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Quadratic Functions

1. Graph y ax bx c In the previous lesson we learned to graph quadratics that did not have the b term Now we will discover how to graph a quadratic function when the b term is present Graphs such as these will not have the vertex on the y axis as we have previously seen Remember that the vertex is the MAXIMUM or MINIMUM point of a parabola For the equation y ax bx c http cms gavirtualschool org Shared Math CCGPS _AnalyticalGeometry 05_ QuadraticFunctions 05_AnalyticalGeometry_QuadraticFunctions SHARED pri 14 34 6 19 2015 Analytical Geometry If a gt O the vertex is a minimum parabola opens up If a lt 0 the vertex is a maximum parabola opens down Find the Vertex The vertex is a point ordered pair Using y ax bx c we can find the x value of the vertex with the formula fy SSS _ 2d To find y plug your value for x into the original equation and solve Using y ax bx c Plug into Recall Axis of Symmetry The axis of symmetry is always equal to the x value of the vertex http cms gavirtualschool org Shared Math CCGPS _AnalyticalGeometry 05_ QuadraticFunctions 05_AnalyticalGeometry_QuadraticFunctions SHARED pri 15 34 6 19 2015 Analytical Geometry ee mmm co om i N i4 6 8 10 o Axis f Symmetry etme eee Solving Quadratic Equations by Graphing We have already learned how to solve quadratic equations by factoring It is
2. Here are the steps for Factoring Completely for your review http cms gavirtualschool org Shared Math CCGPS _AnalyticalGeometry 05_ QuadraticFunctions 05_AnalyticalGeometry_QuadraticFunctions SHARED prin 8 34 6 19 2015 Analytical Geometry Factor out a GCF when possible If nothing can be factored write prime Quiz Group A Graphing Quadratics Graph y ax c Since we know how to factor and solve polynomials let s look at graphing We are going to learn to graph quadratic functions A quadratic function is a polynomial with degree 2 You have already learned to solve quadratics by factoring in the above lessons The standard form of a quadratic function is written as follows y ax bxt c When quadratics are graphed they form a u shaped graph The u shaped graph of a quadratic function is called a parabola http cms gavirtualschool org Shared Math CCGPS _AnalyticalGeometry 05_ QuadraticFunctions 05_AnalyticalGeometry_QuadraticFunctions SHARED prin 9 34 6 19 2015 Analytical Geometry g et 4a gt S 23 5 2 4 The maximum and minimum point on a parabola is called the vertex http cms gavirtualschool org Shared Math CCGPS _AnalyticalGeometry 05_ QuadraticFunctions 05_AnalyticalGeometry_QuadraticFunctions SHARED pri 10 34 6 19 2015 Analytical Geometry The vertical line that passes through the vertex of a parabola dividing the parabola into symmetric parts is
3. 4 1 5 36 20 16 The quadratic will have two real solutions This means the graph crosses the x axis twice Example 5 Find the discriminant and tell the number and type of solutions for x2 6x 9 0 a 1 http cms gavirtualschool org Shared Math CCGPS _AnalyticalGeometry 05_ QuadraticFunctions 05 AnalyticalGeometry_QuadraticFunctions SHARED pri 32 34 6 19 2015 Analytical Geometry 6 4 1 9 36 4 1 9 36 36 0 The quadratic has one real solution This means the parabola s vertex is on the x axis resulting in one x intercept Solving Quadratics Quiz It is now time to complete the Solving Quadratics Quiz quiz You will have a limited amount of time please 2 plan accordingly Module Wrap Up Module Checklist Discussion Factoring Quiz Paula s Peaches Task Parent Graphs Task Solving Quadratics Quiz Acme Factory Project Quadratic Functions Test Review Now that you have completed the initial assessments for this module review the lesson material with the practice activities and extra resources Then continue to the next page for your final assessment instructions Solve the following by factoring Solve the following by completing the square Find the discriminant and use it to determine if the solution has one real two real or two imaginary solution s Then solve for x using the quadratic formula Solve for x using a method of your choice Standardized Test Preparation j
