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Lesson 6.1: Successive Differences in PolynomialsLearning Goals:How can we determine what type of a function a given set of points is represented by?How can we determine the equation for a given set of points?Linear Functionsxy=2x+3First Differences035-3=2157-5=2279-7=23911-9=241113-11=2513What do you notice?1st difference =2 and a=2.y=mx+b y=ax+bLet’s generalize this…xax+bFirst Differences0ba+b-b=a1a+b2a+b-a+b=a22a+b3a-2a=a33a+b914400313690For a linear function the first difference is equivalent to a For a linear function the first difference is equivalent to a y=4x-5y=-10x+51st Difference a=41st Difference a=-10Quadratic FunctionsNumberSquare x2First DifferencesSecond Differences114-1=33-5=2249-4=57-5=23916-9=79-7=241625-16=911-9=252536-25=11636What do you notice?The second differences are all equalLet’s generalize this…#ax2+bx+cFirst DifferencesSecond Differences0ca+b+c-c=a+b3a+b-a+b=2a1a+b+c4a+2b+c-a+b+c=3a+b5a+b-3a+b=2a24a+2b+c9a+3b+c-4a+2b+c=5a+b7a+b-5a+b=2a39a+3b+c16a+4b+c-9a+3b+c=7a+b416a+4b+c742585260985For a quadratic function the second difference is equivalent to 2a 0For a quadratic function the second difference is equivalent to 2a y=2x2y=-5x22nd Difference =2a=22=42nd Difference =2a=2-5=-10161925141605What do you think will happen if we have a cubic function?The third differences will be equal!What do you think will happen if we have a cubic function?The third differences will be equal!Cubic Functions#ax3+bx2+cx+dFirst DifferencesSecond DifferencesThird Differences0d=a+b+c=6a+2b6a1a+b+c+d=7a+3b+c=12a+2b6a28a+4b+2c+d=19a+5b+c=18a+2b327a+9b+3c+d=37a+7b+c464a+16b+4c+d871749113665For a cubic function the third difference is equivalent to 6a 0For a cubic function the third difference is equivalent to 6a y=3x3y=-5x33rd Difference =63=183rd Difference =6a=6-5=-30Example 1:What type of relationship is indicated by the following set of ordered pairs? Look at the differences in the outputs! Explain how you know. Quadratic relationship because the second differences are the samexyFirst DifferencesSecond Differences002-0=28-2=61210-2=814-8=621024-10=1420-14=632444-24=20444Find an equation that all ordered pairs above satisfy.y=3x2-x+02a=6a=34448175-32639000Use your calculator! “regression” = relationshipstat→edit→L1, L2x-list: L1 y-list: L2 stat→calc→5:QuadRegExample 2:What type of relationship does the set of ordered pairs (x,y) satisfy? How do you know?xyFirst DifferenceSecond DifferenceThird DifferenceFourth Difference021-2=-15-(-1)=612-6=6Cubic Relationship116-1=517-5=1218-12=6Because 3rd 2623-6=1735-17=1824-18=6Differences are32358-23=3559-35=24All Equal458117-58=595117304800026924000What is the equation that satisfies the list of ordered pairs above? Round all values to the nearest tenth.stat→calc→6: cubic regressiony=x3+0x2-2x+2y=x3-2x+2Example 3:Show that the set of ordered pairs x, y in the table below satisfied a quadratic relationship. Find the equation of the form y=ax2+bx+c that all of the ordered pairs satisfy.x012345y54-1-10-23-40stat→calc→5:Quad Regression y=-2x2+x+5 Homework 6.1: Successive Differences in Polynomials1. Create a table to find the third differences for the polynomial s3-s2+s for integer values of s from -3 to 3. {Hint: Plug equation into calculator to get the corresponding y-values}2. a) Show that the set of ordered pairs in the table below satisfies a cubic relationship. (Hint: Find the third differences)b) Find the equation that satisfies the ordered pairs listed above. Round all coefficients to