CHAPTER 8 Quadratic Equations, Functions, and Inequalities
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1 CHAPTER Quadratic Equations, Functions, and Inequalities Section. Solving Quadratic Equations: Factoring and Special Forms Section. Completing the Square Section. The Quadratic Formula Section. Graphs of Quadratic Functions Section. Applications of Quadratic Equations Section. Quadratic and Rational Inequalities Review Eercises
2 CHAPTER Quadratic Equations, Functions, and Inequalities Section. Solutions to Even-Numbered Eercises Solving Quadratic Equations: Factoring and Special Forms y 7y yy y 0 y 0 y 0 y u u u u u u u u u 0 u u u 0 u 0 u u 0. z z z z. z. t z z z z ± t z z 0 z ± t ± z z 0 z 0 z z z 0 z. 9z. z 9 z ± 9 z ± 0. ± ± y 0 y y y ± y ± 7
3 Chapter Quadratic Equations, Functions, and Inequalities. 0. y y 0 ± ±0. y 0 ± ±0.9 ± y 0 ± ± 0.9 ± y y ± ± 00 ±7 ± ±00 ± 7 ± ±0 ± ± 0 ± 0.. y 0. v 9 0 ± y v 9 ±i y ± v 9 y ±i v ± 9 v ± i 0.. y 0. y 0 y y ± y ± y ± i ± ±9i ± 9i y ± i y ± i y ± i y ± i ± 9 ± i ± i ± ±i ± i ± i ± i. ±. ±.i
4 Section. Solving Quadratic Equations: Factoring and Special Forms 9. u 9 0. u 9 y y ±. t t 0 tt 0 t 0 t 0 u ± 9 y ± i t 0 t u ±7 i y ± i u ± 7 i y ± i y ± i y 0 7. y 0 0 y y 0 0 y ± y ± y ± y ±i ± ± y 00 0 y y 00 y ±0 y ± 0 y 7. y y 00 y ±0i y ± 0i 0 ± ±. y. y 9 Y X,T, X,T, GRAPH Y 9 GRAPH X,T, 0 0 -intercepts are and intercepts are 0 and. ± 9 0, 0,, 0 ± ± 9, 0, 9, 0
5 90 Chapter Quadratic Equations, Functions, and Inequalities. y 9. y Y 9 X,T, GRAPH Y X,T, GRAPH X,T, -intercepts are and. -intercepts are and ± , 0,, 0 ± ±, 0,, y 9 Y 9 GRAPH X,T, -intercepts are and , 0,, 0 X,T, 9. y Y GRAPH X,T, 0 ± ±i Not real, therefore, there are no -intercepts y 9. Y GRAPH X,T, Y GRAPH X,T, ± ± ± i ±i ± i 0 Not real, therefore, there are no -intercepts. Not real, therefore, there are no -intercepts. 9. y 00. y : y : y y y ± Y X,T, X,T, ENTER GRAPH y 0 y : y : y y y ± Y X,T, ENTER X,T, GRAPH 7
6 Section. Solving Quadratic Equations: Factoring and Special Forms Let u. 0 0 u 0u 0 u u 0 u 0 u 0 u u ± ± 0 0 Let u. 0 0 u u 0 0 u u 0 u 0 u 0 u u ± ± Let u. Let u. 0 0 u u 0 u u 0 u u 0 u u 0 u 0 u 0 u u u u ± ±i ± 9 Check: Check: 9 9? 0? 0 9? 0 0 0? Let u. 0 u u 0 u u 0 u u Check: Check:? 0 9 9? 0? 0 9?
7 9 Chapter Quadratic Equations, Functions, and Inequalities Check: 79 79? 0 Check:? 0 9 i 0 9? 0 Check:? 0 0 0? 0 Check:? 0 0 0? Let u. u u 0 u u 0 u u Let u. u u 0 u 7u 0 u 7 u Let u. 9u u 0 u u 0 u
8 Section. Completing the Square 9. S r. s 0. s r 900 r r 0 t t t t ± t.7 seconds 0 t 00 t 00 t. t..90 sec r r.77 inches P $000, A $7.0 A P r r r, , You must sell 00 units to produce a revenue of $7000. If you sell 0 units, you will also have a revenue of $ r.9 r.07 r 0.7 r 7% r. 00.t 7. If a 0, the equation would not.t be quadratic.. t t 9; Yes. For the quadratic equation 0, the only solution is.. Write the equation in the form u d, where u is an algebraic epression and d is a positive constant. Take the square roots of each side to obtain the solutions u ±d. Section... Completing the Square y y.. u 7u y y. y y s.s.9.9.
