Chapter 10: Quadratic and Exponential Functions

1 522 Chapter 10 Quadratic and Exponential Functions parabola(p. 524) completing the square (p. 539) Quadratic Formula (p. 546) Exponential function (p. 554) geometric sequence (p. 567)Key Vocabulary Lesson 10-1 Graph Quadratic Functions . Lessons 10-2 through 10-4 Solve quadraticequations. Lesson 10-5 Graph Exponential Functions . Lesson 10-6 Solve problems involving Exponential growth and Exponential decay. Lesson 10-7 Recognize and extend geometric andExponentialFunctionsQuadratic Functions and equations are used to solve problems about fireworks, to simulate the flight of golf balls in computer games, to describe arches, to determine hang time in football, and to help with water management. Exponential Functions are used to describe changes in population, to solve compound interest problems, and to determine concentration of chemicals in a body of water after a spill. Exponential decay is one type of Exponential function. Carbon dating uses Exponential decay to determine the age of fossils and will learn about carbon dating in Lesson Chapter 10 Quadratic and Exponential FunctionsChapter 10 Quadratic and Exponential Functions523 Quadratic and Exponential Functions Make this Foldable to help youorganize your notes.

2 Begin with four sheets of grid and WritingAs you read and study the Chapter , write notes and examples for each lessonon each page of the SkillsTo be successful in this Chapter , you ll need to masterthese skills and be able to apply them in problem-solving situations. Reviewthese skills before beginning Chapter Lesson 10-1 Graph FunctionsUse a table of values to graph each equation.(For review, see Lesson 5-3.) x 2x 3x 3y 10 2y 2y 9 For Lesson 10-3 Perfect Square TrinomialsDetermine whether each trinomial is a perfect square trinomial. If so, factor it.(For review, see Lesson 9-6.) 12t 14a 18m 8y 6b 4x 12p 24s 9 For Lesson 10-7 Arithmetic SequencesFind the next three terms of each arithmetic sequence.(For review, see Lesson 4-7.) , 9, 13, 17, .. , 5, 2, 9, ..19. 4, 1, 2, 5, .. , 32, 40, 48, ..21. 1, 6, 11, 16, ..22. 27, 20, 13, 6, .. , , , , .. , , , , ..Fold in HalfFold each sheet in half along the each page with thelesson number as to form a each sheet and tape toform one long Function WordsA Quadratic function can be described by an equation of the formy ax2 bx c, where a 0.

3 ModelsxOyxOyGRAPH Quadratic FUNCTIONSThe function describing the height of the rocket is an example of a Quadratic function. Acan bewritten in the form y ax2 bx c, where a 0. This form of the quadraticfunction is called the standard form. Notice that this polynomial has degree 2 and the exponents are positive. The graph of a Quadratic function is called a .parabolaquadratic functionVocabulary Quadratic function parabola minimum maximum vertex symmetry axis of symmetryGraphing Quadratic Functions524 Chapter 10 Quadratic and Exponential Functions Graph Quadratic Functions . Find the equation of the axis of symmetry and the coordinates of the vertex of a Sky Concert in Peoria, Illinois, is a 4th of July fireworks display set to music. If a rocket (firework) is launched with an initialvelocity of meters per second at a height of meters above the ground, the equation h represents the rocket s height hin meters after rocket will explode at approximately thehighest you coordinate a fireworks display with recorded music?

4 Can you coordinate a fireworks display with recorded music?Graph Opens Upward Use a table of values to graphy 2x2 4x these ordered pairs and connect them with a smooth 2 4 410y 2x2 4x 5 Example1 Example1 Height of Rocket02040 Height (meters)6080 Time (seconds)2468xy 211 110 51 72 531411 Virginia SOL STANDARD The student will solvequadratic equations in one variable both algebraically andgraphically. Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic of ParabolasModel Graphy x2 6x 8 on grid paper. Hold your paper up to the light and fold the parabola in half so that the two sides match exactly. Unfold the a is the vertex of the parabola? an equation of the fold point on the parabola lies on the fold line? a few sentences to describe the symmetry of a parabola based on your findings in this 10-1 Graphing Quadratic Functions525 Notice that the value of ain Example 2 is negative and the curve opensdownward.

5 The highest point, or , of the graph is located at (2, 3). The maximum or minimum point of a parabola is called the .SYMMETRY AND VERTICESP arabolas possess a geometric property called. Symmetrical figures are those in which the figure can be folded in half so that each half matches the other Opens DownwardUse a table of values to graph y x2 4x these ordered pairs and connect them with a smooth x2 4x 1 Example2 Example2 Reading MathThe plural of vertex isvertices. In math, vertexhas several meanings. For example, there are the vertex of an angle, thevertices of a polygon, andthe vertex of a TipThe fold line in the activity above is called the for the parabola. Each point on theparabola that is on one side of the axis of symmetryhas a corresponding point on the parabola on the other side of the axis. The vertex is the only point onthe parabola that is on the axis of symmetry. In the graph of y x2 x 6, the axis of symmetry is x 12.

