Answers (Lesson 10-1) - Merrimack High School

Graphing Quadratic FunctionsGlencoe/McGraw-Hilla function described by an equation of the form f (x) 5 ax 2 1 bx 1 c,where a Þ 0232A2Oyx yGlencoe/McGraw-HillO1. y 5 x2 1 2x22270y12122OyxxOxGlencoe Algebra 1y3. y 5 x2 2 3x 1 2Graph the ordered pairs in the table andconnect them with a smooth curve.579O2. y 5 2x2 2 4y27212223242526xExample 2Use a table of values tograph y 5 2x2 2 6x 2 7.Use a table of values to graph each function.ExercisesGraph the ordered pairs in the table andconnect them with a smooth curve.12214102263yx21Use a table of values tograph y 5 x2 2 4x 1 1.Example 1The degree of a quadratic function is 2, and the exponents are positive. Graphs of quadraticfunctions have a general shape called a parabola. A parabola opens upward and has aminimum point when the value of a is positive, and a parabola opens downward and has amaximum point when the value of a is negative.QuadraticFunctionGraph Quadratic FunctionsExample:y 5 2x 2 1 3x 1 8PERIOD10-1 Study Guide and InterventionNAME DATE2aFor the parabola y 5 ax 2 1 bx 1 c, where a Þ 0,bthe line x 5 2 } is the axis of symmetry.Example: The axis of symmetry ofy 5 x 2 1 2x 1 5 is the line x 5 21.b2a4x 5 2 } 5 212(2)(–1, –1)x 5 –1Oyxd. Graph the function. Glencoe/McGraw-HillOyxx 5 0; (0, 3); min1. y 5 x2 1 3580Oyxx 5 22; (22, 0); max2. y 5 2x2 2 4x 2 4OxGlencoe Algebra 1yx 5 21; (21, 2); min3. y 5 x2 1 2x 1 3Write 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 a minimum.Then graph the function.Exercisesc. Identify the vertex as a maximum ora minimum.Since the coefficient of the x2-term ispositive, the parabola opens upward, andthe vertex is a minimum point.The axis of symmetry is x 5 21.x 5 2}b. Find the coordinates of the vertex.Since the equation of the axis ofsymmetry is x 5 21 and the vertex lieson the axis, the x-coordinate of the vertexis 21.y 5 2x2 1 4x 1 1Original equationy 5 2(21)2 1 4(21) 1 1 Substitute.y 5 2(1) 2 4 1 1Simplify.y 5 21The vertex is at (21, 21).Consider the graph of y 5 2x2 1 4x 1 1.a. Write the equation of the axis ofsymmetry.In y 5 2x2 1 4x 1 1, a 5 2 and b 5 4.Substitute these values into the equationof the axis of symmetry.ExampleThe axis of symmetry contains the minimum or maximum point of the parabola, the vertex.Axis ofSymmetryParabolas have a geometric property called symmetry. Thatis, if the figure is folded in half, each half will match the other half exactly. The vertical linecontaining the fold line is called the axis of symmetry.Graphing Quadratic FunctionsSymmetry and Vertices(continued)PERIOD10-1 Study Guide and InterventionNAME DATEAnswers(Lesson 10-1)Glencoe Algebra 1Lesson 10-1

Glencoe/McGraw-HillOyOyxx4. y 5 2x2 2 4x 1 12. y 5 2x2 1 3OyOyxxPERIODA3 xGlencoe/McGraw-HillOyxx 5 21; (21, 3); max8. y 5 2x2 2 2x 1 2Oyx 5 0; (0, 0); min5. y 5 2x2x581Oyxx 5 21; (21, 24); min9. y 5 2x2 1 4x 2 2Oyx 5 1; (1, 26); min6. y 5 x2 2 2x 2 5xOxGlencoe Algebra 1yx 5 21; (21, 8); max10. y 5 22x2 2 4x 1 6Oyx 5 2; (2, 3); max7. y 5 2x2 1 4x 2 1Write 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 minimum.Then graph the function.3. y 5 x2 2 2x 2 61. y 5 x2 2 4Use a table of values to graph each function.Graphing Quadratic Functions10-1 Skills PracticeNAME DATE(Average)xOy2. y 5 x2 2 6x 1 3xOy3. y 5 22x2 2 8x 2 5xPERIODxOyxx 5 2; (2, 5); max5. y 5 22x2 1 8x 2 3Oxx 5 100Glencoe Algebra 1Glencoe/McGraw-HillAnswers 582Glencoe Algebra 112. An outfielder catches the ball three feet above the ground. How far has the ball traveledhorizontally when the outfielder catches it? 200 ft11. What is the maximum height reached by the baseball? 