the nearest hundredth.3. The distance d feet required to stop a car traveling at 10 v mph under dry asphalt conditions is given by the following table.a) What type of relationship is indicated by the set of ordered pairs?b) Assuming that the relationship continues to hold, find the distance required to stop the car when v=6.4. Show that the set of ordered pairs in the table below satisfies a quadratic relationship. Find the equation of the form y=ax2+bx+c that all ordered pairs satisfy.Lesson 6.2: Linear Systems in Three VariableLearning Goals:What is a linear system of equations?How do we solve a linear system of equations in three variables?Review: Solve the following system of equations algebraically for x and y.1. 5x+3y=72x+y=3Solutions are the intersection of the graphs!Do by elimination method:5x+3y=7 → 5x+3y=7now sub in:52+3y=7-3(2x+y=3) → -6x-3y=-910+3y=7-x=-2 →x=23y=-3 →y=-1x=2 and y=-12. A scientist wants to create 120 ml of a solution that is 30% acidic. To create this solution, she has access to a 20% solution and a 45% solution. How many milliliters of each solution should she combine to create the 30% solution?Create a system of equationsx=ml of 20% solutionx+y=120y=ml of 45% solution.20x+.45y=.30(120)Do by elimination method:-.45x+y=120 → -.45x-.45y=-54sub in: 72+y=120.20x+.45y=.30(120) → .20x+.45y=36y=48-25x=-18 →x=72How does the process change when we try to solve for three variables?3. Determine the values for x, y, and z in the following system?2x+3y-z=54x-y-z=-1x+4y+z=12Create two different systems of two equationsEliminate the same variable from each new systemWith resulting equations, create a new system and eliminate another variableSubstitute answer back into other equations to find other variablesEliminate the z:Eliminate the z:2x+3y-z=54x-y-z=-1x+4y+z=12x+4y+z=123x+7y =175x+3y =11 Now eliminate the x:53x+7y=17 → 15x+35y=85-3(5x+3y=11) → -15x-9y=-3326y=52 →y=23x+72=173x+14=1721+32-z=53x=3 →x=12+6-z=58-z=5 → z=34. 2x+y-z=-5 4x-2y+z=102x+3y+2z=3Eliminate the z:2x+y-z=-522x+y-z=-5 →4x+2y-2z=-104x-2y+z=102x+3y+2z=3 2x+3y+2z=36x-y=5 6x+5y =-7Eliminate the x:Sub in for x:Sub in for last variable (z):6x-y=56x--2=5212+-2-z=-56x+5y=-76x=3 →x=121-2-z=-5-6y=12 → y=-2-1-z=-5 →z=45. x-2y+3z=9-x+3y =-42x-5y+5z=17Eliminate the z:5x-2y+3z=9 → 5x-10y+15z=45-3(2x-5y+5z=17) → -6x+15y-15z=-51 -x+5y=-6Eliminate the y:Sub in for x:Sub in for last variable z:-1-x+3y=-4-x+3-1=-41-2-1+3z=9 -x+5y=-6-x=-1 → x=11+2+3z=92y=-2 →y=-13z=6 → z=2Solving a System of Equations in Three Variables:Pick any two pairs of equations from the system.Eliminate the same variable from each pair using the elimination method. Remember to look for opposites to eliminate a variable.Solve the system of the two new equations using the elimination method. Remember to look for opposites to eliminate a variable.Substitute the solution back into one of the original equations and solve for the third variable.When you solve for a system of 3 linear equations, what does the solution set represent? A pointHomework 6.2: Linear Systems in Three VariableSolve each of the following systems of equations.1. 