1    Chapter 6   •Linear•Quadratic     • L(x) = ax + bwhat happens if…b>0b<0b=0a>0a<0-1<a<1 (a≠0)a = 0a is undefined• Q(x) = ax2+ bx + cwhat happens if…c>0c<0c=0a>0a<0-1<a<1 (a≠0)a = 0a is undefined2 NegativePositiveHorizontal Vertical • Slope Intercept• Standard• Horizontal• Verticaly = mx + bAx + By = Cy = bx = aThis brings us back to the concept: a linear equation has a degree of 1   ! " " #$%&' (• The Equation Form• Direction• Slope• y-intercept• x-intercept• Parallel Slope• Perpendicular Slope1. Slope intercept2. Falling3. -3 4. -75. - -7/(-3) = -7/36. -3 7. -7identify…3    ))  • L(x) = 4x + 7• if you need to “find the zeroes”, x –intercept, solve for 0, etc…• Change the format• L(x) = 4(x + 7/4)• x would have to equal -7/4 for L(x) = 0     • L(x) = ax + bwhat happens if…b>0b<0b=0a>0a<0-1<a<1 (a≠0)a = 0a is undefined• Q(x) = ax2+ bx + cwhat happens if…c>0c<0c=0a>0a<0-1<a<1 (a≠0)a = 0a is undefined")*• Q(x) = ax2+ bx + c• Factored form• Q(x) = a(x - p)(x - q) where p<q• Now what happens?Note: Page 99 –Interval notation4+&,#+& $ ,+& $ ,-.• (-, p )• x = p• (p,q)• x = q• (q, )Note: Page 99 –Interval notation+&,#+& $ ,+& $ ,-.• (-, p )• x = p• (p,q)• x = q• (q, )• Q(x) = 0• Q(x) = 0Note: Page 99 –Interval notation+&,#+& $ ,+& $ ,-.• (-, p )• x = p• (p,q)• x = q• (q, )• Q(x) > 0 pos• Q(x) = 0• Q(x) < 0 neg• Q(x) = 0• Q(x) > 0 posNote: Page 99 –Interval notation5• (-, p )• x = p• (p,q)• x = q• (q, )• Q(x) > 0 pos• Q(x) = 0• Q(x) < 0 neg• Q(x) = 0• Q(x) > 0 posNote: Page 99 –Interval notationZero at p Zero at qPositive PositiveNegativeMake some connections with prior knowledge#+/0%,&121&$36    • The degree is 2.• y = x2• y = ax2+bx+c• y = a(x-h)2+ k• y = a(x-p)(x-q)    • y = x2• y = ax2+bx+c• y = a(x-h)2+ k• y = a(x-p)(x-q)• the basic• Standard Form• Vertex Form• Intercept FormGiven any one of the equations, you should be able to convert it into another form. (except for maybe the basic)4 "#&1• vertex = (0,0)• line of symmetryx = 0939-3424-2111-100yx27 #&12"&2• c y-intercept• -b/2a x-coordinate of the vertexuse substitution to find the y-coordinate• x = -b/2a is the line of symmetry.• Now you can graph it#5&12/.&21• The y-intercept is 2#5&12/.&21• The y-intercept is 2• -10/(2*8) = -5/88#5&12/.&21• The y-intercept is 2• -10/(2*8) = -5/8• y =8(-5/8)2+10(-5/8)+2=-1.125 or -9/8• Vertex (-5/8, -9/8)#5&12/.&21• The y-intercept is 2• -10/(2*8) = -5/8• y =8(-5/8)2+10(-5/8)+2=-1.125 or -9/8• Vertex (-5/8, -9/8)• so with the line of symmetry we can find the third point• (-5/8+-5/8 , 2) = (-10/8,2)#5&12/.&21• Connect the plotted points9• y = (8x+2)(x+1)• y = 2(4x+1)(x+1)This is the intercept form.Equate the quantities to 0 to find the intercepts4x+1 = 0 or x+1 = 0x = -1/4 or x = -1as shown in the graph.6#5&12/.&216   #+&$,+&$,• p & q x-intercepts• a opens up/down wide/skinny• can expand to find the vertex and line of symmetry7)  )) 89:;;• What does “a” denote?• What does “c” denote?• How do these coefficients effect the graph?10) +&$  ,• the quadratic formula will work every time• This should be familiar – now match it to the text.aacbbx242−±−= $ )0 )  aacbbx242−±−=aacbabx2422−+−=<  aacbbx242−±−=aacbabx2422−+−=  11<  aacbbx242−±−=aacbabx2422−+−===aacbabx2422−+−=abxu2+=aacbm2)4(2−==aacbabx2422−+−=abxu2+=aacbm2)4(2−=2244aacbM−=to see proof go to section 1.512 "  *• Q(x) = a ( u2– M)abxu2+=2244aacbM−=4"  • if M < 0 No real roots –if a > 0 Minimum–if a < 0 Maximumabxu2+=2244aacbM−=4"  • if M = 0 or a double rootsame idea regarding max/min–if a > 0 Minimum–if a < 0 Maximumabxu2+=2244aacbM−=134"  • if M > 0 or 2 real