9 9 Chapter Quadratic Equations, Functions, and Inequalities (a) (b) 0 0, ± ± 0 t 0t 0 (a) t 0t t t ± t ± t 0, 0 (b) t 0t 0 tt 0 0 t 0, 0. t 9t 0. (a) (b) t 9t t 9t 0 tt 9 0 t 0 t 9 0 t 9 ± t 9 ± 9 t 9 ± 9 t 9, 0 t 9 0 t 9 y y 0 (a) y y 0 y y y y y y ± y ± y ± y, (b) y y 0 y y 0 y 0 y 0 y y. z z 0 0. t t 0 (a) z z 0 0 (a) t t 0 z z 0 t t z z t t z 9 t 9 z ± 9 t ±9 z ± 7 t ± z ± 7 t ± z, t 9, (b) z z 0 0 (b) t t 0 z z 0 t 9t 0 z 0 z 0 t 9 0 t 0 z z t 9 t
10 Section. Completing the Square (a) 0 (a) ± ± ±,, ± (b) 0, 0 (b) ± ± ± ± ± ±.9,., 9., ± ± ± ± i ± ±.i,.i, 9,. y y 7 0. y y y y ± y ± y.9, z z 0 0 z z z ± z ± ± z ±.,.9 z., 9.
11 9 Chapter Quadratic Equations, Functions, and Inequalities. 0. y y ± y y 9 y y y ± 9 ± 9 ± 0.,. y ± i y.0.i y.0.i ±9 ± 9 0.,.. u u u u 9 9 u 9 u 9 9 u ± 9 u ± i u 0..i u 0..i 0 ± ± ± 9.0, ± 7 0 ± 7 ± 0 ± 0., 0.. z z 0. z z z z 9 9 z z ± z ± z ± i z i z i ± ± 7i i i
12 Section. Completing the Square ± ± ± ± 7 ± 7 ± ± ± 0.,. ± 0 0.,. ± 0.0, ± ± ± ± Not a solution Check:??? Check:???
13 9 Chapter Quadratic Equations, Functions, and Inequalities 0. y. y Y X,T, X,T, GRAPH Y GRAPH X,T, X,T, ± ± ,. ±, ± 9 ± 9 0., 0. ± 9, 0. y. y Y GRAPH X,T, X,T, Y GRAPH X,T, X,T, ±7 ± 7 9., 0. ± 7, 0 ± 9 ± ± Not a solution Check: 0? 0? 0 0 Check: 0? 0? 0
14 Section. The Quadratic Formula 99. (a) Area of square 9 Area of vertical rectangle Area of horizontal rectangle Total area 9 9 (b) Area of small square Total area 9 (c) Verbal Model: Labels: Equations: Area Length Width Length Width feet length feet width 9. Verbal Model: Volume Length Labels: Length Width Width Height Height Not a solution Thus, the dimensions are: length 0 inches, width 0 inches, height inches 9., , , , , ,000 9,79 0, , 00 ±0, 00 ± 00 ±, 9 Thus, 9 or golf clubs must be sold. 9.. Divide the coefficient of the first-degree term by, 9. Yes. 0 and square the result to obtain. 00. True. Given the solutions r and r, the quadratic 0. The student forgot to add to the right side of the equation can be written as r r 0. equation. Correct solution: ± ± Section. The Quadratic Formula
15 00 Chapter Quadratic Equations, Functions, and Inequalities. (a) 7 0 (b) 9 0 ± 7 ± ± ± 9,. (a) (a) ± 9 ± ± ± 9 ± ± 0 9 ±, 7 (b) 0 (b) (a) (a) 0 0. (a) ± ± ± ± ± ± ± 0 ± 0 0 ± 00 0 ± 0 0 ± 0 0, 0 (b) (b), (b)
16 Section. The Quadratic Formula y y 0. 0 ± y ± ± ± y ± ± ± y ± 0 ± 0 ± 7 ± 7 y y ± ± ± ± ± 7 y ± ±. u u u ± 9 ± ± u ± ± ± u ± ± 7 ± 0 u u ± 7 ± 7 ± 7 i ± 7 i ± ± u ± 7 ± ± ± 9 ± ± i ± i y y y ± y ± y ± 9 y ± ± y y ± ± 0 ± 00 0 ± 0 ± 7 0 ± 7 0 ± 7
17 0 Chapter Quadratic Equations, Functions, and Inequalities Divide by. ± 9 ± 9 ± ±,, ± 0 ± 0 ± ±, 0, ± ± ± 09 0 ± ± 0 0 ± ± ± ± ± ± 9 9 ± 9 ± 00 ± 0 ± 0 ± ± b ac 0. b ac b ac Two distinct irrational solutions 0 Two distinct comple solutions 0 Two distinct rational solutions
