NAME DATE PERIOD. Study Guide and Intervention

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1 NAME DATE PERID Stud Guide and Intervention Graph To graph a quadratic inequalit in two variables, use the following steps: 1. Graph the related quadratic equation, = a 2 + b + c. Use a dashed line for < or >; use a solid line for or. 2. Test a point inside the parabola. If it satisfies the inequalit, shade the region inside the parabola; otherwise, shade the region outside the parabola. Eample Graph the inequalit > First graph the equation = B completing the square, ou get the verte form of the equation = ( + 3) 2-2, so the verte is (-3, -2). Make a table of values around = -3, and graph. Since the inequalit includes >, use a dashed line. Test the point (-3, 0), which is inside the parabola. Since (-3) 2 + 6(-3) + 7 = -2, and 0 > -2, (-3, 0) satisfies the inequalit. Therefore, shade the region inside the parabola. Eercises Lesson Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Graph each inequalit. 1. > < > Chapter 4 49 Glencoe Algebra 2

2 NAME DATE PERID Stud Guide and Intervention (continued) Solve Quadratic inequalities in one variable can be solved graphicall or algebraicall. Graphical Method Algebraic Method To solve a 2 + b + c < 0: First graph = a 2 + b + c. The solution consists of the -values for which the graph is below the -ais. To solve a 2 + b + c > 0: First graph = a 2 + b + c. The solution consists of the -values for which the graph is above the -ais. Find the roots of the related quadratic equation b factoring, completing the square; or using the Quadratic Formula. 2 roots divide the number line into 3 intervals. Test a value in each interval to see which intervals are solutions. If the inequalit involves or, the roots of the related equation are included in the solution set. Eample Solve the inequalit First find the roots of the related equation = 0. The equation factors as ( - 3)( + 2) = 0, so the roots are 3 and -2. The graph opens up with -intercepts 3 and -2, so it must be on or below the -ais for Therefore the solution set is { -2 3}. Eercises Solve each inequalit < < < c m 2 - m < < < > > < 0 Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc < < 0 Chapter 4 50 Glencoe Algebra 2

3 NAME DATE PERID Skills Practice Graph each inequalit > Solve each inequalit b graphing > 10 Lesson Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Solve each inequalit algebraicall < > < > > > < 0 Chapter 4 51 Glencoe Algebra 2

4 NAME DATE PERID Practice Graph each inequalit > < Solve each inequalit > > < > > FENCING Vanessa has 180 feet of fencing that she intends to use to build a rectangular pla area for her dog. She wants the pla area to enclose at least 1800 square feet. What are the possible widths of the pla area? 20. BUSINESS A biccle maker sold 300 biccles last ear at a profit of $300 each. The maker wants to increase the profit margin this ear, but predicts that each $20 increase in profit will reduce the number of biccles sold b 10. How man $20 increases in profit can the maker add in and epect to make a total profit of at least $100,000? Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 4 52 Glencoe Algebra 2

5 NAME DATE PERID Word Problem Practice 1. HUTS The space inside a hut is shaded in the graph. The parabola is described b the equation = ( - 1) DAMS The Hoover Dam is a concrete arch dam designed to hold the water of Lake Mead. At its center, the dam s height is approimatel 725 feet, and the dam varies from 45 to 660 feet thick. The dark line on this sketch of the crosssection of the dam is a parabola. Cross-Section of Hoover Dam Lake Mead Write an inequalit that describes the shaded region. Height Dam Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 2. DISCRIMINANTS Consider the equation a 2 + b + c = 0. Assume that the discriminant is zero and that a is positive. What are the solutions of the inequalit a 2 + b + c 0? 3. KISKS Caleb is designing a kiosk b wrapping a piece of sheet metal with dimensions + 5 inches b inches into a clindrical shape. Ignoring cost, Caleb would like a kiosk that has a surface area of at least 4480 square inches. What values of satisf this condition? Thickness a. Write an equation for the Hoover Dam parabola. Let the height be the -value of the parabola and the thickness be the -value of the parabola. (Hint: the equation will be in the form: = k( maimum thickness) + maimum height.) b. Using our equation, graph the parabola of the Hoover Dam for Height Thickness c. Estimate to the nearest foot the thickness of the dam when the height is 200 feet. Lesson Chapter 4 53 Glencoe Algebra 2

6 NAME DATE PERID Enrichment Graphing Absolute Value Inequalities You can solve absolute value inequalities b graphing in much the same manner ou graphed quadratic inequalities. Graph the related absolute function for each inequalit b using a graphing calculator. For > and, identif the -values, if an, for which the graph lies below the -ais. For < and, identif the values, if an, for which the graph lies above the -ais. For each inequalit, make a sketch of the related graph and find the solutions rounded to the nearest hundredth > < < < > Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Chapter 4 54 Glencoe Algebra 2

7 NAME DATE PERID Graphing Calculator Activit and the Test Menu The inequalit smbols, called relational operators, in the TEST menu can be used to displa the solution of a quadratic inequalit. Another method that can be used to find the solution set of a quadratic inequalit is to graph each side of an inequalit separatel. Eamine the graphs and use the intersect function to determine the range of values for which the inequalit is true. Eample 1 Solve Place the calculator in Dot mode. Enter the inequalit into Y1. Then trace the graph and describe the solution as an inequalit. Kestrokes: Y= 2 + 2nd [TEST] 4 6 ZM 4. Use TRACE to determine the endpoints of the segments. Theses values are used to epress the solution of the inequalit, { - 3 or 2 }. [-4.7, 4.7] scl:1 b [-3.1, 3.1] scl:1 Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Eample 2 Eercises Solve each inequalit. Solve Place the left side of the inequalit in Y1 and the right side in Y2. Determine the points of intersection. Use the intersection points to epress the solution set of the inequalit. Be sure to set the calculator to Connected mode. Kestrokes: Y= ENTER 3 ENTER ZM 6. Press 2nd [CALC] 5 and use the ke to move the cursor to the left of the first intersection point. Press ENTER. Then move the cursor to the right of the intersection point and press ENTER ENTER. ne of the values used in the solution set is displaed. Repeat the procedure on the other intersection point. The solution is { }. [-10, 10] scl:1 b [-10, 10] scl:1 [-10, 10] scl:1 b [-10, 10] scl: < < > 0 Lesson > ( + 1)( -3) > 0 Chapter 4 55 Glencoe Algebra 2