4. b and c into the quadratic formula 7 7 a 1 6 2 1 Simplify and solve r http cms gavirtualschool org Shared Math CCGPS_AnalyticalGeometry 05_ QuadraticFunctions 05_AnalyticalGeometry_QuadraticFunctions SHARED pri 30 34 6 19 2015 Analytical Geometry 4 49 4 1 6 oie 49424 A i A Son 73 ane Answers may be kept in simplified radical form or converted to decimal form x 1 2 and x 7 17 Example 2 Use the quadratic formula to find the solutions for 2x2 8x 8 0 y 64 64 4 Example 3 Solve using the quadratic formula x2 2x 5 a 1 b 2 c 5 http cms gavirtualschool org Shared Math CCGPS_AnalyticalGeometry 05_ QuadraticFunctions 05_AnalyticalGeometry_QuadraticFunctions SHARED pri 31 34 6 19 2015 Analytical Geometry 2 V4 20 ity 16 2 y 47 t x 1 2i This function does not have any real solutions On the graph for this quadratic there would be no x intercepts The discriminant is the number underneath the radical of the quadratic formula b2 4ac We can use the discriminant to determine how many and what types of solutions a quadratic function will have When the discriminant is The equation has 2 imaginary solutions zero 1 real solution positive 2 real solutions Example 4 Find the discriminant and tell the number and type of solutions for x2 6x 5 0 b 4ac a 1 b 6 C 5 62 4 1 5 36
5. 6 19 2015 Analytical Geometry You should have seen from the video example that the correct answer is x 2 and x 1 Example 2 Solve the following rational equation by graphing vies You should see from the graph above that the correct answer is x 4 Graphing Quadratics Functions in Vertex Form Quadratic functions are not always written in standard form They can also be written in vertex form y a x h k where h k is the vertex The axis of symmetry for a quadratic in vertex form is x h Vertex Form To graph a quadratic function in vertex form we will use the same y g x h steps for graphing that we learned previously however we will not have to find the vertex since it is given to us in the equation h k IS ve rt Example 3 Graph y x 2 4 ex Vertex 2 4 Axis of S Domain All Real Numbers y Range y 2 4 Axis of Symmetry x 2 http cms gavirtualschool org Shared Math CCGPS _AnalyticalGeometry 05_ QuadraticFunctions 05_AnalyticalGeometry_QuadraticFunctions SHARED pri 20 34 6 19 2015 Analytical Geometry Changing Vertex Form to Standard Form Sometimes you will find it necessary to change quadratic equations from vertex form to standard form This is especially useful if you want to factor the quadratic To change from vertex form to standard form you will simply apply the order of operations and combine like terms Example 4 Convert y 2 x 3 2 3 to
6. The following problems will allow you to apply what you have learned in this module to how you may see 2 questions asked on a standardized test Please follow the directions closely Remember that you may have to Y use prior knowledge from previous units in order to answer the question correctly If you have any questions or concerns please contact your instructor http cms gavirtualschool org Shared Math CCGPS _AnalyticalGeometry 05_ QuadraticFunctions 05_AnalyticalGeometry_QuadraticFunctions SHARED pri 33 34 6 19 2015 Analytical Geometry fl Quiz Group Final Assessments Quadratic Functions Test assignments and the review items and feel confident in your understanding of this material you may begin You will have a limited amount of time to complete your test and once you begin you will not be allowed to restart your test Please plan accordingly Q It is now time to complete the Quadratic Functions Test Once you have completed all self assessments Acme Factory Project Download the Acme Factory Project from the sidebar and complete your answers on a separate word document Please save your work and submit it http cms gavirtualschool org Shared Math CCGPS _AnalyticalGeometry 05_ QuadraticFunctions 05_AnalyticalGeometry_QuadraticFunctions SHARED pri 34 34