18 Section. The Quadratic Formula 0. b ac 0. b ac Two distinct irrational solutions One (repeated) rational solution t t ± t ± 0. 7u 9u yy 7 y 7 0 7uu y 7y 0 7u 0 u 0 u 7 0 u 7 9 ± 9 y 7 0 y 7 y 0 y ± i. 0. y y y y 9 0 y 0 y 9 0 y y 9 ± ± 00 ± 7 ± 7 i ± 0 ± 0 ± ± 7 ± 9 ±
19 0 Chapter Quadratic Equations, Functions, and Inequalities. i i. i 0 i 0 i i 0 i y. y Y X,T, X,T, GRAPH Y X,T, X,T, GRAPH 0 0 ± ± ± , 0,, 0 ± i No -intercepts 90. y 0 9. y Y X,T, X,T, 0 GRAPH Y.7 X,T, 0. X,T,. GRAPH ± 0 0. ± ± ± ± ±. 7. ± 70 0 ± , 0..0, 0, 0., 0 ± ,.7., 0,.7, Y X,T, X,T, GRAPH b ac 7 9 Two real solutions
20 Section. The Quadratic Formula Multiply by. 7 0 Y X,T, X,T, 7 GRAPH b ac No real solutions 9. f h ± 0 ± ± 7 No real values 0 0 ± 9 ± ± ± ± 0. Check:? ± 7 ± ± ± ± ± 0 7 does not check. Check:??..??? 0. 0.
21 0 Chapter Quadratic Equations, Functions, and Inequalities 0. c 0 (a) b ac > 0 (b) b ac 0 (c) b ac < 0 c > 0 c 0 c < 0 c > 0 c 0 c < 0 > c c < c > c c < c 0. c 0 (a) b ac > 0 (b) b ac 0 (c) b ac < 0 c > 0 c 0 c < 0 c > 0 c 0 c < 0 > c c < c > c c < c 0. Verbal Model: Labels: Area Length Width. Length. Width. ± ± inches.. inches. h t 0t 0 (a) 0 t 0t 0 0 t 0t 0 tt t 0 (b) 0 t 0t 0 0 t t 0 t t ± t reject t 0 ± 0 t 0 t. seconds t ± 0. seconds. (a) t t (b) 0 t t 0 tt 9 0 t t t t 9. seconds 0 t t t ± t t ± 9 90 ± 00 t.;. reject t. seconds
22 Section. Graphs of Quadratic Functions 07. (a) t 0t (b) 0 t 0t 0 t 0t 0 t t 0 tt 0 t t 0 0 t t seconds t ± t t ± 7 ± 97 t.; 0.9 reject t. seconds. (a) Y 7.9 X,T,. X,T, 7,7 GRAPH 0,000,000 (b) 99 s million,000,000,000 (in thousands), t.t 7, t.t, t. ±. 7.9, 7.9 t ; a b c 0. Compute b plus or minus the square root of the Solutions are and. quantity b squared minus ac. This quantity divided by the quantity a is the Quadratic Formula. b a, c a. The Quadratic Formula is derived by solving the general quadratic equation a b c 0 by the method of completing the square. Section. Graphs of Quadratic Functions. y (f). y (c). y (a). y 0. y y verte, y. y y 9 9 y verte, y y verte,
23 0 Chapter Quadratic Equations, Functions, and Inequalities. y. y 0 0. y 9 y 0 0 y y 0 0 y y 0 0 y 9 y 0 y verte, 0 verte, y 9 verte, 9 0. f. h. y 9 a, b a, b a 9, b b a b a 7 b a 9 f b a h b a 7 7 y 9 verte, 9 9 verte 7, 9 9 verte,. y. y 0. y < 0 opens downward. > 0 opens upward. < 0 opens downward. verte, verte, verte, 0. y. y > 0 opens upward. < 0 opens downward. y verte 0, y 9 9 y 9 verte, 9. y 9. y y y 7 7 y 0 9 y 0 0 7, 0 and 7, 0 y 9 0, 0 and, 0 y 0 0, 9 0, 0
24 Section. Graphs of Quadratic Functions y. y ± No ± 0 -intercepts y y 0 0 y 0, 0 0, 0 and, 0 0 y 0 y y 0 0, 0. y 0 y 0. y 9 y y y , 0 and, 0 0 y 0 0, 0 ± 0 No ± -intercepts y 9 0, 9. h 9 y 0. f 9 y -intercepts: (, 0) (, 0) (0, 9) 0 -intercepts: ± verte: 0 (, 0) (0, 9) (, 0) verte: f 0 9 h 9 h 0 9. g y. -intercepts: verte: 0 y (0, 0) (, 0) (, ) y -intercepts: verte: y 0 y (, ) (0, 0) (, 0)