6 The vertex is 12 , 6 14 .Notice the relationship between the values aandband the equation of the axis of of symmetryxOyy x2 x 6axis ofsymmetryx 12 Consider the standard form y ax2 bx c. Notice that the value of ainExample 1 is positive and the curve opens upward. The lowest point, or ,of the graph is located at (1, 7).minimumxy 1 60 11223324 15 of the Axis of Symmetry of a Parabola WordsThe equation of the axis of Modelsymmetry for the graph of y ax2 bx c, where a 0,isx 2ba .xOyx b2a526 Chapter 10 Quadratic and Exponential FunctionsVertex and Axis of SymmetryConsider the graph of y 3x2 6x Write the equation of the axis of 3x2 6x 4,a 3 and b 2ba Equation for the axis of symmetry of a parabolax 2( 63) or 1a 3 and b 6 The equation of the axis of symmetry is x Find the coordinates of the the equation of the axis of symmetry is x 1 and the vertex lies on theaxis, the x-coordinate for the vertex is 3x2 6x 4 Original equation y 3( 1)2 6( 1) 4x 1y 3 6 vertex is at ( 1, 7).

7 C. Identify the vertex as a maximum or minimum. Since the coefficient of the x2term is negative, the parabola opens downwardand the vertex is a maximum Graph the can use the symmetry of the parabola to help you draw its graph. On acoordinate plane, graph the vertex and the axis of symmetry. Choose a value forxother than 1. For example, choose 1 and find the y-coordinate that satisfiesthe 3x2 6x 4 Original equation y 3(1)2 6(1) 4 Letx (1, 5). Since the graph is symmetricalabout its axis of symmetry x 1, you canfind another point on the other side of the axis of symmetry. The point at (1, 5) is2 units to the right of the axis. Go 2 units to the left of the axis and plot the point ( 3, 5). Repeat this for several other points. Then sketch the 3x2 6x 4( 3, 5)22( 1, 7)(1, 5)x 1 Example3 Example3 TEACHING TIPC oordinates of VertexNotice that you can find the x-coordinateby knowing the axis ofsymmetry. However, tofind the y-coordinate,you must substitute the value of xinto thequadratic TipYou can determine information about a parabola from its 10-1 Graphing Quadratic Functions527 CHECKDoes ( 3, 5) satisfy the equation?

8 Y 3x2 6x 4 Original equation 5 3( 3)2 6( 3) 4y 5 and x 3 5 5 ordered pair ( 3, 5) satisfies y 3x2 6x 4, and the point is on the Equations and GraphsMultiple-Choice Test ItemRead the Test ItemYou are given a Quadratic function, and you are asked to choose the graph thatcorresponds to the Test ItemFirst write the equation in standard 1 (x 1)2 Original equationy 1 x2 2x 1(x 1)2 x2 2x 1y 1 1 x2 2x 1 1 Subtract 1 from each x2 find the axis of symmetry of the graph of y x2 2ba Equation for the axis of symmetryx 2(21) or 1a 1 and b 2 The axis of symmetry is x 1. Look at the graphs. Since only choices C and Dhave this as their axis of symmetry, you can eliminate choices A and B. Since thecoefficient of the x2term is positive, the graph opens upward. Eliminate choice answer is is the graph of y 1 (x 1)2?xOyDxOyCxOyBxOyATest-Taking TipSometimes you can answera question by eliminatingthe incorrect choices. Forexample, in this testquestion, choices A and Bare eliminated becausetheir axes of symmetry arenot x PracticeSOL/EOC Practice 1.

9 Compare and contrasta parabola with a maximum and a parabola with a ENDEDDraw two different parabolas with a vertex of (2, 1).3. Explainhow the axis of symmetry can help you graph a Quadratic a table of values to graph each x2 x2 4x 5 Write the equation of the axis of symmetry, and find the coordinates of the vertexof the graph of each function. Identify the vertex as a maximum or graph the x2 4x x2 5x (x 2)2 is the graph of y 12 x2 1?xOyDxOyCxOyBxOyA528 Chapter 10 Quadratic and Exponential FunctionsConcept CheckGuided PracticeGUIDED PRACTICE KEYP ractice and ApplyPractice and ApplyUse a table of values to graph each x2 x2 x2 2x x2 4x 3x2 6x 3x2 6x is the equation of the axis of symmetry of the graph ofy 3x2 2x 5? the equation of the axis of symmetry of the graph ofy 4x2 5x the equation of the axis of symmetry, and find the coordinates of the vertexof the graph of each function. Identify the vertex as a maximum or graph the x2 x2 x2 2x x2 6x x2 14x x2 2x 2x2 12x 11 3x2 6x 5 16x 9 8x 2x2 Extra Practice See page Practice See page 151, 216 49352, 534 StandardizedTest PracticeSOL/EOC Practice Lesson 10-1 Graphing Quadratic 3(x 1)2 2(x 4)2 2 x2 10x 1 3x2 12x 5 13 (x 2) 1 23 (x 1) vertex of a parabola is at ( 4, 3).

10 If one x-intercept is 11, what is theotherx-intercept? is the equation of the axis of symmetry of a parabola if its x-intercepts are 6 and 4? diver follows a path that is in the shape of a parabola. Suppose the diver s footreaches 1 meter above the height of the divingboard at the maximum height of the dive. Atthat time, the diver s foot is also 1 meterhorizontally from the edge of the diving is the distance of the diver s foot from thediving board as the diver descends past thediving board? Exercises 39 and 40, use the following information. A carnival game involves striking a lever that forces a weight up a tube. If the weight reaches 20 feet to ring the bell, the contestant wins a prize. The equationh 16t2 32t 3 gives the height of the weight ifthe initial velocity is 32 feet per the maximum height of the a prize be won?PETSFor Exercises 41 43, use the following has 40 meters of fencing to build a pen for her the diagram at the right to write an equation for the area Aof the pen.