f x 9. 10. 11. 12. 13. 14. Ptt P P 16. P y 17. 18. Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 31 ... |
plus 54. o diff nies. Th Th 55. 56. 57. 58. Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 32 2 CHAPTER 4 LINEAR FUNCTIONS LEARNING OBJECTIVES In this section, you will: • Draw and interpret scatter plots. • Use a graphing utility to find the line of best fit. • Distinguish between linear... |
not perfectly so. 10 Selected data from http://classic.globe.gov/fsl/scientistsblog/2007/10/. Retrieved Aug 3, 2010. Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 4.3 FITTING LINEAR MODELS TO DATA 323 Cricket Chirps vs. Temperature ) 90 80 70 60 50 40 0 0 10 20 30 40 50 Cricket... |
the data. Figure 4 compares the two processes for the cricket-chirp data addressed in Example 2. We can see that interpolation would occur if we used our model to predict temperature when the values for chirps are between 18.5 and 44. Extrapolation would occur if we used our model to predict temperature when the value... |
ation. Using our model: T(30) = 30 + 1.2(30) = 66 degrees Based on the data we have, this value seems reasonable. b. The temperature values varied from 52 to 80.5. Predicting the number of chirps at 40 degrees is extrapolation because 40 is outside the range of our data. Using our model: 40 = 30 + 1.2c 10 = 1.2c c ≈ 8.... |
chirp data in Table 2. Solution 1. Enter the input (chirps) in List 1 (L1). 2. Enter the output (temperature) in List 2 (L2). See Table 2. L1 L2 44 80.5 35 70.5 20.4 57 33 66 31 68 35 72 18.5 52 37 73.5 26 53 Table 2 3. On a graphing utility, select Linear Regression (LinReg). Using the cricket chirp data from earlier,... |
and the graph of the data, Figure 7 shows some large data sets with their correlation coefficients. Remember, for all plots, the horizontal axis shows the input and the vertical axis shows the output. 1.0 0.8 0.4 0.0 −0.4 −0.8 −1.0 1.0 1.0 1.0 −1.0 −1.0 −1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Figure 7 Plotted data and relate... |
trend is linear, and if so, find a model for the data. Use the model to predict the consumption in 2008. Year ‘94 ‘95 ‘96 ‘97 ‘98 ‘99 ‘00 ‘01 ‘02 ‘03 ‘04 Consumption (billions of gallons) 113 116 118 119 123 125 126 128 131 133 136 The scatter plot of the data, including the least squares regression line, is shown in ... |
(x) and number of sit-ups a person can do (y). The results of the regression are given below. Use this to predict the number of situps a person who watches 11 hours of TV can do. y = ax + b a = −1.341 b = 32.234 r = −0.896 relationship between the diameter of a tree (x, in inches) and the tree’s age (y, in years). The... |
Stax text for free at http://cnx.org/content/col11759/latest. SECTION 4.3 SECTION EXERCISES 331 20. 10 10 NUMERIC 21. 10 10 22. The U.S. Census tracks the percentage of persons 25 years or older who are college graduates. That data for several years is given in Table 4[14]. Determine whether the trend appears linear. I... |
.8 5 43.1 6 38.8 7 39 8 38 9 32.7 10 30.1 11 29.3 12 27 13 25.8 21 17 100 2000 900 70 25 11 80 1798 988 80 30 2 60 1589 1000 82 31 −1 55 1580 1010 84 40 −18 40 1390 1200 105 50 −40 20 1202 1205 108 27. 28. 29. 30. 31 EXTENSIONS 32. Graph f (x) = 0.5x + 10. Pick a set of 5 ordered 33. Graph f (x) = −2x − 10. Pick a set ... |
year span. The following ordered pairs show the number of units sold in hundreds and the profit in thousands of over the ten-year span, (number of units sold, profit) for specific recorded years: (46, 250), (48, 305), (50, 350), (52, 390), (54, 410). 39. Use linear regression to determine a function y, 40. Predict when... |
each other point-slope form the equation for a line that represents a linear function of the form y − y1 = m(x − x1) slope the ratio of the change in output values to the change in input values; a measure of the steepness of a line slope-intercept form the equation for a line that represents a linear function in the f... |
