Quadratic Polynomials - University of Utah

1 Quadratic PolynomialsIfa>0thenthegraphofax2is obtained by starting with the graph ofx2,and then stretching or shrinking vertically <0thenthegraphofax2is obtained by starting with the graph ofx2,then flipping it over thex-axis, and then stretching or shrinking vertically bythe positive number >0wesaythatthegraphofax2 opens up . Whena<0wesaythat the graph ofax2 opens down .**148 ICiti-ax-ax~S~12 s-aXiS ,c,d2 Randa6= 0, then the graph ofa(x+c)2+dis obtained byshifting the graph ofax2horizontally byc, and vertically byd. (Rememberthatd>0meansmovingup,d<0meansmovingdown,c>0meansmovingleft,andc<0meansmovingright. )Ifa6=0,thegraphofafunctionf(x)=a(x+c)2+ dis called point ( c,d)2R2is called thevertexof the is the parabola that is the graph of 10(x+2)2 3. Itsvertex is ( 2, 3).

2 **149 Ifa,c,d Randa = 0, then the graph ofa(x+c)2+dis obtained byshifting the graph ofax2horizontally byc, and vertically byd. (Rememberthatd>0meansmovingup,d<0meansmovingdown,c>0meansmovingleft,andc<0meansmovingright.)Ifa =0,thegraphofafunctionf(x)=a(x+c)2+dis called point ( c,d) R2is called thevertexof the is the parabola that is the graph of 10(x+2)2 3. Itsvertex is ( 2, 3).**2 Ifa,c,d Randa = 0, then the graph ofa(x+c)2+dis obtained byshifting the graph ofax2horizontally byc, and vertically byd. (Rememberthatd>0meansmovingup,d<0meansmovingdown,c>0meansmovingleft,andc<0meansmovingright. )Ifa =0,thegraphofafunctionf(x)=a(x+c)2+dis called point ( c,d) R2is called thevertexof the is the parabola that is the graph of 10(x+2)2 3. Itsvertex is ( 2, 3).**2 Aquadratic polynomialis a degree 2 polynomial.

3 In other words, a qua-dratic polynomial is any polynomial of the formp(x)=ax2+bx+cwherea,b,c2 Randa6= the squareYou should memorize this equation: (it s calledcompleting the square .)ax2+bx+c=a x+b2a 2+c b24aLet s check that the equation is true:a x+b2a 2+c b24a=a x2+2xb2a+hb2ai2 +c b24a=ax2+a2xb2a+ahb2ai2+c b24a=ax2+bx+ahb24a2i b24a+c=ax2+bx+b24a b24a+c=ax2+bx+cGraphing quadraticsComplete the square to graph Quadratic Polynomials : Ifp(x)=ax2+bx+c,thenp(x)=a x+b2a 2+c b24a. Therefore, the graph ofp(x)=ax2+bx+cis a parabola obtained by shifting the graph ofax2horizontally byb2a,andvertically byc b24a. The parabola opens up ifa>0andopensdownifa< graph 3x2+5x 2, complete the square to find that 3x2+5x 2equals 3(x 56)2+112. To graph this polynomial, we startwith the parabola for 3x2, which opens down since the parabola for 3x2right by56and then up by112.

4 The result is thegraph for 3x2+5x 2. Notice that the graph looks like the graph of 3x2,except that its vertex is the point (56,112).** the parabola for 3x2right by56and then up by112. The result is thegraph for 3x2+5x 2. Notice that the graph looks like the graph of 3x2,except that its vertex is the point (56,112).** the parabola for 3x2right by56and then up by112. The result is thegraph for 3x2+5x 2. Notice that the graph looks like the graph of 3x2,except that its vertex is the point (56,112).**4 DiscriminantThediscriminantofax2+bx+cis defined to be the numberb2 many roots?Ifp(x)=ax2+bx+c, then the following chart shows how the discriminantofp(x)determineshowmanyroots p(x)has:b2 4acnumber of roots>02=01< (x)= 2x2+3x 1. Because 32 4( 2)( 1) =9 8=1ispositive,p(x)= 2x2+3x the discriminant of a Quadratic polynomial tells us about the numberof roots of the polynomial, and why the information from the above chart istrue, will be explained in lectures.

5 **152 Finding rootsIfax2+bx+chas at least one root which is the same as saying thatb2 4ac 0 formulaIfp(x)=ax2+bx+cwitha6=0andifb2 4ac 0,then the roots ofp(x)are b+pb2 4ac2aand b pb2 4ac2aNotice in the Quadratic formula, that we needa6=0tomakesurethatweare not dividing by 0, and we needb2 4ac 0tomakesurethatwearen ttaking the square root of a negative recall that ifax2+bx+chas only one root , thenb2 4ac=0. Thatmeans the two roots from the Quadratic formula are really the same s a good exercise in algebra to check that the Quadratic equation is check that it s true, you need to check thatp b+pb2 4ac2a =0and thatp b pb2 4ac2a = checked above thatp(x)= 2x2+3x 1has2roots,becauseits discriminant equalled 1. The Quadratic formula tells us that those rootsequal 3+p12( 2)= 3+1 4= 2 4=12and 3 p12( 2)= 4 4=1153 ExercisesFor each of the Quadratic Polynomials in problems #1-6: Complete the means rewriteax2+bx+casa x+b2a 2+c b24a.

