Algebra 1 Unit 2 Part 3 - OGLESBY MATH

1Algebra 1Unit 2 Part 3Quadratic FunctionsThursday,March 11thTransformationsof QuadraticFunctionsMonday,March 15thTuesday,March 16thWednesday,March 17thGraphing inStandard FormGraphingCharacteristicsConvertingBetween VertexForm andStandard FormCharacteristicsQuiz Opens at3:30 PMMonday,March 22ndTuesday,March 23rdWednesday,March 24thQuadratic WordProblemsReviewUnit 2 Part 3 Test(during class)Thursday,March 18thFriday,March 12thGraphing inVertex FormCharacteristicsFriday,March 19thQuadratic WordProblemsQuiz Due ByMidnightThursday,March 25thFriday,March 26th

2

3Transformations of Quadratic Functions NotesThe parent function of a function is the simplest form of a function. The parent functionfor a quadratic function is y x2 or f(x) x2. Complete the table and graph the parentfunction below.As you can see, the graph of a quadraticfunction looks very different from thegraph of a linear function.The U-shaped graph of a quadraticfunction is called a .The highest/lowest point (or turning point)on a parabola is called the .Remember, in order for a function to bea quadratic function, one term musthave .The graph above is our parent function – it represents a quadratic function that has notbeen changed in any way. We are going to talk about the transformations ofquadratic functions and how those transformations are represented in the equation ofa quadratic function.Exploring the “k"Answer the following questions about the transformation from the parent graph (solidgraph)to the new function (dotted parabola).𝑦 𝑥2 3𝑦 𝑥2𝑦 𝑥2𝑦 𝑥2 2Describe the transformation:Describe the transformation:What is the vertex of the new function?What is the vertex of the new function?

4Exploring the “h” ValueAnswer the following questions about the transformation from the parent graph (solidgraph)to the new function (dotted parabola).𝑦 𝑥2𝑦 𝑥2𝑦 (𝑥 1)2𝑦 (𝑥 3)2Describe the transformation:Describe the transformation:What is the vertex of the new function?What is the vertex of the new function?Exploring the “a” ValueAnswer the following questions about the transformation from the parent graph (solidgraph)to the new function (dotted parabola).𝑦 3𝑥 2𝑦 𝑥𝑦 𝑥221𝑦 𝑥24Describe the transformation:Describe the transformation:What is the vertex of the new function?What is the vertex of the new function?𝑦 𝑥2Describe the transformation:What is the vertex of the new function?𝑦 𝑥 2

5SummaryVertex Form:VariableSummary of the Effects of the Transformationsahkvertex:Practice1) Given the equations below, describe the transformations from the parent functionand name the vertex:a. 𝑦 (𝑥 4)2 7b. 𝑦 2(𝑥 2)2 51c. 𝑦 (𝑥 3)2 822) Create an equation to represent the following transformations:a. Shifted down 4 units, right 1 unit, and reflected across the x-axisb. Shifted up 6 units, reflected across the x-axis, and stretch by a factor of 3c. Shifted up 2 units, left 4 units, reflected across the x-axis, and shrunk by a factor of ¾.

6Identifying Transformations PracticeEquation𝑦 2𝑥 2 43𝑦 (𝑥 1)221𝑦 (𝑥 2)2 54𝑦 0.4𝑥 22𝑦 (𝑥 3)2 43𝑦 4𝑥 2 2𝑦 (𝑥 1)2 5𝑦 3(𝑥 4)2 11𝑦 𝑥22𝑦 2(𝑥 3)2𝑦 𝑥2 4𝑦 (𝑥 4)2𝑦 1.5𝑥 2 9𝑦 𝑥 2 2𝑦 0.8(𝑥 4)2𝑦 3.2𝑥 2 11a, h, kvaluesReflection?VerticalStretch ?

