March, 07 9 Quadratic Graphs and Their Properties A quadratic function is a function that can be written in the form: Verte Its graph looks like... which we call a parabola. The simplest quadratic function is =. Since all graphs are a variation of this one we called it the "parent" function for quadratics. Dec :7 PM Dec :00 PM The graph is smmetric. (It is the same on both sides.) It will have a minimum or a maimum. The line where ou could fold the graph onto itself is called the ais of smmetr. The ais of smmetr alwas passes through the verte. Apr 9 :07 AM Dec :00 PM What makes the graph open up? What makes the graph skinn? What makes the graph open down? What makes the graph wide? Dec : PM Apr 9 :0 AM
March, 07 What moves the graph up? Give three properties of the graphs of the equations below. What moves the graph down? Apr 9 :0 AM Apr 9 :0 AM Domain and Range of a Quadratic Function Domain values All quadratic function have a domain of all real numbers. ou can plug an number in for. Range values Look at the graph. See the ma/min and use that to create our range. The range is never all real numbers for quadratics. 0 9 7 Domain: Range: Find the domain and range of the following quadratic. 0 9 7 0 0 0 7 9 0 Domain: Range: 0 0 0 Mar 0 0: AM Mar 0 0: AM 9 Graphing Quadratic Functions One important feature of a quadratic function is the location of the ais of smmetr. Find the equation of the ais of smmetr of the following quadratic function. The equation for the ais of smmetr is: Feb 7 :7 AM Feb 7 : AM
March, 07 Find the equation of the ais of smmetr of the following quadratic function. Knowing the location of the ais of smmetr helps ou find the verte of the parabola. ) Find the ais of smmetr. ) Plug our answer from # into the equation for to find the value of. Be sure to use parentheses around negative numbers. ) (,) from and is the location of the verte. Find the verte of the following quadratic function. Feb 7 : AM Feb 7 :9 AM Find the verte of the following quadratic function. The intercept of the quadratic is just the value of. Find the intercept of the following parabola. Feb 7 9:7 AM Feb 7 9:7 AM Find the intercept of the following parabola. Put everthing together. Here are the steps to graphing a quadratic function. a) Find ais of smmetr using = b/a, and then draw it on the graph. b) Plug our answer from part a into the given equation to find the value of the verte. Put the verte in the middle of the table and on our graph. c) The intercept is the value of c. Put it on our graph and table. Mark the matching point on the other side of the graph and table. d) Find one more point b choosing an value not et used on our table. Plug it into the given equation. Put the point on our graph and table. Mark the matching point on the other side. e) Make sure our table has points and our graph has points. Feb 7 9: AM Feb 7 9: AM
March, 07 Graph the quadratic equation be filling in points on the table AND points on the graph. Graph the quadratic equation be filling in points on the table AND points on the graph. 0 0 Feb 7 9: AM Feb 7 9: AM 9 Solving Quadratics Equations when b=0 When can we use this method? When there is no b value. How to use: ) Get the b itself. (add/sub then mult/div) ) Take the square root of both sides ) Answer is the + and of that number The square root of a negative number is not real... thus the answer would be: No solution How to use: ) Get the b itself. ) Take the square root of both sides ) Answer is the + and of that number Feb :9 PM Feb : PM How to use: ) Get the b itself. ) Take the square root of both sides ) Answer is the + and of that number How to use: ) Get the b itself. ) Take the square root of both sides ) Answer is the + and of that number Feb : PM Feb : PM