7. also possible to solve by graphing Remember that we called the solutions to quadratics the roots or zeros of the problem That is because the solutions are the x intercepts or where the graph crosses the x axis If you are given a graph of a quadratic function you do not need to factor to find the solutions if you are able to read the x intercepts from the graph Example 3 Find the solutions for the following graph Since the graph crosses the x axis at 2 and 5 the solutions are x 2 and x 5 Example 4 Find the solutions for the following graph http cms gavirtualschool org Shared Math CCGPS _AnalyticalGeometry 05_ QuadraticFunctions 05_AnalyticalGeometry_QuadraticFunctions SHARED pri 16 34 6 19 2015 Analytical Geometry This graph does not have any x intercepts therefore there are no solutions Rule for checking solutions To determine if a value is a solution plug it into the original equation and see if both sides are equal Example 4 Is x 3 a solution for the equation 0 2x2 5x 1 0 2 3 4 5 3 1 0 2 9 5 3 1 0 18 15 1 0 4 ANSWER Example 5 Is x 3 a solution for the equation 0 x2 6x 9 0 32 6 3 9 0 9 6 3 9 0 9 18 9 0 0 ANSWER Solving Quadratics using a Graphing Calculator Solving quadratics by graphing is not always the most efficient method to solve because oftentimes graphs are difficult to read This is especially true if the solutions are not whole numbers To make it ea
8. called the axis of symmetry 10 et 10 8 6 4 2 14 6 8 10 2 i i 4 i Axis of Symmetry 6p i of 10 4 To graph a quadratic function we can use a table of values and plot the points The Parent Quadratic Function http cms gavirtualschool org Shared Math CCGPS _AnalyticalGeometry 05_ QuadraticFunctions 05_AnalyticalGeometry_QuadraticFunctions SHARED pri 11 34 6 19 2015 Analytical Geometry 2 The parent quadratic function is y x Notice the domain is all real numbers and the range is y 2 0 Transformations of the Parent Function Review the transformations that you learned from your previous course http cms gavirtualschool org Shared Math CCGPS _AnalyticalGeometry 05_ QuadraticFunctions 05_AnalyticalGeometry_QuadraticFunctions SHARED pri 12 34 6 19 2015 Analytical Geometry Ifa is negative e the graph reflects across the x axis If cis positive e the graph vertically translates up If c is negative e the graph vertically translates down jaa clesteit Ve V ei the graph vertically stretches Graph ais greater than 1 looks skinnier iia eleste Vite Ve er the graph vertically shrinks Graph looks ais between 0 and 1 wider The following quadratics are graphed below Pay close attention to each equation and how they compare to the graph of the parent function Domain and Range Graph the function y 3x2 4 and identify its domain and range Compare it to the parent
9. standard form y 2 x 3 x 3 3 y 2 x 6x 9 3 y 2x 12x 18 3 y 2x2 12x 21 Example 5 Convert y 3 x 4 1 to standard form 3x2 24x 49 Graphing Quadratic Functions in Intercept Form Intercept form is y a x p x q where p and q are the intercepts The steps to graphing are the same after we find the vertex Notice that intercept form is the same as factored form The axis of symmetry for a quadratic in intercept PHG oo form is 2 To find the vertex in intercept form PEG PEG EAF Graph y x 2 x 2 e Intercepts x 2 and x 2 e Vertex 0 4 e Domain All real numbers e Range y 2 4 e Axis of Symmetry x 0 http cms gavirtualschool org Shared Math CCGPS _AnalyticalGeometry 05 QuadraticFunctions 05_AnalyticalGeometry_QuadraticFunctions SHARED pri 21 34 6 19 2015 Analytical Geometry Changing Intercept Form to Standard Form To change from intercept form to standard form simply foil distribute and combine like terms Change y x 3 x 2 to standard form y x2 x 6 e y x x 6 Change y 2 x 5 x 2 to standard form e y 2x2 6x 20 Graphing Quadratic Inequalities Graphing inequalities is just like graphing any other equation There are two differences First you have to check to see if your graph will be solid or dashed Second you have to decide where to shade Graph y 2 x2 4x 1 Remember 2 and lt represe