25 0 Chapter Quadratic Equations, Functions, and Inequalities. y -intercepts: 0 0 verte: f y (, 0). y 0. y -intercepts: -intercepts: 0 0 ± ± ± y verte: 0 y ± verte: ± y (, 0) (, ) ( +, 0) y 9 0 (, 9) (, 0) (, 0). f 7. y y verte: f 7 f -intercepts: 0 7 ± 7 ± ± (, ) y -intercepts: 0 ± ± No -intercepts verte: ± y (, ) No -intercepts
26 Section. Graphs of Quadratic Functions. y 7 -intercepts: verte: y 0 7 ± 7 ± y (, ) ( +, 0) (, 0) ± ± ±. y 7 -intercepts: verte: 0 7 y 7 y ± 7 ± 7 7 ± 0 7 (, ) no -intercepts 70. f 7. h -intercepts: y 0 y ± verte: (, 0) (, 0) (0, ) Vertical shift units down f f 0
27 Chapter Quadratic Equations, Functions, and Inequalities 7. h 7. h 7. h y y y Horizontal shift units right Vertical shift units down Reflection in the -ais Horizontal shift units right Vertical shift units down 0. y 0 Y 0 GRAPH verte Check: y a, b a y, b X,T, 0 X,T,. y Y.7 7. GRAPH verte,. Check: y a 0.7, b a y X,T, b X,T, 7 0
28 Section. Graphs of Quadratic Functions. verte, 0; point 0,. verte, ; point 0, a0 0 y 0 y a y a y a0 y a y a y a y a. verte, ; a 90. verte, ; point 0, 0 y a h k y a h k y y a y y a y 0 a0 y 9 0 a y a a y y y 9. verte, ; point 0, 9. verte, ; point 0, y a h k y a y a a0 a a y y y 0 y 0 y a h k y a a0 a a a y y 0 y y 9. y (a) y 0 0 (b) y (c) 0 feet Maimum height feet 0 0 ± 0 ± 0 0 ± ±. feet
29 Chapter Quadratic Equations, Functions, and Inequalities 9. y y 70 (a) (b) (c) y y 9 feet y y Maimum height feet ± 0 ± 0 ±.; 0. reject. feet (a) (b) (c) y y feet y y Maimum height 7 feet ± ± 9, ± 0,0 0.99; 0.99 reject 0.99 feet 0. y 0 A y 0 y Maimum height.0 feet Y 00 X,T, 0 when A is maimum. 000 X,T, GRAPH P 0 0s s Y 0 0. GRAPH X,T, P 0 0s s X,T, 00 P s 0s 0 P s 0s P s 0 0 Amount of advertising that yields a maimum profit $000 Maimum profit $0,
30 Section. Applications of Quadratic Equations a To find the verte of the graph of a quadratic function, 00 a0,000 use the method of completing the square to write the function in standard form f a h k. The 00 0,000 a verte is located at point h, k. 00 a y y 00. The graph of a quadratic function f a b c opens upward if a > 0 and opens downward if a < 0.. It is not possible for the graph of a quadratic function to have two y-intercepts. All functions must have only one y-value for each -value. Section. Applications of Quadratic Equations. Verbal Model: Selling price per computer Cost per computer Profit per computer 7,000 7, ,000 7, ,000,000 7, , computers Selling price 7,000 $ Verbal Model: Selling price per sweatshirt Cost per sweatshirt Profit per sweatshirt sweatshirts reject Selling price 0 $