53 ft10. What is the equation of the axis of symmetry?BASEBALL For Exercises 10–12, use the following information.A player hits a baseball at a 45 angle with the ground with an initial velocity of 80 feet persecond from a height of three feet above the ground. The equation h 5 20.005x2 1 x 1 3 givesthe path of the ball, where h is the height and x is the horizontal distance the ball travels.of their target.9. Will her brother be able to catch the keys? Explain. No, the keys will be 8 ft short8. How high do the keys reach? 30 ft7. How long does it take the keys to reach their highest point? 1.25 sMiranda throws a set of keys up to her brother, who is standing on a third-story balconywith his hands 38 feet above the ground. If Miranda throws the keys with an initial velocityof 40 feet per second, the equation h 5 216t2 1 40t 1 5 gives the height h of the keys aftert seconds.yx 5 22; (22, 27); min6. y 5 2x2 1 8x 1 1PHYSICS For Exercises 7–9, use the following information.Oyx 5 0; (0, 3); max4. y 5 2x2 1 3Write 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 minimum.Then graph the function.Oy1. y 5 2x2 1 2Use a table of values to graph each function.Graphing Quadratic Functions10-1 PracticeNAME DATEAnswers(Lesson 10-1)Lesson 10-1

PERIODGlencoe/McGraw-Hill3.A4OyxparabolaB., the value of b isx(21, 22)25, and. This point is theofvertexIt opens downward if a0.Glencoe/McGraw-Hill583point at which the parabola turns upward or downward.Glencoe Algebra 16. Look up the word vertex in a dictionary. You will find that it comes from the Latin wordvertere, which means to turn. How can you use the idea of “to turn” to remember whatthe vertex of a parabola is? Sample answer: The vertex of a parabola is the0.of the parabola. This line goes1 bx 1 c, the parabola opens upward if aof the parabola.axis of symmetryax2,5. For a quadratic function y 5through theyou fold the parabola is the4. If you fold a parabola along a line to get two halves that match exactly, the line wherevertex. This point is the(maximum/minimum) point of the graph.3. The maximum or minimum point of a parabola is called thethe parabola.minimum.(21, 4)Oy(maximum/minimum) point of the graph.c. The lowest point of graph B is located atmaximumb. The highest point of graph A is located ata. Each graph is a curve called aA.Helping You Remember 2function is y 5 ax2 1 bx 1 c. For the2. The graphs of two quadratic functions are shown below. Complete each statement aboutthe graphs.the value of c isfunction y 5 2x2 2 5x 1 3, the value of a is1. The standard form for aquadratic80 meters 4 seconds after being launched.How can you coordinate a fireworks display with recorded music?Read the introduction to Lesson 10-1 at the top of page 524 in your textbook.According to the graph, at what height does the rocket explode and in howmany seconds after being launched? It explodes at a height ofReading the LessonPre-ActivityGraphing Quadratic Functions10-1 Reading to Learn MathematicsNAME DATExOy2. y 5 x2 1 2x Glencoe/McGraw-HillOy4. y 5 (x 2 1)2x584Oy5. y 5 (x 2 3)2xxxxGlencoe Algebra 1Oy6. y 5 (x 