10. x2 10x 24 Final Answer x 6 x 4 Self Check fl Factoring Trinomials with a Leading Coefficient not Equal to 1 What do we do when we have to factor a trinomial ax bx c where a does not equal 1 This will change our steps since we will need to consider the leading coefficient Depending on what the leading coefficient is there may be different steps to take Self Check A Once we factor the trinomial s we can use the zero product property to solve Remember that if the two binomials multiplied together equal zero than one or both of them must also equal zero Quiz Group fl Quadratic Functions Discussion il It is now time to complete the Quadratic Functions discussion Post one polynomial that should be factored by grouping After you have posted your problem factor another classmate s problem Once a classmate has factored the problem you created reply letting them know if the answer is correct or not You will have a total of 3 posts when you are done with this discussion Factor Polynomials Completely Polynomials sometimes require multiple steps to factor completely For example you might have to factor out a GCF from a trinomial and then continue to factor the trinomial into two binomials Check out the following video Example 1 Factor completely 4x3 x This example can be factored both by dividing by a GCF x and then by factoring the resulting binomial x 4x2 1 Notice that there a
11. 6 19 2015 Analytical Geometry Analytical Geometry Quadratic Functions Quadratic Functions What does the accelerator in a car a kitchen sink an airplane the spots on a giraffe a cell phone and insect population have in common They all use quadratic functions Quadratics are used in the creation and understanding of an infinite number of things in the world predicted and outlined before it is even born through the use of quadratic equations We would not have cell phone mathe calculations to test early prototypes In this module you Wi algebraically and graphically Essential Questions The spots on a cheetah can actually be s as efficient as they are today if it weren t for maticians using quadratics in mathematical ll learn how to interpret polynomial functions both What are the parts of a polynomial How can we use quadratic equations and inequalities to solve problems How can we use a system of equations to solve problems What are the features of a graph of a quadratic How can a quadratic equation be solved algebraically Uses for What happens to a quadratic equation when the Quadratic graph is transformed How do quadratic functions compare to linear Function and exponential functions How can we use quadratic regression to find equations that fit real world problems Module Minute A Quadratic functions can be solved algebraically by taking the square root factoring complet
12. aula s Peaches Task Assignment from the sidebar and complete your answers on a separate word document Please save your work and submit your completed assignment Average Rates of Change for Graphs To find the average rate of change between two points simply find the slope of the line that connects those two points Poan paap Slope m 7 al siad Example 1 Find the average rate of change of f x x 3x from x4 to x2 given that x4 2 and x2 0 f 2 2 3 2 f 2 8 3 2 f 2 8 6 f 2 2 f 0 0 3 0 f 0 0 3 0 0 0 0 f 0 0 2 0 2 3 0 a http cms gavirtualschool org Shared Math CCGPS_AnalyticalGeometry 05_QuadraticFunctions 05_AnalyticalGeometry_QuadraticFunctions SHARED pri 24 34 6 19 2015 Analytical Geometry Example 2 Compare the rates of change for f x 2x and g x ya from x4 0 to xo 4 Only use positive value since there is not a negative sign in from of the y4 w nan ws PJ Y E I The two graphs both have positive average rates of change However f x has a higher average rate of http cms gavirtualschool org Shared Math CCGPS_AnalyticalGeometry 05_ QuadraticFunctions 05_AnalyticalGeometry_QuadraticFunctions SHARED pri 25 34 6 19 2015 Analytical Geometry change so it is increasing faster than g x Example 3 Calculate the average rate of change of the function y 2x 1 on the interval 1 lt
13. call that the distributive property allows us to multiply x by each term in the above binomial The opposite of the distributive property is factoring We can use monomial factoring to do the reverse of the above problem y2 2x x x 2 Notice that the two terms have x in common so we can factor the x out and write what is left inside parenthesis Monomial factoring is the opposite of distribution Steps to Monomial Factoring Steps to Solve by Factoring Find the GCF of the polynomial Divide the GCF into the original polynomial 1 Set the equation equal to zero 2 Factor completely 3 Find the zeros for each factor Rewrite as a product of the GCF and the new divided polynomial Factoring Trinomials with a Leading Coefficient of 1 A standard trinomial will