31 Chapter Quadratic Equations, Functions, and Inequalities. Verbal Model: Length Width Perimeter. Verbal Model: Length Width Area.w w 0.w w 7w w 0.w 9w 0 w w 0 w l.w 70 l.w Verbal Model: Length Width Area Verbal Model: Length Width Perimeter 70 0 A m A P 0 centimeters P 0. Verbal Model: Length Width. Verbal Model: Length Width Perimeter Area l l 700 l l 0 l 700 l l 0 l 00 l 0 l 0 l 0 w l w l Verbal Model: Length Width Perimeter Verbal Model: Length Width Area 0 P 0 A 0 inches P 70 ft A. Verbal Model: Area Length 00 w w 00 w w 0 w w Width 0 w w 0 w w 0 reject l w Verbal Model: Length Width Perimeter 0 P 90 feet P. Verbal Model: Labels: Area Height Base Height Base inches reject inches. Verbal Model: Length Width Area 0,00 0,00 0, The lot is feet 0 feet.
32 Section. Applications of Quadratic Equations 7 0. (a) Verbal Model: Area Length Width Not real therefore cannot enclose a rectangular region. (b) Verbal Model: Area 0 ± 00 0 A 00 A 00 A 79 square feet Yes, can enclose a circular area ± 0 0 Radius A 0 r 0 ± 0 w w w 00 w 0 w 0 C r 00 r 00 r 0 r. Verbal Model: Labels: Length Width Area Length Width reject inches Each side should be reduced by inches.. A P r r. r.0 r 0.0 r or % A P r r 0.90 r r.0 r 0.0 r % r A P r r.07 r.00 r 0.00 r or.%
33 Chapter Quadratic Equations, Functions, and Inequalities 0. Verbal Model: Cost per member Number of members $0 Labels: Number of members Number going to game Verbal Model: Investment per person; current group Investment per person; new group 000 Labels: Number in current group Number in new group 0,000 0,000 0, ,000 0,000 0, , ,000 0,000 0, ,000 0, investors. Common formula: a b c , ± ± 0, ± yards,. yards
34 Section. Applications of Quadratic Equations 9. (a) d 00 h d 0,000 h 00 (b) Y 0,000 GRAPH X,T, (c) When d 00 feet h is approimately 7. feet (d) h d d 0,000 0 d 0, d 0, d 0, ,000 0,000 0,000 0,000 0,000 90, ,000 00, Verbal Model: Work done by Person Work done by Person One complete job ± 0 0 ± 00 0 ± 0 ± 7 ± 7.,. Thus, it would take person one. hours to complete the task alone and it would take person two.. hours. 0. Verbal Model: Rate Company A Rate Company B Rate together Labels: Time Company A Time Company B 0 ± ± ± 7. days. 9. days
35 0 Chapter Quadratic Equations, Functions, and Inequalities. h h 0 t t t 79 t. t.7 or seconds h h 0 t. 0 9 t t 9 t. t 7. seconds t t 0 t t t ± ± 9 ±. t., 0.0 t t t ±. seconds will pass before you hit the ball.. h t t (a) t t (b) 0 t t 0 t t t ± t t ± ± 7 0. second and 0.9 second t ± t t ± 0 ± 7 t 0. reject t. seconds (c) t h.9 feet Maimum height is.9 feet. 0. Verbal Model: Integer Integer Product n n 0 n n 0 0 n n 0 n 0 n 0 n n reject n n. Verbal Model: Even integer Even integer nn 0 n n 0 n n 0 0 n n 70 0 n 7n 0 Product n 7 0 n 0 n 7 n reject n n