7Writing Equations in Vertex Form PracticeWrite the equation for a quadratic function which has been 1) reflected across the x-axis and translated down 3 units.2) vertically stretc or valley of afunction.Think:What is the lowest point onmy graph?Extrema:Extrema:Extrema:Extrema:Write:y k(y-value of the vertex)

14– end behavior –Define:Behavior of the ends of the function (what happens to the y-values or f(x)) as xapproaches positive or negative infinity. The arrows indicate the function goes onforever so we want to know where those ends go.Think:Write:As x goes to the left (negative infinity), whatAs x - , f(x) direction does the left arrow go?Think:Write:As x goes to the right (positive infinity), whatAs x , f(x) direction does the right arrow go?As x - , f(x) As x , f(x) As x , f(x) As x - , f(x) As x - , f(x) As x - , f(x) As x , f(x) As x , f(x)

15– interval of increase and decrease –Interval of IncreaseDefine:Think:Write:The part of the graph that isFrom left to right, is my[left, right] of portionrising as you read left to right.graph going up?going upInterval of DecreaseDefine:Think:Write:The part of the graph that isFrom left to right, is my[left, right] of portionfalling as you read from left tograph going down?going downright.Interval of Increase:Interval of Increase:Interval of Decrease:Interval of Decrease:Interval of Decrease:Interval of Decrease:Interval of Increase:Interval of Increase:

16Identify the listed characteristics for the following graph.Domain:Range:Vertex:Max or Min:Extrema Value:Axis of al of Increase:Interval of Decrease:As x , f(x) As x , f(x)

17Average Rate of Change NotesAverage Rate of Change (AROC): The change in the value of a quantity divided by theelapsed time. For a function, this is the change in the y-value divided by the change inthe x-value for two distinct points on the graph.Finding AROC from a graph.Using the problem, find the two points for which you are trying to find the average rateof change between. Then, use the formula to find the AROC.Find the AROC of the interval [ 4, 1].Find the AROC between 𝑥 1 and 𝑥 5.Finding AROC from a graph.Using the problem, plug in the two x-values (one at a time) to find the two points forwhich you are trying to find the average rate of change between. Then, use theformula to find the AROC.Given 𝑦 (𝑥 2)2 6, find the averagerate of change between 𝑥 3 and𝑥 2.Given 𝑦 4𝑥 2 6𝑥 11, findthe AROC of the interval [0,5].

18Average Rate of Change Practice1) Find the average rate of changeover the interval [-1, 3].2) Find the average rate of changeover the interval -3 x 2.3) Using the equation 𝑦 4(𝑥 2)2 6, find the average rate of change from x -2 tox 1.4) Using the equation 𝑦 𝑥 2 6𝑥 2, find the average rate of change for the interval[-6, -2].

19Characteristics PracticeDomain:Range:Int. of Increase:Int. of Decrease:Max/Min:Extrema Value:Zeros:Y-Int:X-Int:As 𝑥 , 𝑓(𝑥) As 𝑥 , 𝑓(𝑥) Vertex:Axis of Symmetry:Vertex:X-Int:Int. of Decrease:Zeros:Range:As 𝑥 , 𝑓(𝑥) Max/min:Axis of Sym:Domain:Y-Int:As 𝑥 , 𝑓(𝑥) Int. of Increase:Int. of Constant:

20Draw a graph that has the following characteristics: Vertex at (3, 4) End behavior of as 𝑥 , 𝑓(𝑥) Two zeros A y-intercept of (0, -2) A domain of ( - , )Then, identify the following:Axis of Symmetry:Range:Interval of Increase:Interval of Decrease:

21Writing Equations in Vertex Form When Given a GraphSteps: Find the vertex Find stretch/shrink/reflection (AROC from vertex to one pointto the right) Plug values into equation

22Graphing in Vertex Form – Practice1) Determine the equation for thefunction graphed on the left.a) Domain:b) Range:c) Extrema:d) Axis of Symmetry:e) Increasing:f) Decreasing:g) As 𝑥 , 𝑓(𝑥) h) AROC 3 𝑥 1.1) Determine the equation for the functiongraphed on the left.g) As 𝑥 , 𝑓(𝑥) i) AROC between 𝑥 1 and 𝑥 4.a) Domain:b) Range:c) Extrema:d) Axis of Symmetry:e) Increasing:f) Decreasing:h) As 𝑥 , 𝑓(𝑥)