1 2)2Oy3. y 5 x2 2 2These equations have the form y 5 (x 2 b)2. Graph each equation.Oy1. y 5 x2 1 1OyPERIODThese equations have the form y 5 x 2 1 c. Graph each equation.y 5 (x 2 4)2 1 3Then slide up 3 units.y 5 (x 2 4)2Slide to the right 4 units.Start with y 5 x2.The graph of a quadratic equation in the formy 5 (x 2 b)2 1 c is a translation of the graph of y 5 x2.When a figure is moved to a new position withoutundergoing any rotation, then the figure is said tohave been translated to that position.Translating Quadratic Graphs10-1 EnrichmentNAME DATEAnswers(Lesson 10-1)Glencoe Algebra 1Lesson 10-1

PERIODGlencoe/McGraw-Hillan equation of the form ax 2 1 bx 1 c 5 0, where a Þ 0A5x Glencoe/McGraw-Hill23; 24Of (x )1. x2 1 7x 1 12 5 0xf (x )Glencoe Algebra 14Answers585212282424 O4, 23284Ox8xxGlencoe Algebra 1no real rootsOf (x )3. x2 2 4x 1 5 5 0To solve x2 2 6x 1 9 5 0, you need to knowwhere the value of f(x) 5 0. The vertex of theparabola is the x-intercept. Thus, the onlysolution is 3.2. x2 2 x 2 12 5 0Solve each equation by graphing.ExercisesTo solve x2 1 4x 1 3 5 0, you need to knowwhere the value of f(x) 5 0. This occurs atthe x-intercepts, 23 and 21.The solutions are 23 and 21.Of (x )the vertex and several other points on eitherside of the axis of symmetry.Graph the vertex and several other points oneither side of the axis of symmetry.f (x )x 5 } or 3. The vertex is at ( 3, 0). Graph62(1)Solve x2 2 6x 1 9 5 0 bygraphing.Graph the related function f(x) 5 x2 2 6x 1 9.The equation of the axis of symmetry isExample 2x 5 2 } or 22. The vertex is at ( 22, 21).42(1)Solve x2 1 4x 1 3 5 0 bygraphing.Graph the related function f(x) 5 x2 1 4x 1 3.The equation of the axis of symmetry isExample 1The solutions of a quadratic equation are called the roots of the equation. The roots ofa quadratic equation can be found by graphing the related quadratic functionf(x) 5 ax2 1 bx 1 c and finding the x-intercepts or zeros of the function.Quadratic EquationSolve by GraphingSolving Quadratic Equations by Graphing10-2 Study Guide and InterventionNAME DATE23221232221Of (x )x Glencoe/McGraw-Hill21 , x , 0,4,x,5Of (x )4. x2 2 4x 2 1 5 026 , x , 25,22 , x , 21Of (x )1. x2 1 7x 1 9 5 0xx5860 , x , 1,2,x,3Of (x )5. 4x2 2 12x 1 3 5 022 , x , 21,2,x,3Of (x )2. x2 2 x 2 4 5 0xxxxGlencoe Algebra 122 , x , 21,3,x,4Of (x )6. x2 2 2x 2 4 5 0no real rootsOf (x )3. x2 2 4x 1 6 5 0Solve each equation by graphing. If integral roots cannot be found, estimate theroots by stating the consecutive integers between which the roots lie.ExercisesThe x-intercepts of the graph are between 25 and 24 and between 22 and 21.So one root is between 25 and 24, and the other root is between 22 and 21.2224Graph the related function f(x) 5 x2 1 6x 1 6.xf (x ) Notice that the value of the function changesfrom negative to positive between the x-values251of 25 and 24 and between 22 and 21.ExampleSolve x2 1 6x 1 6 5 0 by graphing. If integral roots cannot be found,estimate the roots by stating the consecutive integers between which the roots lie.The roots of a quadratic equation may not be integers. If exactroots cannot be found, they can be estimated by finding the consecutive integers betweenwhich the roots lie.Estimate SolutionsSolving Quadratic Equations by Graphing(continued)PERIOD10-2 Study Guide and InterventionNAME DATEAnswers(Lesson 10-2)Lesson 10-2