2x+3y=312x-15y=-42. A chemist needs to make 40 ml of a 15% acid solution. He has a 5% acid solution and a 30% acid solution on hand. If he uses the 5% and 30% solutions to create the 15% solution, how many ml of each does he need?3. x+2y+3z=53x+2y-2z=-135x+3y-z=-11Lesson 6.3: Linear Systems in Three VariableLearning Goals:What is a linear system of equations?How do we solve a linear system of equations in three variables?1. x+y+z=25x+5y+5z=34x+y-3z=-6-5x+y+z=2 -5x-5y-5z=-105x+5y+5z=3 5x+5y+5z=30≠-7 FALSE so NO SOLUTIONS3790950429260What is different in this system of equations that you have not seen before? What does it mean?No Solutions.00What is different in this system of equations that you have not seen before? What does it mean?No Solutions.2. x-y+z=-3x-y-z=-35x-5y+z=-15x-y+z=-3x-y-z=-3-3(2x-2y=-6)x-y-z=-35x-5y+z=-156x-6y=-182x-2y=-66x-6y=-180=0 (infinite solutions)4352925449580What is different in this system of equations that you have not seen before? What does it mean?Infinite Solutions.00What is different in this system of equations that you have not seen before? What does it mean?Infinite Solutions.Practice:matrix→edit→x, then 2nd Quit→rref3. a+3b+c=32a-3b+2c=3-a+3b-3c=1a=3 b=13 c=-1 4.x+y+z=-14x+4y-4z=-23x+2y+z=0x=74 or 1.75 y=-52 or -2.5 z=-14 or -.25 5. p+q+3r=4 2q+3r=7p-q-r=-2p=25 q=5 r=1 6.-2x-3y+z=-6x+y-z=57x+8y-6z=31Infinite many solutions!Homework 6.3: Linear Systems in Three VariableSolve each of the following systems of equations algebraically.1.3x-5y+z=9x-3y-2z=-85x-6y+3z=152.4x-3y+2z=12x+y-z=3-2x-2y+2z=53.4x+2y-z=21-x-2y+2z=133x-2y+5z=704. x+3y-z=2x+y-z=03x+2y-3z=-1Lesson 6.4: Transformations of FunctionsLearning Goal: What is a translation and how do we represent that translation of a function in its equation?Parent FunctionsFunctions that belong to the same family share key characteristics. The parent function is the most basic function in a family. Functions in the same family are transformations of their parent function. Below are the four parent functions we will work with in this unit. (There are others!)Quadratic Absolute Value CubicSquare Rooty=x2y=x y=x3y=xUsing your calculator, sketch the graph of each of the equations below. Explain what happened to the parent function of fx=x2 in each graph.fx=x2+3 fx=x2-3 fx=x+32 fx=x-32Moved up 3Moved down 3Moved left 3Moved right 3Summary of Translations fx=yfx+aVertical Up afx-aVertical Down af(x+a)Horizontal Left af(x-a)Horizontal Right aIt is a vertical or horizontal movement of a graphfx-a+ba is the horizontal opposite and b is the vertical movementExample 1: State the transformation that occurs and then sketch the graph of the function without using your calculator.a) fx=x-1+4 Parent: y=x b) gx=x+2-1 Parent: y=x Horizontal: Right 1Horizontal: Left 2Vertical: Up 4Vertical: Down 1 c) hx=x+33+2 Parent: y=x3d) fx=x-52 Parent: y=x2Horizontal: Left 3Horizontal: Right 5Vertical: Up 2Vertical: Stays in same spot a) b) c) d) e) gx=x-2-3 Parent: y=xf) hx=x+4 Parent: y=xHorizontal: Right 2Horizontal: Stays in same spotVertical: Down 3Vertical: Up 439528754635500Example 2: a) Given the graph of f(x), graph gx=fx-3+1. State the value of g0.gx is a translation of f(x)Horizontal: Right 3Vertical: Up 1g0=2 360045028575000b) Given the graph of g(x), graph hx=gx+2-3. State the value of h(-2)hx is a translation of g(x)Horizontal: Left 2Vertical: Down 3h-2=-3372427521971000c) Given the graph of f(x), graph gx=fx-2-1. State the value of g(3).gx is a translation of f(x)Horizontal: Right 2Vertical: Down 1g3=2Example 3: State the parent function and the translation that is occuring in each of the given functions.FunctionParent FunctionTranslationsfx=x2-10y=x2Down 10gx=x+2y=xLeft 2hx=x-1+6y=xRight 1, Up 6fx=x-43-5y=x3Right 4, Down 5Example 4: If the parabola is f(x) and the vertex is (-2, 5), state the vertex after the given translations:a) fx-2+2(-2, 5)b) fx-5(-2, 5)Right 2, Up 2+2, +2Down 5 -5(0, 7)(-2, 0)c) fx+1-3(-2, 5)d) f(x-6)(-2, 5)Left 1, Down 3-1, -3Right 6+6, (-3, 2)(4, 5)Homework 6.4: Transformations of Functions1. Sketch the graph of2. Sketch the graph offx=x-1+4fx=x+3-2 3. Given the function f(x) shown below, 4. If a quadratic function f(x) hascreate a graph for hx=fx+2. a turning point at (-3, 7) then Find the value of h2.where does the quadratic function g defined by76200952500gx=fx+4+5 have a turning point?