have the same slope. Perpendicular lines have negative reciprocal slopes, assuming neither is vertical. See Example 18. • A line parallel to another line, passing through a given point, may be found by substituting the slope value of the line and the x- and y-values of the given point into the equation, f (x) = mx + b... |
text for free at http://cnx.org/content/col11759/latest. 33 6 CHAPTER 4 LINEAR FUNCTIONS CHAPTER 4 REVIEW EXERCISES LINEAR FUNCTIONS 1. Determine whether the algebraic equation is linear. 2. Determine whether the algebraic equation is linear. 2x + 3y = 7 6x 2 − y = 5 3. Determine whether the function is increasing or ... |
. 2x − 6y = 12 −x + 3y = 1 14. 1 __ x − 2 y = 3 3x + y = − 9 For the following exercises, find the x- and y-intercepts of the given equation 15. 7x + 9y = −63 16. f (x) = 2x − 1 Download the OpenStax text for free at http://cnx.org/content/col11759/latest. CHAPTER 4 REVIEW 337 For the following exercises, use the descr... |
function of the year, t. When will no one be afflicted? For the following exercises, use the graph in Figure 1 showing the profit, y, in thousands of dollars, of a company in a given year, x, where x represents years since 1980. y 12,000 10,000 8,000 6,000 4,000 2,000 0 5 10 15 20 25 30 x Figure 1 27. Find the linear ... |
600 1995 5,950 2000 6,300 2005 6,600 2010 6,900 Table 3 36. Eight students were asked to estimate their score on a 10-point quiz. Their estimated and actual scores are given in Table 4. Plot the points, then sketch a line that fits the data. Predicted Actual 6 6 7 7 7 8 8 8 Table 4 7 9 9 10 10 10 10 9 37. Draw a best-f... |
://cnx.org/content/col11759/latest. 34 0 CHAPTER 4 LINEAR FUNCTIONS CHAPTER 4 PRACTICE TEST 1. Determine whether the following algebraic equation can be written as a linear function. 2x + 3y = 7 2. Determine whether the following function is increasing or decreasing. f (x) = −2x + 5 3. Determine whether the following f... |
the pair of lines parallel, perpendicular, or neither? Line 1: Passes through (−2, −6) and (3, 14) Line 2: Passes through (2, 6) and (4, 14) Download the OpenStax text for free at http://cnx.org/content/col11759/latest. CHAPTER 4 PRACTICE TEST 341 15. Write an equation for a line perpendicular to f (x) = 4x + 3 and pa... |
75. Assume the population is changing linearly. a. How much did the population drop between the year 2004 and 2012? b. What is the average population decline per year? c. Find an equation for the population, P, of the school t years after 2004. Download the OpenStax text for free at http://cnx.org/content/col11759/late... |
obsolete by digital photography. (credit “film”: modification of work by Horia Varlan; credit “memory cards”: modification of work by Paul Hudson) CHAPTER OUTLINE 5.1 Quadratic Functions 5.2 Power Functions and Polynomial Functions 5.3 Graphs of Polynomial Functions 5.4 Dividing Polynomials 5.5 Zeros of Polynomial Functi... |
working with higher degree functions, so they provide a good opportunity for a detailed study of function behavior. Recognizing Characteristics of Parabolas The graph of a quadratic function is a U-shaped curve called a parabola. One important feature of the graph is that it has an extreme point, called the vertex. If... |
a, b, and c are real numbers and a ≠ 0. If a > 0, the parabola opens upward. If a < 0, the parabola opens downward. We can use the general form of a parabola to find the equation for the axis of symmetry. b2 − 4ac −b ± √ The axis of symmetry is defined by x = − # b _______________ __, to solve 2a 2a ax 2 + bx + c = 0 ... |
and the vertex is a maximum. Figure 5 represents the graph of the quadratic function written in standard form as y = −3(x + 2)2 + 4. Since x − h = x + 2 in this example, h = −2. In this form, a = −3, h = −2, and k = 4. Because a < 0, the parabola opens downward. The vertex is at (−2, 4). Vertex y 4 2 y = −3(x + 2)2 + ... |
