Graphs of quadratics functions are parabolas opening up if a > 0, and down if a < 0. Examples:

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1 Quadratic Functions ( ) = a + b + c Graphs o quadratics unctions are parabolas opening up i a > 0, and down i a < 0. Eamples: = = + = = 0 MATH 80 Lecture B o 5 Ronald Brent 07 All rights reserved.

2 Notes: ) Quadratic unctions alwas have an ais o smmetr, that is a vertical line about which the graph is smmetric. This line is = b a ) The also have a point where the turn around. This point is called the verte and it occurs where a b =, and b = c a. ) The deinitel intersect the -ais at (0,c). The ma or ma not intersect the ais. This depends upon the discriminant, b a c. How to graph a quadratic unction ( ) = a + b + c ) Check the sign o a. I a > 0, the graph is curved up, or a < 0 it s curved down. ) b Determine the line o smmetr b computing = a ) Determine the verte using b = a. ) Find the -intercept. (Set = 0, and so = c.) 5) Determine -intercepts b solving a + b + c = 0. MATH 80 Lecture B o 5 Ronald Brent 07 All rights reserved.

3 Eample: Graph ( ) = +. b Here a =, b =, and c =, so = =. Since a > 0, the graph is a a parabola opening upwards. The ais o smmetr is =. The verte is when =, in which case =. So, it is the point (, ). The -intercept is (0, ). The -intercepts occur where + = 0, or =, and =. MATH 80 Lecture B o 5 Ronald Brent 07 All rights reserved. Now one can graph: (, 0) (0, ) (, ) (, 0)

4 Eample: Graph ( ) = + 6. b ( ) Here a =, b =, and c = 6, so = = a. Since a < 0, the graph is a ( ) parabola opening downwards. The verte is when =, in which case = 8. So, it is the point (, 8). The -intercept is (0, 6). The -intercepts occur where + 6 = 0, or =, and =. Now one can graph: (-, 8) 0 (0, 6) 5 (-, 0) (, 0) = - -5 MATH 80 Lecture B o 5 Ronald Brent 07 All rights reserved. -0

5 Scaling Functions Vertical Scaling: Suppose k > 0, then one ma obtain the graph o k () graph verticall o () () = b stretching the = i k >, or compressing the graph (verticall) o = i 0 < k <. I k < 0, then one ma obtain the graph o = k () b irst rotating the graph about the -ais, then b stretching the graph o = () verticall i k >, or compressing the graph o = () i 0 < k <. Eamples: 5 = ( ) = () = ( ) MATH 80 Lecture B 5 o 5 Ronald Brent 07 All rights reserved. -5

6 = = = 0 = 0 = = 0 MATH 80 Lecture B 6 o 5 Ronald Brent 07 All rights reserved.

7 Horizontal Scaling: Suppose k > 0, then one ma obtain the graph o = ( k ) b stretching the graph o = () horizontall i 0 < k <, or compressing the graph o = () i k >. I k < 0, then one ma obtain the graph o = ( k ) b irst rotating the graph about the -ais, then b stretching the graph o = () horizontall i 0 < k <, or b compressing the graph o = () i k >. Eample: = () = ( ) 0 0 MATH 80 Lecture B 7 o 5 Ronald Brent 07 All rights reserved.

8 = () = ( / ) 0 0 = () = ( ) 0 0 MATH 80 Lecture B 8 o 5 Ronald Brent 07 All rights reserved.

9 Shiting Functions Vertical Shits: Suppose a > 0, then one ma obtain the graph o = ( ) + a b shiting the graph o = () up a units. The graph o = ( ) a is obtained rom the graph o = () b shiting it down a units. Eamples: = ( ) + = () 0 = ( ) MATH 80 Lecture B 9 o 5 Ronald Brent 07 All rights reserved.

10 Horizontal Shits: Suppose a > 0, then one ma obtain the graph o = ( a ) b shiting the graph o = () right a units. The graph o = ( + a) is obtained rom the graph o = () b shiting it let a units. Eample: = ( +) = = ( ) 0 0 MATH 80 Lecture B 0 o 5 Ronald Brent 07 All rights reserved.

11 Putting It All Together: Eample: Given the unction (), shown in black below, graph the unction g ( ) = (( + )). ) First graph ( ) Vertical stretch o. ) Then graph ( ) Horizontal compression b. ) Then graph ( ( + ) ) Horizontal shit let units. ) Finall graph g ( ) = (( + )) Vertical shit down unit. () 0 MATH 80 Lecture B o 5 Ronald Brent 07 All rights reserved.

12 Even and Odd Functions: A unction is even i ( ) = ( ) or all in its domain. A unction is odd i ( ) = ( ) or all in its domain. Eample: ( ) = ( ) = = (. ( ) = is even since ) ( ) = = =. ( ) = is odd since ( ) ( ) Notes: ) Even unctions are smmetric about the -ais. ) Odd unctions are smmetric through the origin. ) An even unction times an even unction is also even. ) An even unction times an odd unction is ODD! 5) An odd unction times an odd unction is EVEN! 6) A unction can be neither even nor odd. Take or eample ( ) = + MATH 80 Lecture B o 5 Ronald Brent 07 All rights reserved.

13 Periodic Functions: A unction is periodic with period P i ( + P) = ( ) holds or all in the domain o. The smallest such number P is called the period o. Eamples: 5 () Period = g () Period = h () - - Period = -5 MATH 80 Lecture B o 5 Ronald Brent 07 All rights reserved.

14 Final Function and Graph Summar: Vertical Line Test: Since a unction must have one and onl one value associated with each element in its domain, i a an vertical line crosses a graph more than once, the graph cannot be that o a unction. Intercepts: An place that the graph o a unction intersects the -ais is called the -intercept. I the point (0, b) is on the graph o the unction one sas either (0, b) is the -intercept, or = b is the - intercept. This can happen at most once. An place that the graph o a unction intersects the -ais is called the -intercept. I the point (a, 0) is on the graph o the unction one sas either (a, 0) is the -intercept, or = a is the - intercept. This can occur an number o times. Roots An time the graph o the unction = () has an -intercept represents a place where ( ) = 0. This place is called a root o the unction. This is useul in graphing unctions. MATH 80 Lecture B o 5 Ronald Brent 07 All rights reserved.

15 Summar o Operations With Constants Stretching: Let k > 0. = k () causes vertical stretching or k >. = ( k ) causes horizontal stretching or k <. Compressing Let k > 0. = k () causes vertical compression or k <. = ( k ) causes horizontal compression or k >. Relection Let k < 0 = k () causes vertical relection ollowed b compression or k >. = ( k ) causes horizontal relection then compression or k <. = k () causes vertical relection ollowed b stretching or k <. = ( k ) causes horizontal relection ollowed b stretching or k >. Shiting Let a > 0. = ( ) + a causes vertical shit up a units. = ( ) a causes vertical shit down a units. = ( a) causes horizontal shit right a units. = ( + a) causes horizontal shit let a units. MATH 80 Lecture B 5 o 5 Ronald Brent 07 All rights reserved.