(-7, 12)(-4, 5)1, 12(4, 5)5. State the parent function and transformation that is occurring in each function below in the order it is occurring.FunctionParent FunctionTranslationsfx=x2+3gx=x-2-1hx=x-1+6fx=x+43Lesson 6.5: Reflections of GraphsLearning Goal: What are reflections and how do we represent the reflection of a graph in its equation?Warm-Up: State the transformation that is occurring, then sketch the graph of the function without using your calculator.a) fx=x-32+2b) gx=x+1-3 y=x2y=xRight 3, Up 2Left 1, Down 3fx-a+b where a is the horizontal translaton and b is the vertical translation781050-113030REFLECTING FUNCTIONS IN THE X AND Y AXESThe function -f(x) is a reflection of f(x) in the x-axis.The function f(-x) is a reflection of f(x) in the y-axis.REFLECTING FUNCTIONS IN THE X AND Y AXESThe function -f(x) is a reflection of f(x) in the x-axis.The function f(-x) is a reflection of f(x) in the y-axis.rx-axis=negate yry-axis =negate xExample 1 a) Given the graph of f(x) below, sketch the graph of f(-x). b) Given the graph of g(x) below, sketch the graph of -g(x). c) Given the graph of f(x) below, sketch the graph of -f(x).d) Given the graph of g(x) below, sketch the graph of g(-x).Example 2: After a reflection in the y-axis, the quadratic function gx=4x2-7x+2 would have the equation(1) y=-4x2+7x+2ry-axis =negate x or x→-x(2) y=-4x2+7x-2y=4x2-7x+2 (3) y=4x2+7x+2y=4-x2-7-x+2 (4) y=4x2+7x-2y=4x2+7x+2 When doing a combination of translations and reflections, the following order should be used:1) Horizontal translation2) Reflection3) Vertical translationExample 3: State the transformations in the order in which they occur.FunctionsTransformation H – R - Vfx=-gx-2+3Right 2Reflect x- axisUp 3fx=g-x-5Reflect y- axisDown 5fx=g(-x+2)Left 2Reflect y-axisfx=-gx-4+1Right 4Reflect x-axisUp 1fx=-g-x+1-4Left 1Reflect x-axisReflect y-axisDown 4Example 4: Write the equation of the following functions, given the parent function and the transformation performed.a) fx=x2, reflected in the y-axis, translated 5 units to the right and 3 units down.Fill in the blanks of parent graph: fx=____( _____x _____)2___fx=(-x-5)2-3b) fx=x, reflected in both the x-axis and y-axis, translated 11 units to the left and 4 units up.Fill in the blanks of parent graph: fx=_________x ________fx=--x+11+4c) gx=x, reflected in the x-axis, translated 21 units to the right.Fill in the blanks of parent graph: fx=_________x ______ __fx=-x-21Example 5: For each of the following functions, state the transformation and sketch the graph without using your calculator.-25717534607500a) fx=-x+3-4b) gx=-x-2+33571875103505002755900000-57150-51308000c) hx=--x-13d) fx=-x+22+1Homework 6.5: Reflections of Graphs1. Write the equation of the following function, given the parent function and the transformation performed.fx=x3, translated right 2 units, reflected in the x-axis, and translated down 4 units.2. Write the equation of the parent function fx=x that has been translated 9 units to the left and reflected in the y-axis.State the order of the transformations occurring and sketch the graph of3. gx=-x-1+34. fx=-x+32-25. If fx=-2x2+5x-3 and g(x) is a reflection of f(x) across the y-axis, then an equation of g is which of the following?