CTIONS 347 This is the axis of symmetry we defined earlier. Setting the constant terms equal: ah2 + k = c k = c − ah2 = c − a ( b2 __ 4a = c − 2 b ) __ 2a In practice, though, it is usually easier to remember that k is the output value of the function when the input is h, so f(h) = k. forms of quadratic functions A qua... |
formula in the form g(x) = a(x −(−2))2 − 3 = a(x + 2)2 − 3. Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 34 8 CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS Substituting the coordinates of a point on the curve, such as (0, −1), we can solve for the stretch factor. −1 = a(0 + 2)2 − 3 2 = ... |
k = f (h) = f ( − __ 2a b __. 2a Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 5.1 QUADRATIC FUNCTIONS 349 Example 3 Finding the Vertex of a Quadratic Function Find the vertex of the quadratic function f (x) = 2x 2 − 6x + 7. Rewrite the quadratic in standard form (vertex form).... |
a quadratic function The domain of any quadratic function is all real numbers unless the context of the function presents some restrictions. The range of a quadratic function written in general form f(x) = ax2 + bx + c with a positive a value is ), ∞#). ), or [ f ( − #b f (x) ≥ f ( − #b __ __ 2a 2a ) ]. ), or ( −∞, f ... |
__ 10 2 9 ) ) = −5 ( __ 10 61 ___ 20 = 9 + 9 ( __ 10 ) − 1 The range is f (x) ≤ 61 ___ 20, or ( −∞, 61 ]. ___ 20 Try It #3 Find the domain and range of f (x) = 2 ( x −# #4 ) __ 7 2 + 8 __. 11 Determining the Maximum and Minimum Values of Quadratic Functions The output of the quadratic function at the vertex is the max... |
of L. W = 80 − 2L Now we are ready to write an equation for the area the fence encloses. We know the area of a rectangle is length multiplied by width, so A = LW = L(80 − 2L) A(L) = 80L − 2L2 This formula represents the area of the fence in terms of the variable length L. The function, written in general form, is A(L)... |
the demand for the item will usually decrease. For example, a local newspaper currently has 84,000 subscribers at a quarterly charge of $30. Market research has suggested that if the owners raise the price to $32, they would lose 5,000 subscribers. Assuming that subscriptions are linearly related to the price, what pr... |
100 Analysis This could also be solved by graphing the quadratic as in Figure 12. We can see the maximum revenue on a graph of the quadratic function. Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 5.1 QUADRATIC FUNCTIONS 353 (31.80, 2,528.1) y 3,000 2,500 2,000 1,500 1,000 500 )... |
2 In this case, the quadratic can be factored easily, providing the simplest method for solution. 1 __, 0 ) and (−2, 0). So the x-intercepts are at ( 3 0 = (3x − 1)(x + 2) 0 = 3x − 1 1 __ x = 3 0 = x + 2 or x = −2 Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 35 4 CHAPTER 5 POLYNOMIAL A... |
h = − # b __ 2a = − # 4 ___ 2(2) = −1 k = f (−1) = 2(−1)2 + 4(−1) − 4 = −6 f(x) = 2(x + 1)2 − 6 We can now solve for when the output will be zero. 0 = 2(x + 1)2 − 6 6 = 2(x + 1)2 3 = (x + 1)2 x + 1 = ± √ — 3 The graph has x-intercepts at (−1 − √ 3, 0) and (−1 + √ — — 3 x = −1 ± √ 3, 0). — Download the OpenStax text fo... |
? b. What is the maximum height of the ball? c. When does the ball hit the ground? Solution a. The ball reaches the maximum height at the vertex of the parabola. h = − # 80 ______ 2(−16) = 80 ___ 32 5 __ = 2 = 2.5 The ball reaches a maximum height after 2.5 seconds. Download the OpenStax text for free at http://cnx.org... |
rock hit the ocean? Access these online resources for additional instruction and practice with quadratic equations. • Graphing Quadratic Functions in General Form (http://openstaxcollege.org/l/graphquadgen) • Graphing Quadratic Functions in Standard Form (http://openstaxcollege.org/l/graphquadstan) • Quadratic Functio... |
x + 1 19. f(x) = 2 20. f(x) = − #1 __ x 2 − 2x + 3 3 For the following exercises, determine the domain and range of the quadratic function. 21. f(x) = (x − 3)2 + 2 22. f(x) = −2(x + 3)2 − 6 23. f(x) = x 2 + 6x + 4 24. f(x) = 2x 2 − 4x + 2 25. k(x) = 3x 2 − 6x − 9 For the following exercises, use the vertex (h, k) and a... |