23Graphing and Characteristics of Quadratic Functions[standard form]To graph a quadratic function that is in standard form, follow these steps: Create an x-y table with 5 rows Find the vertex – this goes in the middle rowTo find the x-value of the vertex: 𝑥 𝑏2𝑎Then plug the x-value into the equation to get the y-value Fill out the two x-values before and after the vertex Use your calculator to find the y-values and graph**Note: the y-intercept of a quadratic function in standard form is **For the following problems, find the vertex and graph the function.1) 𝑦 𝑥 2 2𝑥 12) 𝑦 3𝑥 2 6𝑥

2413) 𝑓(𝑥) 𝑥 6𝑥 94) 𝑦 2 𝑥 2𝑥 65) 𝑓(𝑥) 1.2𝑥 2 86) 𝑦 2𝑥 2 10𝑥 322

25For the graphs below, find the characteristics listed.Domain:Range:Zeros:Y-Intercepts:Interval of Increase:As x , f(x) Extrema:Range:X-Intercepts:Max or Min:As x , f(x) Interval of Increase:Interval of Decrease

26Graphing in Standard Form – PracticeGraph the following.1) 𝑦 2𝑥 2 6𝑥 313) 𝑦 2 𝑥 2 4𝑥 32) 𝑦 𝑥 2 2𝑥 14) 𝑦 2𝑥 2 8𝑥 6

27Converting Between Vertex and Standard FormConverting From Standard Form to Vertex Form1) Identify a, b, and c from the equation2) Find the x-value of the vertex by using 𝑥 𝑏2𝑎3) Find the y-value of the vertex by plugging in the x-value from step #24) Plug a (from original equation), h (the x-value of vertex), and k (the y-value of thevertex) into vertex form1) 𝑦 𝑥 2 12𝑥 322) 𝑓(𝑥) 𝑥 2 8𝑥 93) 𝑓(𝑥) 𝑥 2 10𝑥 34) 𝑦 𝑥 2 6𝑥 15

28Converting from Vertex Form to Standard Form1) Re-write the binomial squared as the product of a binomial multiplied by itself2) Use the distributive property to multiply3) Distribute the coefficient, if there is one4) Combine like terms1) 𝑓(𝑥) 2(𝑥 5)2 833) 𝑦 2 (𝑥 6)2 22) 𝑦 3(𝑥 1)2 44) 𝑓(𝑥) 0.75(𝑥 16)2 12

29Converting Between Forms PracticePart One: Convert from standard form to vertex form.1) 𝑦 𝑥 2 8𝑥 152) 𝑦 𝑥 2 4𝑥3) 𝑦 2𝑥 2 12𝑥 74) 𝑦 2𝑥 2 8𝑥 17Part Two: Convert from vertex form to standard form.5) 𝑦 (𝑥 4)2 56) 𝑦 (𝑥 3)2 27) 𝑦 2(𝑥 2)2 38) 𝑦 (𝑥 8)2 612

30Quadratic Keywordsfrom scaffoldedmath.com

31Applications of Quadratic Functions1) This graph represents the height of a diver above the water vs. the time after thediver jumps from a springboard. Answer the following questions based on theinformation.a) How long did it take the diver to hitthe water?b) How tall was the diving board?c) What was the maximum heightreached by the diver?But what do we do if the graph isn’t given to us? If we are not given a graph, we will begiven the equation that represents the scenario.You will need to determine whether the problem is asking you to find the vertex, thex-intercept(s), or the y-intercept. Vertex: maximum, minimum, highest, lowestVertex form: 𝑦 𝑎(𝑥 ℎ)2 𝑘 vertex at (ℎ, 𝑘)Standard form: 𝑦 𝑎𝑥 2 𝑏𝑥 𝑐 x-value of vertex found using 𝑥 then plug that in to find y-value 𝑏2𝑎and X-Intercept: ending, landing, ground level, se

The parent function of a function is the simplest form of a function. The parent function for a quadratic function is y x2 2or f(x) x. Complete the table and graph the parent function below. As you can see, the graph of a quadratic function looks very different from the graph of a linear function. The U-shaped graph of a quadratic