be in the form ax bx c where a b and c are integers In the past we have used multiplication and the distributive property to get these trinomials Now let s determine how to factor them We will begin by factoring trinomials that have a leading coefficient of 1 This means the a value in our standard form will be 1 Recall the FOIL method that we used when multiplying two binomials x 1 x 2 x2 2x x 2 x2 x 2 Let s factor the trinomial x x 25 We know that this trinomial will result in the product of two binomials based on the previous FOIL example We also know that the factors of x2 are x and x We then need to think of two numbers that have a prod
14. function y x2 We can see both from the equation and the graph that there are three transformations that have taken place The first transformation is the negative sign in front of the equation This tells us that the function has been reflected across the x axis Next the number 3 tells us that the function has been vertically stretched by a factor of 3 Finally the 4 that is added to the end of the equation tells us that the function has been translated up 4 units The domain is still all real numbers however the range has become y lt 4 Graph the function y 2x2 2 and identify its domain and range Compare it to the parent function y x2 The domain is all real numbers The range is y 2 2 The graph has a vertical shrink by a factor of 2 and has been translated down 2 units from the parent function http cms gavirtualschool org Shared Math CCGPS_AnalyticalGeometry 05_ QuadraticFunctions 05_AnalyticalGeometry_QuadraticFunctions SHARED pri 13 34 6 19 2015 Analytical Geometry Graph the function y 5x2 2 and identify its domain and range Compare it to the parent function y x2 The domain is all real numbers The range is y lt 2 The graph has been reflected across the x axis vertically stretched by a factor of 5 and translated up 2 units from the parent function Quiz Group 4 Factoring Quiz _ It is now time to complete the Factoring Quiz quiz You will have a limited amount of time please plan Q accordingly
15. hs Assignment from the sidebar and complete your answers on a separate word document Please save your work and submit it Solving Quadratics by Completing the Square You will notice that some quadratics have perfect squares as the c value We call these perfect square trinomials Try multiplying x 5 x 5 x 5 x 10x 25 This is called a perfect square trinomial because the c value 25 is a perfect square Multiply x 9 x 9 x 9 x 18x 81 This is also a perfect square trinomial because 81 is a perfect square e Which value of c will x 6x be a perfect square trinomial e Which value of x will x 12x be a perfect square trinomial Now we will learn a new method for solving quadratic equations We have already learned to solve by factoring and by graphing This method is called completing the square http cms gavirtualschool org Shared Math CCGPS _AnalyticalGeometry 05_ QuadraticFunctions 05_AnalyticalGeometry_QuadraticFunctions SHARED pri 28 34 6 19 2015 Analytical Geometry When should completing the square be used e When the quadratic cannot be factored e When the b value is even preferred e When the a value is 1 preferred The last two options are preferences but not required for completing the square We will learn another method for solving which is easier than completing the square under those two circumstances Example 1 Solve 0 x2 8x 4 by completing the squa
16. ing the The square or using the quadratic formula Sometimes the solution will result in a complex number od le _ Quadratics can also be solved graphically When looking at a graph of a quadratic we can identify the Mi v Le extrema values intercepts intervals of increase and decrease symmetries and end behavior When X nu i applied to real world problems this presents a wealth of information Many real world situations are better modeled by a parabola rather than a linear or exponential equation The structure of a quadratic equation can tell us whether the graph will open up or down as well as provide us with information about how the graph may have transformed from its parent function Key Words Horizontal shift A rigid transformation of a graph in a horizontal direction Vertex form of a quadratic function A quadratic equation written in the form f x a x h k where ais a nonzero constant and h k is the vertex of the graph Discriminant The number b 4ac when a quadratic equation is written in the form y ax bx C Quadratic Formula The solutions of the quadratic equation in the form y ax bx c when a is nonzero I r b p dae and b and c are real numbers Da http cms gavirtualschool org Shared Math CCGPS _AnalyticalGeometry 05_ QuadraticFunctions 05_AnalyticalGeometry_QuadraticFunctions SHARED prin 1 34 6 19 2015 Analytical Geometry A handout of these key words and definitions
