Lecture Notes Graph of a Parabola - 2 page 1
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1 Lecture Notes Graph of a Parabola - page Eample. Graph the parabola = +. Clearl label the coordinates of ve points of the parabola, including verte and intercepts. Solution: We arrange the terms b degree to obtain the polnomial form, = + +. From the polnomial form, we obtain the intercept b substituting =. If =, then = + + =. And so the intercept is (; ). We factor out the leading coe cient and complete the square to obtain the standard form of the equation. = + + = ( ) = + = {z + } = ( ) distribute = ( ) + The equation = ( ) + is the standard form of the equation of the parabola. It is the form we use to determine the verte of the parabola. The verte has to do with the complete square being zero. For the complete square to be zero, we solve ( ) = = = And so the coordinate of the verte is. And if the complete square is zero, then the coordinate of the verte can be easil found: = ( ) + = + = so the verte is (; ). Notice that the epression ( ) is alwas negative or zero, and so now is the greatest value that the epression ( ) + achieves. Indeed, if the leading coe cient is negative, the parabola opens downward. We now factor the epression to nd the coordinates of the intercepts. For the intercepts, we solve the equation =? so that = =? so that + + = We continue the computation from the second last line and factor via the di erence of squares theorem. ( ) = ( ) = ( + ) ( ) = ( + ) ( ) = =) = = Thus, there are two intercepts, ( ; ) and (; ). We will compute a few more points before graphing the parabola. We will work with coordinates close to that of the verte, and use the standard form of the parabola, = ( ) +. c Hidegkuti, Powell, Last revised: March,
2 Lecture Notes Graph of a Parabola - page if =, then = ( ) + = =) found ( ; ) if =, then = ( ) + = =) found ( ; ) if =, then = ( ) + = 7 =) found (; 7) if =, then = ( ) + = 7 =) found (; 7) if =, then = ( ) + = =) found (; ) if =, then = ( ) + = =) found (; ) if = 7, then = (7 ) + = =) found (7; ) We are read to graph: we have the following points, listed left to right: ( ; ) intercept ( ; ) ( ; ) intercept (; ) (; 7) verte (; ) (; 7) (; ) (; ) intercept (; ) (7; ) Eample. Graph the parabola = + 7. Clearl label the coordinates of ve points of the parabola, including verte and intercepts. Solution: = + 7 polnomial form =) intercept: (; 7) We factor out the leading coe cient = + ( + ) = + + = + {z + } = ( + ) = ( + ) 7 standard form =) verte: ( ; 7) = ( + ) = ( + + ) ( + ) = ( + 7) ( ) factored form =) intercepts ( 7; ) ; (; ) c Hidegkuti, Powell, Last revised: March,
3 Lecture Notes Graph of a Parabola - page We nd a few additional points, close to the verte. left to right: We are read to graph: we have the following points, listed ( ; ) intercept ( 7; ) ( ; ) ( ; ) ( ; ) ( ; 7) verte ( ; 7) intercept (; 7) (; ) (; ) (; ) (; ) intercept (; ) (; ) Eample. Graph the parabola = including verte and intercepts.. Clearl label the coordinates of ve points of the parabola, Solution: Since the leading coe cient is negative, the graph will be that of a downward opening parabola. = + polnomial form =) intercept: (; ) We factor out the leading coe cient, = + ( ) = + As we are getting read to complete the square, we notice that the epression in the parentheses is a complete square. This means that = ( ) is the standard form and the factored form all in one. For the verte, we can think of the equation as = ( ) + and so the verte is (; ). For the intercepts, we can think of the equation as = ( ) ( ) and so there is one intercept, (; ) that is also the verte. c Hidegkuti, Powell, Last revised: March,
4 Lecture Notes Graph of a Parabola - page We nd a few additional points, close to the verte and then graph the parabola. intercept (; ) 9 ; (; ) (; ) 9 ; (; ) 7; verte and intercept (; ) 9; (; ) (; ) ; Eample : Graph the parabola =. Clearl label the coordinates of ve points of the parabola, including verte and intercepts. Solution: Since the leading coe cient is negative, the graph will be a downward opening parabola. = =) intercept: (; ) = + + ( + ) = + + = + {z + } + = ( + ) = ( + ) + =) verte: ( ; ) = ( + ) = ( + ) ( + ) = ( + ) ( + ) =) intercepts ( ; ) ; ( ; ) We nd a few more points close to the verte and graph the parabola. ( ; 99) ( ; ) ( ; 7) intercept ( ; ) ( 9; ) ( ; ) ( 7; ) verte ( ; ) ( ; ) ( ; ) ( ; ) intercept ( ; ) ( ; 7) intercept (; ) c Hidegkuti, Powell, Last revised: March,