following exercises, write the equation for the graphed function. 42. y y 5 4 3 2 1 –4 –3 –2 –1 –1 –2 –3 –4 –5 40. 436 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 –6 –7 NUMERIC 416 –5 –4 –3 –2 –1 –1 –2 1 2 3 4 x 44. y 45. 7 6 5 4 3 2 1 –3 –2 –1 –1 –2 –2 –1 –1 –7 –6 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5 For the following exercises, use the ... |
determine the height of the object when it has traveled 100 feet away horizontally. 55. A suspension bridge can be modeled by the quadratic function h(x) = 0.0001x 2 with −2000 ≤ x ≤ 2000 where ∣ x ∣ is the number of feet from the center and h(x) is height in feet. Use the [TRACE] feature of your calculator to estimat... |
the same size given 500 feet of fencing. 69. Among all of the pairs of numbers whose sum is 6, find the pair with the largest product. What is the product? 70. Among all of the pairs of numbers whose difference is 12, find the pair with the smallest product. What is the product? 71. Suppose that the price per unit in ... |
icity and end behaior • rite a polynomial function that satisfies certain criteria. 5.2 POWER FUNCTIONS AND POLYNOMIAL FUNCTIONS Suppose a certain species of bird thrives on a small island. Its population over the last few years is shown in Table 1. Figure 1 (credit: Jason Bay, Flickr) Year Bird Population 2009 800 201... |
f (x) = x f (x) = x 2 f (x) = x 3 f (x) =# #1 __ x 1 __ x2 — f (x) = √ x — x f (x) = f (x) = 3 √ Constant function Identify function Quadratic function Cubic function Reciprocal function Reciprocal squared function Square root function Cube root function Solution All of the listed functions are power functions. The co... |
be symbolically written as x → ∞, we are describing a behavior; we are saying that x is increasing without bound. With the positive even-power function, as the input increases or decreases without bound, the output values become very large, positive numbers. Equivalently, we could describe this behavior by saying that... |
−∞, f (x) → ∞ and x → ∞, f (x) → −∞ y Negative constant k < 0 x x x → −∞, f (x) → ∞ and x → ∞, f (x) → −∞ x → −∞, f (x) → −∞ and x → ∞, f (x) → −∞ Figure 4 Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 5.2 POWER FUNCTIONS AND POLYNOMIAL FUNCTIONS 363 How To… Given a power funct... |
–8 –5 –4 –3 –2 21 3 4 5 x Figure 6 Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 36 4 CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS Analysis We can check our work by using the table feature on a graphing utility. x −10 −5 0 5 10 f (x) 1,000,000,000 1,953,125 0 −1,953,125 −1,000,000,000 T... |
real number other than zero. Each expression ai x i is a term of a polynomial function. Example 4 Identifying Polynomial Functions Which of the following are polynomial functions? f (x) = 2x 3 ċ 3x + 4 g(x) = −x(x2 − 4) h(x) = 5 √ — x + 2 Solution The first two functions are examples of polynomial functions because th... |
. Identify the coefficient of the leading term. Example 5 Identifying the Degree and Leading Coefficient of a Polynomial Function Identify the degree, leading term, and leading coefficient of the following polynomial functions. f (x) = 3 + 2x 2 − 4x 3 g(t) = 5t 5 − 2t 3 + 7t h(p) = 6p − p 3 − 2 Solution For the function... |
2x2 + 1 3x5 –5 –4 –3 –2 f (x) = −6x3 + 7x2 + 3x + 1 −6x3 –5 –4 –3 –2 Table 3 y 6 5 4 3 2 1 –1 –1 –2 –3 –4 –5 –6 y 6 5 4 3 2 1 –1 –1 –2 –3 –4 –5 –6 y 6 5 4 3 2 1 –1 –1 –2 –3 –4 –5 –6 y 6 5 4 3 2 1 –1 –1 –2 –3 –4 –5 –6 21 3 4 5 x 21 3 4 5 x 21 3 4 5 x 21 3 4 5 x Download the OpenStax text for free at http://cnx.org/cont... |
1)(x + 4), express the function as a polynomial in general form, and determine the leading term, degree, and end behavior of the function. Solution Obtain the general form by expanding the given expression for f (x). f (x) = −3x2(x − 1)(x + 4) = −3x2(x2 + 3x − 4) = −3x4 − 9x3 + 12x2 The general form is f (x) = −3x4 − ... |