17. is also available in the sidebar What To Expect e Quadratic Functions Discussion e Factoring Quiz e Paula s Peaches Task e Parent Graphs Task e Solving Quadratics Quiz e Acme Factory Project e Quadratic Functions Test To view the standards from this unit please download the handout from the sidebar Solving Polynomial Equations In the previous lesson we learned how to simplify polynomial expressions We will now learn how to find the solutions of the expressions Before we start solving we need to review a very important property the Zero Product Property Zero Product Property tells us that duct Property sro it will become awe terms f a lyin a when multip ni ero Ofer the first or second ei gi E gi Ii We will soon look at graphing new functions When we graph we will refer to the graph s roots or zeros The zeros of a function are any values for x when y equals zero Using those two pieces of information we can solve the following examples Example 1 Solve x 1 x 3 0 http cms gavirtualschool org Shared Math CCGPS_AnalyticalGeometry 05_ QuadraticFunctions 05 AnalyticalGeometry_QuadraticFunctions SHARED _ prin 2 34 6 19 2015 Analytical Geometry Notice that this problem is made up of two binary expressions that are multiplied together to equal zero This means that at least one of those binary expressions must be zero We will se
18. ng squared in the equation 2 Get the squared item by itself 3 Take the square root of both sides do not forget the square root is both positive and negative Example Solve for the variable http cms gavirtualschool org Shared Math CCGPS _AnalyticalGeometry 05_ QuadraticFunctions 05_AnalyticalGeometry_QuadraticFunctions SHARED pri 18 34 6 19 2015 Analytical Geometry X 3 Self Check Quiz Group i Solving Systems of Equations by Graphing When given two or more equations system of equations it is possible to solve by graphing using a graphing calculator The solutions to a system of equations are the point s where the graphs intersect The following are the steps to solving using a TI graphing calculator Solve Systems Using a Graphing Calculator Move the Move the On your cursor as cursor as graphing Set two OaE N close as you close as you equation COR Graphthe Barn can to the can to the Press ae Select 5 point of point of slain hs eat equations equations Trace Intersect intersection inte rsection Saa in for y and to find the to find the the second first curve second equation in and press curve and for y enter press enter time to get your answer equal to Example 1 Solve the following rational equation by graphing x2 1 x 3 http cms gavirtualschool org Shared Math CCGPS_AnalyticalGeometry 05_ QuadraticFunctions 05_AnalyticalGeometry_QuadraticFunctions SHARED pri 19 34
19. nt solid lines lt and gt represent dashed lines A dashed line means the solution to the function is not on the line Since our example has is 2 we will graph it as normal with a solid line http cms gavirtualschool org Shared Math CCGPS_AnalyticalGeometry 05_ QuadraticFunctions 05_AnalyticalGeometry_QuadraticFunctions SHARED pri 22 34 6 19 2015 Analytical Geometry This is the graph of the equation Now to graph the gt part we need to shade all of the possible solutions To figure out which side of the graph to shade simply select a point that is not on the graph and test it in your equation If the point is true then shade that side If it is false shade the other side Let s test the point 1 1 1 1 4 1 1 121 4 1 1 121 4 1 1 2 4 True shade the region that contains this point Match the following inequalities with their graphs http cms gavirtualschool org Shared Math CCGPS_AnalyticalGeometry 05_ QuadraticFunctions 05_AnalyticalGeometry_QuadraticFunctions SHARED pri 23 34 6 19 2015 Analytical Geometry Self Check Solving Systems of Inequalities Recall that a system of inequalities is simply two or more inequalities graphed on the same coordinate plane The solutions to the system will be contained in the overlapping shaded region the region shaded by both inequalities Graph the system e y gt x 2x 1 e y lt x2 3x 1 Self Check fl Paula s Peaches Task Download the P