5 Lecture Notes Graph of a Parabola - page Eample. Graph the parabola = 9. Clearl label the coordinates of ve points of the parabola, including verte and intercepts. Solution: Since the leading coe cient is negative, the graph will be a downward opening parabola. = + 9 polnomial form =) intercept: (; 9) Because the linear term is missing, we do not need to complete the square, the polnomial form is also the standard form. If it helps, we can think of + 9 as ( ) + 9 to see that the verte is (; 9). = + 9 standard form =) verte: (; 9) For the factored form, we factor out the leading coe cient and factor via the di erence of squares theorem. = 9 = = ( + ) ( ) factored form =) intercepts ( ; ) ; (; ) We nd a few additional points close to the verte and then graph the parabola. ( ; ) ( ; 7) intercept ( ; ) ( ; ) ( ; ) verte and intercept (; 9) (; ) (; ) intercept (; ) (; 7) (; ) Eample. Graph the parabola = + 7. Clearl label the coordinates of ve points of the parabola, including verte and intercepts. Solution: Since the leading coe cient is positive, the graph will be an upward opening parabola. We rst factor out the leading coe cient and then complete the square to obtain the standard form. = + 7 =) intercept: (; 7) = + ( ) = + 9 = {z + 9 } 9 + = ( ) + = ( ) + =) verte: (; ) This parabola has no intercepts. This is because it is an upward opening parabola with its verte above the ais. Indeed, the epression ( ) + is alwas at least. Consequentl, this epression is never zero. Algebraicall, this means that the factored form can not eist, because it would certainl guarantee intercepts. Indeed, this parabola does not have a factored form of its equation because the sum of squares in the parentheses in ( ) + can not be factored. Recall that the sum of two squares can never be factored. c Hidegkuti, Powell, Last revised: March,
6 Lecture Notes Graph of a Parabola - page We nd a few more points close to the verte and graph the parabola. 9 7 ( ; ) intercept (; 7) (; ) (; ) verte (; ) (; ) (; ) (; ) Practice Problems Graph each of the parabolas given below. In each case, clearl label the coordinates of ve points of the parabola, including verte and intercepts..) =.) = 7.) = ( ).) = + 7.) =.) =.) =.) = + + c Hidegkuti, Powell, Last revised: March,
7 Lecture Notes Graph of a Parabola - page 7 Practice Problems - Answers.) = polnomial form: = + intercept: (; ) standard form: = ( + ) + verte: ( ; ) factored form: = ( + 7) ( ) intercepts: ( 7; ) and (; ) additional points: ( ; ), ( ; ), ( ; ), ( ; ), (; ), (; ), (; 7) ) = + 7 polnomial form: = + 7 intercept: (; 7) 9 7 standard form: = ( ) verte: (; ) factored form: = ( ) intercept: (; ) additional points: (; ), (; 7), (; ), (; ), (; ), (7; ), (; 7) ) = polnomial form: = + intercept: (; ) standard form: = ( ) verte: (; ) factored form: doesn t eist, no intercepts additional points: ( ; ), ( ; ), (; ), (; ), (; ), (; ), (; ) ) = polnomial form: = + intercept: (; ) standard form: = ( ) + 9 verte: (; 9) factored form: = ( ) intercepts: (; ) and (; ) - - additional points: ( ; 7), (; ), (; ), (; ), (; ), (7; 7), (; ) c Hidegkuti, Powell, Last revised: March,
8 Lecture Notes Graph of a Parabola - page.) = polnomial form: = intercept: (; ) standard form: = verte: (; ) factored form: = ( + ) ( ) intercepts: ( ; ) and (; ) additional points: ( ; ), ( ; ), ( ; 9), (; 9), (; ), (; ), (; ) ) = + + polnomial form: = + + intercept: (; ) standard form: = ( + ) + verte: ( ; ) factored form: doesn t eist, no intercepts additional points: 7;, ( ; ), ; ; ; ; ( ; ) ; -, ; 9 ; (; ) - 7.) = ( ) polnomial form: = + intercept: (; ) standard form: = ( ) verte: (; ) factored form: = ( ) intercept: (; ) - - additional points: ( ; 9), ( ; ), (; ), (; ), (; 9), (; ) ) = polnomial form: = intercept: (; ) standard form: = ( + ) + verte: ( ; ) factored form: = ( + ) ( + ) intercepts: ( ; ) and ( ; ) additional points: ( 7; ), ( ; ), ( ; ), ( ; ), (; ), (; ) For more documents like this, visit our page at and click on Lecture Notes. questions or comments to mhidegkuti@ccc.edu. c Hidegkuti, Powell, Last revised: March,
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