(1) gx=-2x2-5x-3 (2) gx=-2x2+5x+3 (3) gx=2x2+5x-3 (4) gx=2x2+5x+36. State the order of the transformations that are occurring in each of the given equations.FunctionTransformationsfx=-x+22-1gx=-x-5+1hx=--x+3+2fx=-x-43Lesson 6.6: Dilations of FunctionsLearning Goals:What are dilations and how do we represent a dilation of a function in its equation?How can we state the transformation of a graph given its equation in the correct order?How can we graph a function given its equation using our knowledge of transformations?How can we write the equation of a function given the parent function and transformations performed?Warm-Up:a) The graph of f(x) is given below. b) The graph of f(x) is given below.Sketch the graph of gx=fx+1-4 Sketch the graph of hx=-fx+2. 104775-219075VERTICAL SCALINGThe function hx=k?f(x) represents the vertical scaling of a function by a factor of k.{Compression when 0<k<1 and stretch when k>1}00VERTICAL SCALINGThe function hx=k?f(x) represents the vertical scaling of a function by a factor of k.{Compression when 0<k<1 and stretch when k>1}Really means multiply y by scale factor.024130HORIZONTAL SCALINGThe function f(kx) represents a horizontal scaling of fx by a factor of 1k. {compression}The function f1kx represents a horizontal scaling of f(x) by a factor of k. {stretch}00HORIZONTAL SCALINGThe function f(kx) represents a horizontal scaling of fx by a factor of 1k. {compression}The function f1kx represents a horizontal scaling of f(x) by a factor of k. {stretch}Really means multiply x by scale factor. Scale factor is reciprocal of what you see in the question.Example 1:a) Consider the graph below.b) Consider the graph below.Sketch the graph of gx=12f(x).Sketch the graph of gx=f12x. Example 2: a) The graph of g(x) is shown on the gridb) The graph of f(x) is shown on below. Sketch the graph of 2fx.the grid below. Sketch a graph of f2x. Summary of TransformationsFunctionTransformationfx+aVertical translation, up afx-aVertical translation, down af(x+a)Horizontal translation, left af(x-a)Horizontal translation, right a-f(x)Reflection, negate y, rx-axisf(-x)Reflection, negate x, ry-axisk?fx or 1k?f(x)Vertical, Dilation, multiply y by either k or 1kfk?x or f1k?xHorizontal, Dilation, multiply x by either k or 1kPerforming multiple transformations:If a function has multiple transformations, they are applied in the following order:H orizontal translation D ilation R eflectionV ertical translationFor example: fx=-x-23-4; this function will be translated RIGHT 2, then REFLECTED over the x-axis, and then translated DOWN 4.Example 4: State the parent function and the order in which the transformations are occurring in each function.FunctionTransformation H D R Vfx=2x+12-4H: Left 1,D: Multiply x by 12,R:V: Down 4gx=14-x+2H:D: Multiply y by 14,R: ry-axis,V: Up 2hx=-3x-2-1H: Right 2,D: Multiply y by 3R: rx-axisV: Down 1Example 5: a) Write the equation of the function whose parent function is fx=x2 and has been translated 3 units to the right, reflected over the x-axis, horizontally scaled by a factor of 2, and translated 4 units up.? gx=-12x-32+4b) Write the equation of the function whose parent function is fx=x and has been vertically scaled by a factor of 2, horizontally scaled by a factor of 5, and translated 10 units to the left.? gx=215x-10c) Write an equation of the function whose parent function is fx=x and has been vertically scaled by a factor of 7, reflected in the x-axis, reflected in the y-axis, translated 5 units to the right and translated 2 units up.? gx=-7-x-5+2Homework 6.6: Dilations of Functions1. The graph of fx=10-x2 represents the graph of fx=x2 after(1) a vertical shift upwards of 10 units followed by a reflection in the x-axis.