March, 07 Not ever equation will have a solution: Eample: How to use: ) Get the b itself. ) Take the square root of both sides ) Answer is the + and of that number How to use: ) Get the b itself. ) Take the square root of both sides ) Answer is the + and of that number Ma 7 : PM Ma 7 :9 PM How to use: ) Get the b itself. ) Take the square root of both sides ) Answer is the + and of that number How to use: ) Get the b itself. ) Take the square root of both sides ) Answer is the + and of that number Feb : PM Feb : PM 9 Solving Quadratics Using the Factoring Method Zero Product Propert When two numbers multipl to be 0, then one of the numbers must be 0. We should write all quadratic problems with a value of b in their factored form so that we can use this propert. This onl works when the equation is =0. Ma : PM Ma :0 PM
March, 07 Ma :0 PM Ma :0 PM How are these different than the ones we did esterda? esterda there was no b value. Toda there is a b value. Ma : PM Ma : PM How do the answers relate to the graph? The answers we get are where the graph, when equal to 0, will cross the -ais. Solve the equation Check the answers Sketch the graph Ma :7 PM Apr 7 0: AM
March, 07 Summar of how to solve when there is a b-value: ) Make sure one side is =0 (add/sub to make it happen) ) Factor using the ac method ) Set each factor (thing in parentheses) equal to 0 9 Completing the Square and Verte Form Section we used square roots to solve. That section was simple in comparison to section. Eample: = *Don't forget plus and minus Apr 7 0: AM Feb 7 9:0 AM So we can appl that same knowledge to the following equation. (+) = *Don't forget plus and minus AFTER ou square root both sides. If possible write the following as a quantit squared. What if the equation is NOT written as a quantit squared? Then we need to force it to be that we so that we can appl our simple process from section. Let's take a look at writing epressions as "something" squared. Feb 7 0: AM Apr :9 PM So what do we do with the things that were not squared values. MAKE them squared while still following our rules of algebra. Completing the Square Steps: ) Make sure a = (if not div b a to all terms) ) Get c b itself on the other side ) Add to both sides. ) Factor must be the same factors (will alwas be b/) ) Square Root ( + or ) ) Solve resulting two equations. Feb 7 0: AM Feb 7 0: AM 7
March, 07 Solve b completing the square. Solve b completing the square. Feb 7 : PM Feb 7 : PM Solve b completing the square. Solve b completing the square. Ma : PM Feb 7 :0 AM Verte Form: Find the verte: (h,k) is the verte of the quadratic Ma :7 PM Ma : PM
March, 07 Find the verte: Find the verte: Ma : PM Ma : PM If not in the right form, then ou must complete the square to find the verte. Write in verte form and state the verte: Write in verte form and state the verte: Ma : PM Ma : PM 9 The Quadratic Formula Solve using the quadratic formula. The equation must be =0 in order to use. a is b is c is There are two seperate answers here. The first time ou do + the square root and get an answer. The second time ou do the square root and get an answer. Feb 7 : AM Apr :00 PM 9
March, 07 Solve using the quadratic formula. (Must be set = 0 to use the quadratic formula.) Solve using the quadratic formula. Feb 9 : PM Feb 9 :0 PM 9 7 Linear, Quadratic, and Eponential Models Linear Quadratic Eponential Are the following graphs linear, quadratic or eponential? 0 0 0 0 0 0 0 0. 0 0. 0 0 0 7 7 Linear the same amount each time Quadratic The differences are the same. Eponential the same amount each time Feb 7 : AM Feb 7 :9 PM Are the following graphs linear, quadratic or eponential? Are the following graphs linear, quadratic or eponential? 0 0 0 0 0 0 0 0 0 0 Feb 7 :9 PM Feb 7 :9 PM 0
March, 07 Which tpe of function best models the data in each table? Which tpe of function best models the data in each table? Tpe of function: Equation from calc: Tpe of function: Equation from calc: 0 7 0 0 Feb 7 :0 PM Feb 7 :0 PM Which tpe of function best models the data in each table? Tpe of function: Equation from calc: 9 Solving Sstems Using a Graphing Device ) Plug both equations into calculator. (Be sure each equation is solved for or ou can't plug it in.) ) Adjust window so that ou can see the intersections of the graphs. ) Find the intersections of the graphs. Remember the value of where the cross is our answer. 0 Feb 7 :0 PM Mar 7: AM Use a graphing device to solve for the value(s) of. Use a graphing device to solve for the value(s) of. Mar 7:7 AM Mar 7: AM