36 Section. Applications of Quadratic Equations. Verbal Model: Odd integer Odd integer n n n n n n 0 0 n n 0 0 n 0n 0 n 0 0 n 0 n 0 n reject n 7 The integers are 7 and 9. Product n 9. Verbal Model: Time for part 00 r 00 rr rr r r 00r r r r 00r 00 r r r r 0 0 r 0 Time for part r 0 r 0r 00 0 r r 00 0 r 0r r 0 r reject Thus, the average speed for the first part of the trip was 0 miles per hour.. d y y 9 ± ±,, y (, ) (,) y = d d (, ) S.t 0.t 7 (a) Y. X,T, 0. X,T, 7 GRAPH,000 (b) Sales were approimately $. billion (00 million) in the year t 0.t 7 0.t 0.t 7 Use the Quadratic Formula to solve for t The four strategies that can be used to solve a quadratic equation are factoring, the Square Root Property, completing the square and the Quadratic Formula.. 0 feet minute seconds feet minute 0 seconds
37 Chapter Quadratic Equations, Functions, and Inequalities Section. Quadratic and Rational Inequalities. 0. 9y 0. yy y y y 0 y y 0 0 y 0 y 0 y 0 y 0 Critical numbers 0, y y y y y y Critical numbers:, Critical numbers, , Negative:, Critical numbers, Critical numbers:,.. 7. Negative:, Negative:, 0, 0, Negative:, 9, Negative:,, 0.. > 0 ± ± ± 0 ± 0 ± 0 ± 0 Negative: 0, 0 0, 0, Critical numbers: 0,, 0 Negative: 0,, Solution:, 0, 0. > 0. z 9 Critical numbers: 0, z 9 0 z z 0 Negative:, 0 0, Negative:, Solution: 0, 0 Critical numbers: z,, Negative:,, 0 z Solution:,
38 Section. Quadratic and Rational Inequalities. 7 < t t > 7 < 0 0 t t > 0 Critical numbers: 7, Critical numbers: 0, t t > 0 Critical numbers:,, 7, 0 Negative: 7, Negative: 0,,,, Negative:, Solution: 7, Solution:, 0,, Solution:,, t 0. t t 0 < 0. 0 > 0. y y > 0 t 0t < 0 Critical numbers:, 0, Negative:, 0 ± 0 ± ± i y y > 0 Critical numbers: y,, Negative:, 0, Solution:, 0 ± i No critical numbers, Solution:,, 0 t 0 is greater than 0 for all values of. Solution:, 0 y ± ± 0 ± ± ± ± 0 Critical numbers: ±, Negative: Solution: 0,,,, + 0
39 Chapter Quadratic Equations, Functions, and Inequalities. < 0. y y 0. t t 0 0 < 0 y 0 t t 0 is not less than zero for any value of. Solution: none Critical number: y, Solution: Critical numbers: t, Negative:,, y, 0 0 Solution:,, 0 t. < < 0 0 Critical numbers: 7, 9 0 Negative: 7, 9, 7 9, Solution: 7, Critical numbers:,, Negative:,, Solution:,, y 0 0 y 0 for all real numbers. 0 for all real numbers. Solution:, Solution:, 0 0 y. y y y y y 0 y y 9 0 y y 0 y ± ± ± ± ± y ± Critical numbers: ±, Negative: Solution: +,,,, 0 y
40 Section. Quadratic and Rational Inequalities Critical numbers:, 0, 0 0 Critical numbers:, Negative:,, 0 0 Negative: 0,, Negative:, 0, Solution:, 0,, Solution:,.. Y X,T, X,T, TEST 0 GRAPH Y X,T, X,T, TEST 0 GRAPH > 0 > 0 < 0 9 < 0 Critical numbers:, Critical numbers: 0, 9 Negative:,, 0 Negative:, 0 0, 9 0, 9, Solution:, 0, Solution: 0, 9. Y X,T, X,T, 9 TEST GRAPH 9 < 7 < 0 7 < 0 Critical numbers:, 7, Negative:, 7 7, Solution:, Y X,T, X,T, TEST GRAPH > > 0 < 0 < 0 Critical numbers:,, Negative:,, Solution:, or Critical number: Critical numbers:, 0