(2) a reflection in the x-axis followed by a vertical shift of 10 units upwards.(3) a leftward shift of 10 units followed by a reflection in the y-axis.(4) a reflection across the x-axis followed by a rightward shift of 10 units.4619625321310002. Which of the following equations represents the graph shown below?(1) y=x+32+4(2) y=-x+32+4(3) y=-x-32+4(4) y=-x-32-43. Consider the graph of f(x) below. Sketch the graph of 2f(x) on the same set of axes.4. Write the equation of the function whose parent function is fx=x2 and has been vertically stretched by a factor of 3, translated 10 units to the left, and reflected in the y-axis.5. Write the function whose parent function is fx=x and has been horizontally scaled by a factor of 5, translated 6 units to the right, reflected in the x-axis, and translated 2 units down.3429000127635006. Consider the function f(x) graphed on the grid.a) Sketch the graph of gx=f12x on the same set of axes.b) How would you describe its graph compared to the graph of fx.7. State the order of the transformations being performed in the given equations.FunctionTranslationsfx=-x2+3gx=2-x-2-1hx=-12x-1+6Lesson 6.7: The Definition of a ParabolaLearning Goals:What is the definition and properties of a parabola?What are the equations of a parabola?How can we sketch the graph of a parabola given its equation?Warm-Up: State the parent function and the translation that is occurring in each of the given functions.FunctionParent FunctionTranslationsfx=x-42-5Vertex = 4, -5y=x2Vertex =(0, 0)Right 4Down 5On the following diagram, draw and label the vertex and axis of symmetry. Discuss with a partner: Suppose you are viewing the cross-section of a flat mirror and a parabola-shaped mirror. Where would the incoming light be reflected in each type of design? Sketch your ideas below. ParabolaGeometry Review: When given a point in a plane, what is the set of all points that are equidistant from that point? Demonstrate by drawing a picture.Definition of a Parabola: When given a line and a point not on the line, the set of all points in the plane that are equidistant from the point and the line is called a parabola.FOCUS: is the point located "p" units away from the vertex inside the parabola on the axis of symmetry.DIRECTRIX: is the line perpendicular to the axis of symmetry located "p" units from the vertex.VERTEX FORM EQUATIONS OF PARABOLASParabolas can be horizontal or vertical.In all parabolas, the coordinate of the vertex is located at (h, k).The standard form for the equation of a parabola is y=ax2+bx+c. In this form, it is difficult to graph the parabola without the use of a graphing calculator.The vertex form for the equation of a parabola is y=a(x-h)2+k, where a=14p. In this form, it is eacy to graph the parabola by hand.In all parabolas, the variable p represents the distance from the vertex to the focus OR the distance from the vertex to the directrix.SUMMARYBefore you can determine the properties of a parabola, how must you express its equation? Vertex Form = non-squared variable (x or y) is isolated.When given the equation, how can you tell which direction a parabola will open?Horizontal x= and vertical y=look at sign (+ or -) of coefficient to determine specific direction!When given the equation, what are the coordinates of the vertex? Vertex =(h, k)What does the distance p represent on your parabola?P is the distance from vertex to focus. P is the distance from vertex to directrix.Directions: Identify the vertex and the distance of p for each parabola. Also identify which direction the parabola opens.Vertical Parabolasy=±14p(x-h)2+kHorizontal Hyperbolasx=±14p(y-k)2+ha) x-3=-18y2b) -16y=(x-1)2x=-18y2+3-16y-16=(x-1)2-16x=-18y+02+3y=-116x-12+0Opens leftV=(3, 0)Opens down V=(1, 0)4p=84p=16p=2p=4Based on the lesson so far, how would you graph the parabolas below?Vertical Parabolasy=±14p(x-h)2+kHorizontal Hyperbolasx=±14p(y-k)2+h1. Sketch 12x+2=(y+3)2 on the grid below and label the focus and directrix.12(x+2)12=(y+3)212x+2=112y+32Opens right with a vertex at V=-2, -3x=112(y+3)2-24p=122p=23=6p=3 How do you find the width of the parabola? Vertical Parabolasy=±14p(x-h)2+kHorizontal Hyperbolasx=±14p(y-k)2+h10763252536825002. Sketch -8y+1=(x-2)2 on the grid below and label the focus and directrix. -8(y+1)-8=-18(x-2)2Opens Down with a Vertex at V=(2, -1)y+1=-18x-22y=-18x-22-12p=22=44p=8p=2 Summary of How to Graph Parabolas1. Express parabola in vertex form.Vertical Parabolasy=±14p(x-h)2+kHorizontal Hyperbolasx=±14p(y-k)2+h2. Locate the vertex (h, k) on your graph.3. Decide whether the parabola opens up, down, left, or right.4. To locate the axis of symmetry, draw a line through the vertex going in the same direction the parabola opens.5. To locate the focus, count p units from the vertex on the axis of symmetry in the same direction the parabola opens {see picture below}.6. To locate the directrix, count p units from the vertex on the axis of symmetry in the opposite direction of the focus {see picture below}.7. Plot two points located 2p units from the focus in a direction that is parallel to the directrix {see picture below}.8. To draw the parabola start at the vertex and draw to each endpoint of the width. 3086100-86042500Homework 6.7: The Definition of a Parabola1. Sketch the parabolas below, labeling its focus and directrix.a) y=-14x2+1b) x=18(y-2)2 c) y-1=-14x2d) -8x+1=(y-2)2 e) Focus: (2, 4) Directrix: y=-2f) Focus: (0, 5) Directrix: x=-1 Lesson 6.8: Equations of Parabolas and Congruent ParabolasLearning Goals:How can we write the equation of a parabola given the focus and directrix?How can we determine the equation of a congruent parabola?Do now: Answer the following questions to help prepare for today’s lesson.1. What is the definition of a parabola? (Use the diagram to help with your explanation)A parabola is the set of all points in a plane that are equidistant from a point (the focus) and a line (the directrix).2. What is the vertex form of the equation of a vertical parabola?y=±14p(x-h)2+k3. What is the vertex form of the equation of a horizontal parabola?x=±14p(y-k)2+h4. What do the values of (h, k) and p represent about a parabola? h, k=vertexp=distance from vertex to focus OR vertex to directrixPractice Identifying the Vertexy=124x2 x=-18y2+3 x=120(y-1)2 y=124(x-0)2+0x=-18(y-0)2+3x=120(y-1)2+0V=(0, 0)V=(3, 0)V=(0, 1)y=-14x-22-2 y=112x2+5x=-116y+62-3V=(2, -2)V=(0, 5)V=(-3, -6)Vertical Parabolasy=±14p(x-h)2+kHorizontal Hyperbolasx=±14p(y-k)2+h933450-95885How To Write the Equation of a Parabola1. Draw a picture2. Find the vertex (h, k)3. Determine the value of "p"4. Determine which way parabola opens5. Write the equation0How To Write the Equation of a Parabola1. Draw a picture2. Find the vertex (h, k)3. Determine the value of "p"4. Determine which way parabola opens5. Write the equationDirections: Write the equation of the following parabolas in vertex form.425767553975001. x=±14p(y-k)2+hopens right!V=-6, 0 p=2x=18(y-0)2-6x=18y2-637242750002. x=±14p(y-k)2+hopens left!V=0, 0 p=3x=-112y23. Vertex: (2, 4) Focus: (2, 0)4. Vertex: (0, 0) Directrix: x=3Draw a rough sketchDraw a rough sketchopens down!Opens left!y=±14p(x-h)2+kx=±14p(y-k)2+hV=2, 4 p=4V=0, 0 p=3y=-116x-22+4x=-112y-02+0 or x=-112y23. 