the graph changes direction from increasing to decreasing or decreasing to increasing. The y-intercept is the point at which the function has an input value of zero. The x-intercepts are the points at which the output value is zero. How To… Given a polynomial function, determine the intercepts. 1. Determine the y-inte... |
-intercepts occur when the output is zero. To determine when the output is zero, we will need to factor the polynomial. f (x) = x4 − 4x2 − 45 = (x2 − 9)(x2 + 5) = (x − 3)(x + 3)(x2 + 5) 0 = (x − 3)(x + 3)(x2 + 5) x − 3 = 0 or x + 3 = 0 or x2 + 5 = 0 x = 3 or x = −3 or (no real solution) The x-intercepts are (3, 0) and ... |
n x-intercepts and n − 1 turning points. Example 10 Determining the Number of Intercepts and Turning Points of a Polynomial Without graphing the function, determine the local behavior of the function by finding the maximum number of x-intercepts and turning points for f (x) = −3x 10 + 4x 7 − x 4 + 2x 3. Solution The p... |
0 + 3)(0 − 4) = 0 The y-intercept is (0, 0). The x-intercepts are found by determining the zeros of the function. 0 = −4x(x + 3)(x − 4) x = 0 or x + 3 = 0 or x − 4 = 0 x = 0 or x = −3 or x = 4 The x-intercepts are (0, 0), (−3, 0), and (4, 0). The degree is 3 so the graph has at most 2 turning points. Try It #9 Given th... |
x5 9. f (x) = x2 _____ x2 − 1 7. f (x) = (x2)3 8. f (x) = x − x4 10. f (x) = 2x(x + 2)(x − 1)2 11. f (x) = 3x + 1 For the following exercises, find the degree and leading coefficient for the given polynomial. 12. −3x 15. x(4 − x2)(2x + 1) 13. 7 − 2x2 16. x 2 (2x − 3)2 14. −2x2 − 3x5 + x − 6 For the following exercises... |
.2 SECTION EXERCISES 373 34. 37. y 5 4 3 2 1 –1 –1 –2 –3 –4 –5 y 6 5 4 3 2 1 –1 –1 –2 –3 –4 –5 –6 –5 –4 –3 –2 –5 –4 –3 –2 35. 21 3 4 5 x –5 –4 –3 –2 38. 21 3 4 5 x –5 –4 –3 –2 y 5 4 3 2 1 –1 –1 –2 –3 –4 –5 y 6 5 4 3 2 1 –1 –1 –2 –3 –4 –5 36. 21 3 4 5 x –5 –4 –3 –2 y 5 4 3 2 1 –1 –1 –2 –3 –4 –5 21 3 4 5 x 21 3 4 5 x For... |
3 + x2 + 2x 59. f (x) = x3 − 2x2 − 15x 57. f (x) = x4 − 81 60. f (x) = x3 − 0.01x EXTENSIONS For the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or −1. There may be more than one correct answer. 61. The y-intercept is (... |
Express the volume of the cube as a function of m, the number of minutes elapsed. 68. A rectangle has a length of 10 inches and a width of 6 inches. If the length is increased by x inches and the width increased by twice that amount, express the area of the rectangle as a function of x. 69. An open box is to be constr... |
the local behavior of quadratics, a special case of polynomials. In this section we will explore the local behavior of polynomials in general. Recognizing Characteristics of Graphs of Polynomial Functions Polynomial functions of degree 2 or more have graphs that do not have sharp corners; recall that these types of gr... |
form. 3. Technology is used to determine the intercepts. How To… Given a polynomial function f, find the x-intercepts by factoring. 1. Set f (x) = 0. 2. If the polynomial function is not given in factored form: a. Factor out any common monomial factors. b. Factor any factorable binomials or trinomials. 3. Set each fac... |
out the common factor. (x 2 − 1)(x − 5) = 0 Factor the difference of squares. (x + 1)(x − 1)(x − 5) = 0 Set each factor equal to zero. x + 1 = 0 or x − 1 = 0 or x − 5 = 0 x = −1 x = 1 x = 5 There are three x-intercepts: (−1, 0), (1, 0), and (5, 0). See Figure 4. y 18 12 6 f (x) = x3 − 5 x2 − x + 5 –6 –4 –2 2 4 6 x –6 ... |
to be factorable using techniques previously discussed. Fortunately, we can use technology to find the intercepts. Keep in mind that some values make graphing difficult by hand. In these cases, we can take advantage of graphing utilities. Looking at the graph of this function, as shown in Figure 6, it appears that the... |