20. re Rewrite the problem so that it is equal to c Subtract 4 from both sides 4 r Bt Divide the b term by 2 a 554 Square the result from step 2 4 16 Add the result from step 3 to both sides of the equal sign 4 16 r Br 16 Simplify and factor 12 r 8r 16 2 12 r 4 Solve for x 4 y12 x 4 4 23 r r 536 and x 1 46 http cms gavirtualschool org Shared Math CCGPS _AnalyticalGeometry 05_ QuadraticFunctions 05_AnalyticalGeometry_QuadraticFunctions SHARED pri 29 34 6 19 2015 Analytical Geometry Answers may be written in simplified radical form or with decimals Example 2 Solve 0 x2 14x 4 4 2 14x 14 __ 3 7 49 4 49 x 14x 49 45 x 7 35 x 7 74 3y5 2 X 13 7 and x 292 Using the Quadratic Formula and Discriminant We have now learned 3 ways to solve quadratic functions We can solve by graphing factoring or completing the square Each of these methods is best used for certain types of quadratics There is one last method that can be used every single time for any quadratic function The final method for solving is using the quadratic formula lt 4 V b dac A Quadratic Formula at Learn the Quadratic Formula song to help you remember it Example 1 Use the quadratic formula to find the solutions of x2 7x 6 First put the quadratic in standard form and identify a b and c x2 7x 6 0 a 1 b 7 c 6 Plug a
21. re only two terms however since x is squared we can still factor We can pretend that there is a middle term and it happens to be zero x 4x2 Ox 1 Now think of factors for 1 that add up to 0 x 2x 1 2x 1 http cms gavirtualschool org Shared Math CCGPS _AnalyticalGeometry 05_ QuadraticFunctions 05_AnalyticalGeometry_QuadraticFunctions SHARED prin 7 34 6 19 2015 Analytical Geometry There are three factors for this polynomial Factor by Grouping Occasionally polynomials will be written in a way that looks as if someone started to factor but did not finish Example 2 Factor by Grouping x 3x 1 2 3x 1 This problem looks as if someone started to factor using the six step method but then stopped at step 5 All we need to do is finish that last step We can write this polynomial as two binomials because the parenthesis contain the same expression x 2 3x 1 Example 3 Factor by grouping 3x 4 y 6 4 y 3x 6 4 y Example 4 Factor by grouping x2 3x 4x 12 Notice that this example does not have the parenthesis in place This is as if we are at step 3 of the 6 step process Remember when adding parenthesis always have a sign between them x2 3x 4x 12 Now we can factor out the GCF from each binomial and continue the factoring process x x 3 4 x 3 x 4 x 3 Example 5 Factor by grouping 3n 3n2 n 1 3n 3n2 n 1 3n2 n 1 n 1 3n2 1 n 1
22. sier you can use a graphing calculator The following are instructions for a TI graphing calculator 83 84 Plus If you have another http cms gavirtualschool org Shared Math CCGPS_AnalyticalGeometry 05_ QuadraticFunctions 05_AnalyticalGeometry_QuadraticFunctions SHARED pri 17 34 6 19 2015 Analytical Geometry brand of calculator you can search the internet or read the user manual for instructions on how to find the zero of a graph Solve Using a Graphing Calculator Steps For the right bound move your Press cursor toa enter i oint to again to Trace aske fane bee the right ss a he left of of the guess 7ero and press enter For the left Set the bound Finally the zero will be displayed equation Eria equal to Plug the Selon Select Calc move your zero and equation aI 274 Graph Select 2 cursor toa write in into Y standard meeen the zero and press enter Using Square Roots to Solve Quadratic Equations Do you remember learning about square roots Positive numbers have both a positive square root and negative square root The positive square root is called the principle square root You cannot take the square root of a negative number The square root symbol is called a radical The number under the square root is called the radicand Using square roots is another method that can be used to solve quadratic equations Solving Quadratic Equations Steps 1 Determine what is bei