41 Chapter Quadratic Equations, Functions, and Inequalities 7. > 0 7. > 0 Critical number: Critical number:, 0, 0 Negative:, Negative:, Solution:, Solution:, Critical numbers:, 0. < 0 Critical numbers:,, Negative:,,, Negative:, Solution:,,, 0 0 Solution:,. y y 0. < 0 Critical numbers: y, Negative:,,, 0 Solution:,, 0 y Critical numbers: Negative:,,,, Solution:, 0. u u 0. t > 0 t Critical numbers: Critical numbers: u, t,, Negative:,, Solution:, 7 u Negative:,, Negative:, Solution:, 0 t
42 Section. Quadratic and Rational Inequalities > 0 > 0 > 0 7 > 0 > Critical numbers: 7 9., Negative:, 7 7,, Solution:, 7, < < 0 0 < 0 < 0 0 < 0 Critical numbers:, 0 Negative: 0,, Solution: 0, 0 0 0, Critical numbers: Negative:,, Negative:, Solution:,, 7 9. < 0 Y TEST 0 GRAPH < 0 < 0 X,T, < 0 > 0 Critical numbers: Negative:, 0 0,, 0, Solution:, 0, 9. 0 Y TEST 0 GRAPH X,T, X,T, 0 0 Critical numbers:,, 0 0 Negative:,, Solution:, 9 9 0
43 Chapter Quadratic Equations, Functions, and Inequalities 00. < TEST GRAPH X,T, X,T, Y < < 0 Critical numbers:,, 0 < 0 < 0 Negative:,, Solution:, 9 9 < 0 0. > Y TEST GRAPH X,T, > Critical numbers: 0., 0, 0. > 0, 0. > 0 Negative: 0., 0, 0, 0. 0., Solution:, 0. 0., 0. y (a) y 0 (b) y 0. y (a) y (b) y 0 GRAPH Y GRAPH X,T, X,T, Y Solution: Solution: (a), Look at -ais and vertical asymptote. (b), (Graph y and find intersection.) (a) (b), (Graph y and find intersection.) 0, (Look at -ais.) X,T, X,T,
44 Section. Quadratic and Rational Inequalities 9 0. h t t 0. t t 0 > 0 t t < 0 Critical numbers: t 7, Negative: Solution: t t > 0 height > 0 7 7, 7 7, 0.,. 7, t r > 0 r > 0 r r > 0 0 0r 0r > 0 0r 0r > 0 Critical numbers: r r > 0 00 r cannot be negative Solution: 0 0.0,, r >.% , 0 0,,. Area 00 l0 l 00 0l l 00 0 l 0l 00 l 0 ± 0 00 l l l l 0 ± ± 00 0 ± 0 ± Critical numbers: l ± 0, Negative:,, 0 Solution:, l ±. Verbal Model: Inequality: Profit P , ,000,000, ,, ± ,, Revenue , ,000. ± 7., ,90; 70,0 Cost. ± Critical numbers: 7,90, 70,0, 7,90 Negative: 7,90, 70,0 70,0, Solution: 7,90, 70,0
45 0 Chapter Quadratic Equations, Functions, and Inequalities. (a) Y..7 X,T,. X,T,.00 X,T, GRAPH (b) Let y 00 and find the intersection of the graphs. Solution:.,.,. t The direction of the inequality is reversed, when both sides are multiplied by a negative real number. 0. An algebraic epression can change signs only at the -values that make the epression zero or undefined. The zeros and undefined values make up the critical numbers of the epression, and they are used to determine the test intervals in solving quadratic and rational inequalities.. < 0 is one eample of a quadratic inequality that has no real solution. Any inequality of the form c < 0, c any positive constant or c > 0, c any positive constant will not have a real solution. Review Eercises for Chapter. u u 0. z uu 0 z u 0 u 0 z z 0 0 u z 0 z z z ±9 ±7. y u y ±0 u ± y ± ± 0 u ±i y ± 7, ± ± ±i ± i 9 ±9 ± ± ±