6 What s the vertex of the corresponding parabola?The vertex of the parabola fora(x+c)2+dis the point ( c,d). Is its parabola opening up, or opening down?The parabola opens up if the leading coe cient of the quadraticpolynomial is positive. The parabola opens down if the leadingcoe cient is negative. What s its discriminant?The discriminant ofax2+bx+cisb2 4ac. How many roots does it have?There are two roots if the discriminant is positive, one root if thediscriminant equals 0, and zero roots if the discriminant is negative. What are its roots (if it has any)?Ifax2+bx+chas roots, they are b+pb2 4ac2aand b pb2 4ac2a. Match its graph with one of the lettered graphs on the next ) 2x2 2x+122.)x2+2x+13.) 3x2 9x+64.) 4x2+16x 195.)x2+2x 16.) 3x2+6x+5154A.)B.)C.)D.)E.

7 F.)155az~j)1)3) I2:3 ,(z,.-3)s)1+ 3 C)(-1,2)s )I) I z-z)s )I) I z-z)s )I) I z-z)s )I) I z-z)s )I) I z-z)s )I) I z-z)az~j)1)3) I2:3 ,(z,.-3)s)1+ 3 C)(-1,2)I)- 4az~j)1)3) I2:3 ,(z,.-3)s)1+ 3 C)(-1,2)s )I) I z-z)s )I) I z-z)s )I) I z-z)s )I) I z-z)s )I) I z-z)s )I) I z-z)az~j)1)3) I2:3 ,(z,.-3)s)1+ 3 C)(-1,2)I)- 4az~j)1)3) I2:3 ,(z,.-3)s)1+ 3 C)(-1,2)s )I) I z-z)s )I) I z-z)s )I) I z-z)s )I) I z-z)s )I) I z-z)s )I) I z-z)az~j)1)3) I2:3 ,(z,.-3)s)1+ 3 C)(-1,2)I)- 4az~j)1)3) I2:3 ,(z,.-3)s)1+ 3 C)(-1,2)s )I) I z-z)s )I) I z-z)s )I) I z-z)s )I) I z-z)s )I) I z-z)s )I) I z-z)az~j)1)3) I2:3 ,(z,.-3)s)1+ 3 C)(-1,2)I)- 4az~j)1)3) I2:3 ,(z,.-3)s)1+ 3 C)(-1,2)s )I) I z-z)s )I) I z-z)s )I) I z-z)s )I) I z-z)s )I) I z-z)s )I) I z-z)az~j)1)3) I2:3 ,(z,.-3)s)1+ 3 C)(-1,2)I)- 4az~j)1)3) I2:3 ,(z.

8 -3)s)1+ 3 C)(-1,2)s )I) I z-z)s )I) I z-z)s )I) I z-z)s )I) I z-z)s )I) I z-z)s )I) I z-z)az~j)1)3) I2:3 ,(z,.-3)s)1+ 3 C)(-1,2)I)- 4az~j)1)3) I2:3 ,(z,.-3)s)1+ 3 C)(-1,2)s )I) I z-z)s )I) I z-z)s )I) I z-z)s )I) I z-z)s )I) I z-z)s )I) I z-z)az~j)1)3) I2:3 ,(z,.-3)s)1+ 3 C)(-1,2)I)- 4az~j)1)3) I2:3 ,(z,.-3)s)1+ 3 C)(-1,2)s )I) I z-z)s )I) I z-z)s )I) I z-z)s )I) I z-z)s )I) I z-z)s )I) I z-z)az~j)1)3) I2:3 ,(z,.-3)s)1+ 3 C)(-1,2)I)- 47.) Suppose you shoot a feather straight up into the air, and thattis thetime measured in seconds that follow after you shoot the feather into the the height of the feather at timetis given by 2t2+20tfeet, then whatis the maximum height that the feather reaches? How many seconds does ittake for the feather to reach its maximum height?

9 8.) Let s say you make cogs for a living. After accounting for the cost ofbuilding materials, you earn a profit ofx2 10x+45centsonthex-th cogthat you make. Which cog do you earn the least amount of profit for making?How much profit do you earn for that cog?Solve forxin the equations given in # )x2+x+12x+1=110.)x2+3x2+2x+5=111.) 3x 5x2= 2156