+ 3)(x − 2)2(x + 1)3 2 4 x y 40 30 20 10 –10 –20 –30 –40 –4 –2 Figure 7 Identifying the behavior of the graph at an x-intercept by examining the multiplicity of the zero. The x-intercept x = −3 is the solution of equation (x + 3) = 0. The graph passes directly through the x-intercept at x = −3. The factor is linear (h... |
zero Zero with multiplicity 2 Zero with multiplicity 3 Figure 8 For higher even powers, such as 4, 6, and 8, the graph will still touch and bounce off of the horizontal axis but, for each increasing even power, the graph will appear flatter as it approaches and leaves the x-axis. For higher odd powers, such as 5, 7, a... |
zero occurs at x = −1. The graph looks almost linear at this point. This is a single zero of multiplicity 1. The last zero occurs at x = 4. The graph crosses the x-axis, so the multiplicity of the zero must be odd. We know that the multiplicity is likely 3 and that the sum of the multiplicities is likely 6. Try It #2 ... |
x → ∞, f (x) → −∞ x → −∞, f (x) → −∞ End Behavior: x → ∞, f (x) → −∞ x → −∞, f (x) → ∞ Figure 11 Understanding the Relationship Between Degree and Turning Points In addition to the end behavior, recall that we can analyze a polynomial function’s local behavior. It may have a turning point where the graph changes from ... |
+ 2x2) an = −(x 2)(2x 2) − 2x 4 Then, identify the degree of the polynomial function. This polynomial function is of degree 4. The maximum number of turning points is 4 − 1 = 3. Graphing Polynomial Functions We can use what we have learned about multiplicities, end behavior, and turning points to sketch graphs of poly... |
end behavior is that of a vertically reflected cubic, with the outputs decreasing as the inputs approach infinity, and the outputs increasing as the inputs approach negative infinity. See Figure 13. y Figure 13 x To sketch this, we consider that: • As x → −∞ the function f (x) → ∞, so we know the graph starts in the s... |
Intermediate Value Theorem states that for two numbers a and b in the domain of f, if a < b and f (a) ≠ f (b), then the function f takes on every value between f (a) and f (b). (While the theorem is intuitive, the proof is actually quite complicated and require higher mathematics.) We can apply this theorem to a speci... |
least one real zero between 3 and 4. We have shown that there are at least two real zeros between x = 1 and x = 4. Analysis We can also see on the graph of the function in Figure 18 that there are two real zeros between x = 1 and x = 4. –5 –4 –3 –2 y 10 8 6 4 2 –1 –2 –4 –6 –8 –10 f (1) = 5 is positive f (4) = 2 is pos... |
FUNCTIONS 387 Example 10 Writing a Formula for a Polynomial Function from the Graph Write a formula for the polynomial function shown in Figure 19. y 6 4 2 –6 –4 –2 2 4 6 x –2 –4 –6 Figure 19 Solution This graph has three x-intercepts: x = −3, 2, and 5. The y-intercept is located at (0, −2). At x = −3 and x = 5, the g... |
8 CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS local and global extrema A local maximum or local minimum at x = a (sometimes called the relative maximum or minimum, respectively) is the output at the highest or lowest point on the graph in an open interval around x = a. If a function has a local maximum at a, then f (a... |
latest. SECTION 5.3 GRAPHS OF POLYNOMIAL FUNCTIONS 389 Notice, since the factors are w, 20 − 2w and 14 − 2w, the three zeros are 10, 7, and 0, respectively. Because a height of 0 cm is not reasonable, we consider the only the zeros 10 and 7. The shortest side is 14 and we are cutting off two squares, so values w may ta... |
a zero of a polynomial function f? 2. If a polynomial function of degree n has n distinct zeros, what do you know about the graph of the function? 3. Explain how the Intermediate Value Theorem can 4. Explain how the factored form of the polynomial assist us in finding a zero of a function. helps us in graphing it. 5. ... |
= x3 − 9x, between x = 2 and x = 4. 26. f (x) = x5 − 2x, between x = 1 and x = 2. 27. f (x) = −x4 + 4, between x = 1 and x = 3. 28. f (x) = −2x3 − x, between x = −1 and x = 1. 29. f (x) = x3 − 100x + 2, between x = 0.01 and x = 0.1 For the following exercises, find the zeros and give the multiplicity of each. 30. f (x... |