23. t each expression equal to zero in order to find what solutions cause this statement to be true Solve x Kx 3 0 fi ie x 1 0 x F3 0 i H H Te E The two solutions or zeros are x 1 and x 3 Example 2 Example 3 http cms gavirtualschool org Shared Math CCGPS _AnalyticalGeometry 05 QuadraticFunctions 05_AnalyticalGeometry_QuadraticFunctions SHARED prin 3 34 6 19 2015 Analytical Geometry Each of the above examples was written in factored form This means that the expressions were written as terms that were multiplied together Sometimes we will have problems that are not in factored form which means that we need to factor them first Recall that the greatest common factor GCF is the largest common divisor of each term of a polynomial Finding Greatest Common Factor Steps 1 Find the largest factor of each coefficient 2 For variables the variable must be present in each term Choose the variable with the smallest exponent Finding GCF examples Example 1 Axsy 2xy9 12x9 2x is the GCF since 2 is the largest number that can be factored out of 4 2 and 12 and each of the monomials has at least one x Example 2 m n 4m n 8m4n 1 m3n is the GCF Monomial Factoring http cms gavirtualschool org Shared Math CCGPS _AnalyticalGeometry 05_ QuadraticFunctions 05_AnalyticalGeometry_QuadraticFunctions SHARED prin 4 34 6 19 2015 Analytical Geometry f x 2 x2 2X Re
24. uct to equal our last term 2 but a sum to equal our middle term 1 If we list the factors of a our last term 2 we get http cms gavirtualschool org Shared Math CCGPS _AnalyticalGeometry 05_ QuadraticFunctions 05_AnalyticalGeometry_QuadraticFunctions SHARED prin 5 34 6 19 2015 Analytical Geometry 1 2 1 2 These are the only two factors that have a product of FOIL Method muliphy Fint uter nnet Last 2 Which of these pairs has a sum of 1 ANSWER Using these two numbers write our trinomial as the product of two binomials E X 3 S F d x 1 x 2 3 x T 2 O xt nn Ti cae 0 2 p Steps for Factoring Trinomials with a Leading Coefficient of 1 L x 3 x 2 For the polynomial ax bx c 1 Determine all the factors of c 2 Determine which factors of c equal b when added or subtracted 3 Rewrite your trinomial as the product of two binomials Example 1 Factor x2 5x 6 Factors of 6 1 6 1 6 2 3 2 3 Which of these factors add up to 5 ANSWER So our final answer would be x 2 x 3 Example 2 Factor x 11x 24 Final Answer x 3 x 8 Example 3 Factor x 7x 18 Final Answer x 9 x 2 Example 4 Factor x 9x 36 Final Answer x 12 x 3 http cms gavirtualschool org Shared Math CCGPS _AnalyticalGeometry 05_ QuadraticFunctions 05_AnalyticalGeometry_QuadraticFunctions SHARED prin 6 34 6 19 2015 Analytical Geometry Example 5 Factor
25. x lt 0 Plug in 1 y 2 1 2 1 y 2 1 1 y 2 1 y 1 1 1 Plug in 0 y 2 0 2 1 y 2 0 1 y 0 1 y 1 0 1 Find the average rate of change slope formula o 1 Identifying Intervals of Increase and Decrease Recall the intervals of increase and decrease are intervals where a graph increase or decreases along the x axis The intervals are determined by x Example 4 Graph the y x2 2x 1 Determine the vertex and the intervals of increase and decrease fi Vertex x 2a y 1 2 2 1 1 y 142 1 1 y 1 2 1 http cms gavirtualschool org Shared Math CCGPS _AnalyticalGeometry 05_ QuadraticFunctions 05_AnalyticalGeometry_QuadraticFunctions SHARED pri 26 34 6 19 2015 Analytical Geometry Increasing x gt 1 Decreasing x lt 1 Example 5 Graph the y x 1 x 3 Determine the vertex and the intervals of increase and decrease 4 3 Vertex x 2 2 x 2 x 1 y 1 1 1 3 y 2 2 yoo 1 4 Increasing x lt 1 Decreasing x gt 1 Now try this one on your own http cms gavirtualschool org Shared Math CCGPS _AnalyticalGeometry 05_ QuadraticFunctions 05_AnalyticalGeometry_QuadraticFunctions SHARED pri 27 34 6 19 2015 Analytical Geometry Graph the y 3x2 6x 4 Determine the vertex and the intervals of increase and decrease Vertex Increasing Decreasing Parent Graphs Assignment Download the Parent Grap

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