46 Review Eercises for Chapter. 0 Check: 9 9? 0 0 9? Not a solution Check:? 0? Check:? 0 0? 0 0 0? 0 0 0? 0 0 Not a solution 9 Check: 9 9? 0? 0? 0? y 0y ±0 ± 0 0.;. u u 0 u u u u ± u ±,,
47 Chapter Quadratic Equations, Functions, and Inequalities. t t 0. t t t t 7 t ± 7 t ±7 t ± 7 ± 7 t t 0.7; ± 7 ± ± 9 ± 7, ± 0 ± 9 0 ± 9 ±, y y 9 0. r r 0 ± ± ± 9 b ac One repeated rational solution b ac 0 0 Two distinct irrational solutions ± ± y 0. 0 b ac 7 b ac Two distinct comple solutions Two distinct irrational solutions i i i 0 i 0 i 0 i 0 i i 0 i
48 Review Eercises for Chapter. g 9. 9 Verte:, y 9 9 Verte:, 9. y y 70. -intercepts: Verte: y y 9 9 9, 9 9, (0, 0) (, 0) ( ( y 0 -intercepts: Verte: y , 9 y 0 (, 0) (, 0) 0 ( 9, ( 7. Verte:, 0 y-intercept: 0, y a h k y a 0 a0 a y 7. Verte:, Point: 0, y a h k y a y a a0 a a a y 7. (a) Y. X,T,. X,T, 9 GRAPH CONTINUED
49 Chapter Quadratic Equations, Functions, and Inequalities 7. CONTINUED (b) Maimum number of bankruptcies,,000 occuring in the year 99 N ,,0 Maimum occurs when t. 7. Verbal Model: Selling price per computer Cost per computer Profit per computer Labels: Number computers sold Number computers purchased 700 7, ,000,000 7, reject 7,000 7,000 Selling price of each computer 7, ,000 7, , computers $ A P r,9.,000 r.0 r.0 r 0.0 r.%. Formula: c a b Labels: c 9 a b a b b b feet and feet. (a) t t 9 (b) 0 t t 9 0 t t 0 t t 0 t t seconds 0 t t 0 t t t 0 discard t t 0 t seconds
50 Review Eercises for Chapter Critical numbers: 0, 0 0 Critical numbers:, > 0 Critical numbers: 0, 0 > 0 > 0 Negative:, 0 Critical numbers:, 0, Negative: 0, Solution:, 0 0,, Negative:, 0 0, Solution:,, 9. > 0 > 0 Critical numbers:, Negative:,,, Solution:,, 0 9. > 0 Critical numbers: Negative:,,,, Solution:,, C C 00, , Critical numbers:, 9 00, <, Negative:, 9 9, ,000 00,000. Critical numbers:. < 0 < 0 0, 90,909 Solution:, 9 must be positive. 0, 90,909 Negative: 90,909, Solution: 90,909, ; > 90,909
Applications of Quadratic Equations
33 Chapter 6 Quadratic Equations and Inequalities Section 6. Applications of Quadratic Equations. Verbal model: Selling price per doz eggs.6 Number eggs sold Number eggs purchased 6.6 6.6.3 6.6 9.6.6.3.8
Graph is a parabola that opens up if a 7 0 and opens down if a 6 0. a - 2a, fa - b. 2a bb
238 CHAPTER 3 Polynomial and Rational Functions Chapter Review Things to Know Quadratic function (pp. 150 157) f12 = a 2 + b + c Graph is a parabola that opens up if a 7 0 and opens down if a 6 0. Verte:
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