x + 2)(x − 4) Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 5.3 SECTION EXERCISES 391 For the following exercises, use the graphs to write the formula for a polynomial function of least degree. 48. f(x) 49. f(x) 50. f(x) 5 4 3 2 1 –1–1 –2 –3 –4 –5 y 5 4 3 2 1 –1–1 –2 –3 –4 –5 21... |
y-intercept at (0, 9) 60. Degree 4. Root of multiplicity 2 at x = 4, and roots of multiplicity 1 at x = 1 and x = −2. y-intercept at (0, −3). Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 39 2 CHAPTER 5 POLYNOMIAL AND RATIONAL FUNCTIONS 61. Degree 5. Double zero at x = 1, and triple ze... |
–2 2 4 6 x –300 –200 –8 –16 –24 f(x) 2·105 1·105 (100, 0) (–300, 0) x 100 200 (0, –90,000) –400 –300 –200 –100 –1·105 –2·105 –3·105 –4·105 REAL-WORLD APPLICATIONS For the following exercises, write the polynomial function that models the given situation. 75. A rectangle has a length of 10 units and a width of 8 units.... |
ċ w ċ h = 61.5 ċ 40 ċ 30 = 73,800 So the volume is 73,800 cubic meters (m3). Suppose we knew the volume, length, and width. We could divide to find the height. h = V ____ l ċ w = 73,800 _______ 61.5 ċ 40 = 30 As we can confirm from the dimensions above, the height is 30 m. We can use similar methods to find any of the... |
to long division of whole numbers. We can write a polynomial dividend as the product of the divisor and the quotient added to the remainder. The terms of the polynomial division correspond to the digits (and place values) of the whole number division. This method allows us to divide two polynomials. For example, if we... |
der Writing the result in this manner illustrates the Division Algorithm. Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 5.4 DIVIDING POLYNOMIALS 395 the Division Algorithm The Division Algorithm states that, given a polynomial dividend f (x) and a non-zero polynomial divisor d(x... |
x − 2) 0 Multiply x + 1 by 5x. Subtract. Bring down the next term. −2x divided by x is −2. Multiply x + 1 by −2. Subtract. The quotient is 5x − 2. The remainder is 0. We write the result as or 5x2 + 3x − 2 __________ x + 1 = 5x − 2 5x2 + 3x − 2 = (x + 1)(5x − 2) Analysis This division problem had a remainder of 0. This... |
3x − 2 • the quotient is 2x2 + 5x − 7 • the remainder is 1 Try It #1 Divide 16x 3 − 12x 2 + 20x − 3 by 4x + 5. Using Synthetic Division to Divide Polynomials As we’ve seen, long division of polynomials can involve many steps and be quite cumbersome. Synthetic division is a shorthand method of dividing polynomials for ... |
more clear in Example 3. synthetic division Synthetic division is a shortcut that can be used when the divisor is a binomial in the form x − k where k is a real number. In synthetic division, only the coefficients are used in the division process. How To… Given two polynomials, use synthetic division to divide. 1. Wri... |
2, and repeat until the last column is reached. −2 4 4 −6 −20 10 −8 −4 20 2 −10 0 The result is 4x 2 + 2x − 10. The remainder is 0. Thus, x + 2 is a factor of 4x3 + 10x2 − 6x − 20. Analysis The graph of the polynomial function f (x) = 4x3 + 10x2 − 6x − 20 in Figure 2 shows a zero at x = k = −2. This confirms that x + 2... |
in Figure 3. Height Width x − 2 Length 3 x Figure 3 We can now write an equation by substituting the known values into the formula for the volume of a rectangular solid. V = l ċ w ċ h 3x4 − 3x3 − 33x2 + 54x = 3x ċ (x − 2) ċ h To solve for h, first divide both sides by 3x. 3x ċ (x − 2) ċ h ____________ 3x = 3x4 − 3x3 −... |
degree of the quotient? ALGEBRAIC For the following exercises, use long division to divide. Specify the quotient and the remainder. 3. (x2 + 5x − 1) ÷ (x − 1) 6. (4x2 − 10x + 6) ÷ (4x + 2) 9. (2x2 − 3x + 2) ÷ (x + 2) 12. (x3 − 3x2 + 5x − 6) ÷ (x − 2) 4. (2x2 − 9x − 5) ÷ (x − 5) 7. (6x2 − 25x − 25) ÷ (6x + 5) 10. (x3 −... |
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