4. 5. 6. 5. Focus: (0, 5) Directrix: y=-16. Focus: (4, -1) Directrix: x=-4Draw a rough sketchDraw a rough sketchopens up!opens right!y=±14p(x-h)2+kx=±14p(y-k)2+hV=0, 2 p=3V=0, -1 p=4y=112(x-0)2+2 or y=112x2+2x=116y+12+0 or x=116(y+1)24181475356235007. Find the equation of the set of points which are equidistant from (4, -2) and the line y=4.They do not use the words focus, directrix, vertex or even parabola!!We know to write the equation of a parabola because the question is the definition of a parabola.y=-112(x-4)2+1V=4, 1 p=3Geometry Review: When are two graphs considered congruent to each other?When they are the same size and shape-332740147320Congruent ParabolasAll parabolas with the same distance between the vertex and focus (p) will be congruent. For example, the following parabolas are congruent:y=14x2 y=-14(x+3)2 y=14x2+3 y=-14x-12+2x=14y2 x=-14(y+3)2 x=14y2+3 x=-14y-12+2These all have p=1Why would having the same distance (p) make any two parabolas congruent?the width of the parabolas will be equal, making parabolas congruent00Congruent ParabolasAll parabolas with the same distance between the vertex and focus (p) will be congruent. For example, the following parabolas are congruent:y=14x2 y=-14(x+3)2 y=14x2+3 y=-14x-12+2x=14y2 x=-14(y+3)2 x=14y2+3 x=-14y-12+2These all have p=1Why would having the same distance (p) make any two parabolas congruent?the width of the parabolas will be equal, making parabolas congruent3486150300355001. Determine which equation represent the graph of a parabola that is congruent to the parabola shown to the right.a. y=120x2b. y=110x2+3c. y=-120x2+8d. y=15x2+5e. x=110y2f. x=15(y-3)2g. x=120y2+1Look at the p distance. p=5. 14p=14(5)=120ANSWERS: A, C, GHow to Write the Equation of Congruent ParabolasWhen are two parabolas congruent to each other?Same p distance.4162425164465002. Write the equation for two different parabolas that are congruent to the parabola with focus point (0, 3) and directrix line y=-3.p=3y=112x2+1x=112y2+1x=-112y2+63. Parabola P is given below.Which equation below is congruent to parabola P?(1) x=-13y2 (2) y=112(x+1)2 (3) x+3=16y2 (4) y=-13x22524125579755004. Parabola A is given below. Write an equation of a parabola that is congruent to parabola A, but with a vertex at (2, 5). y=14(x-2)2+5 y=-14(x-2)2+5 Equal p-values! p=1 V=(2, 5) Homework 6.8: Equations of Parabolas and Congruent Parabolas1. Sketch the parabolas below, labeling its focus and directrix.a) y=18x2b) x=-112(y-2)2 c) -16y+2=x2d) Vertex: (0, 0) Focus: (0, -3) 389890314325002. Write the equation of the following parabolas in vertex form.a) b) Vertex: 5, 0 Focus: 8, 0c) Vertex: (-1, -3) Directrix: x=43. Let P be the parabola with focus (0, 0) and directrix y=-2.a) Write an equation whose graph is a parabola congruent to P with the focus 0, 4.b) Write an equation whose graph is a parabola congruent to P with the same directrix y=-2, but different focus.4. The vertex of a parabola, f(x), has coordinates (-3, 5). Determine the coordinates of the vertex of the parabola defined by f(x+3)(1) -3, 8 (2) -6, 2 (3) 0, 5 (4) (-6, 5) ................
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