© Teachers Teaching with Technology (Scotland)

©Teachers Teaching with Technology (Scotland)

Teachers Teaching with Technology

T3 Scotland

Quadratic FunctionsSymmetry of Graphs

QUADRATIC FUNCTION

AimTo demonstrate how the TI-83 can be used to explore quadratic graphs and their axis of symmetry.

ObjectivesMathematical objectivesBy the end of this topic you should be able to

• Find the equation of the axis of symmetry of a quadratic graph• Use the axis of symmetry to calculate the maximum or minimum turning point

Calculator objectivesBy the end of this topic you should be able to

• Use the ZOOM key to graph quadratic functions• Use the DRAW menu to draw vertical lines

T3 Scotland Quadratic Function: Symmetry of Graphs Page 1 of 9

QUADRATIC FUNCTION

Calculator Set Up

Use the key and come down to 4: as shown followed by to set the initial window range.

Plot the function Graph cuts the y-axis atand sketch it on the blank screen shown coordinate

Plot the function Graph cuts the y-axis atand sketch it on the blank screen shown coordinate

Plot the function Graph cuts the y-axis atand sketch it on the blank screen shown coordinate

Plot the function Graph cuts the y-axis atand sketch it on the blank screen shown coordinate

What do you notice about the coordinates and the number in the function?______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

will be?______________________________________________________________________________

� T3 Scotland Quadratic Function: Symmetry of Graphs Page 2 of 9

ZOOM

ENTER

y x= 2

y x= + � �2 2

y x= + � �2 3

y x= − � �2 2

What do you think the coordinates of the point where the graph cuts the y-axis y x= − � �2 4

Plot the function Graph cuts the y-axis atand sketch it on the blank screen shown coordinate

Plot the function Graph cuts the y-axis atand sketch it on the blank screen shown coordinate

Plot the function Graph cuts the y-axis atand sketch it on the blank screen shown coordinate

Plot the function Graph cuts the y-axis atand sketch it on the blank screen shown coordinate

Complete the following statements:

1. All the points on the y-axis have their x coordinate equal to __________________________

2. The y-axis is an axis of __________________ for all of the above graphs

T3 Scotland Quadratic Function: Symmetry of Graphs Page 3 of 9

y x= − 2

y x= − +2 2

y x= − +2 3

y x= − −2 2

Turning points

The parabola with equation is shown here. Maximum Turning PointThe curve has a maximum turning point at (0,2) because the maximum Y value that the equation can have is +2.Notice the maximum value occurs on the axis of symmetry.

Minimum Turning

The parabola with equation is shown here. PointThe curve has a minimum turning point at (0,-3) because the minimum Y value that the equation can have is -3.Notice the minimum value occurs on the axis of symmetry

NoteEvery parabola has an axis of symmetry.For every parabola the axis of symmetry passes through the turning point.

� T3 Scotland Quadratic Function: Symmetry of Graphs Page 4 of 9

y x= − � � +�2 2

y x= − � �2 3

Calculator Set Up

Use the ZOOM key and come down to 6: as shown

followed by ENTER to set the initial window range.

Plot the following functions and sketch them on the screens shown.Use the table facility on the calculator if necessary (see Hint Sheet).

Multiply out the brackets Multiply out the bracketsgives gives

y = y =

Cuts x-axis at ( , ) and ( , ) Cuts x-axis at ( , ) and ( , )

Multiply out the brackets Multiply out the bracketsgives gives

y = y =

Cuts x-axis at ( , ) and ( , ) Cuts x-axis at ( , ) and ( , )

Multiply out the brackets Multiply out the bracketsgives gives

y = y =

Cuts x-axis at ( , ) and ( , ) Cuts x-axis at ( , ) and ( , )� T3 Scotland Quadratic Function: Symmetry of Graphs Page 5 of 9

y x x� � �( )( )2 1 y x x� � �( )( )3 1

y x x= + � �( )( )2 4 y x x �( )( )1 3

y x x= − � � � �( )( )2 3 y x x= − � �( )5

+

+

+ + + −

− −

−

= =

=

Sketch the following graphs and rewrite as “bracket” equations.You will need to use zoom decimal and zoom standard

y = ( )( ) y = ( )( )

y = ( )( ) y = ( )( )

y = ( )( ) y = ( )( )

y = ( )( ) y = ( )( )

T3 Scotland Quadratic Function: Symmetry of Graphs Page 6 of 9

y x x= + +2 3 2 y x x= + −2 2 3

y x x= − +2 5 6 y x x= + +2 5 4

y x x= − − +2 4 5 y x= −2 16

y x x= − −2 2 y x x= + −2 6

For each of the following state whether it has a maximum or minimum turning point (by ticking the box) and find the axis of symmetry. Sketch the axis of symmetry on the screens shown.

a) b)

c) d)

e) f)

What do you notice about the point where the axis of symmetry cuts the x-axis when you compare it to the x coordinates of the points where the graph cuts the x-axis? __________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

T3 Scotland Quadratic Function: Symmetry of Graphs Page 7 of 9

Cuts x-axis at ( , ),( , )

Axis of symmetry cuts x-axis at ( , )

Coordinates of max / min turning point ( , )

Cuts x-axis at ( , ),( , )

Axis of symmetry cuts x-axis at ( , )

Coordinates of max / min turning point ( , )

Cuts x-axis at ( , ),( , )

Axis of symmetry cuts x-axis at ( , )

Coordinates of max / min turning point ( , )

Cuts x-axis at ( , ),( , )

Axis of symmetry cuts x-axis at ( , )

Coordinates of max / min turning point ( , )

Cuts x-axis at ( , ),( , )

Axis of symmetry cuts x-axis at ( , )

Coordinates of max / min turning point ( , )

Cuts x-axis at ( , ),( , )

Axis of symmetry cuts X axis at ( , )

Coordinates of max / min turning point ( , )

y x x= + −2 6 y x x= + −2 2 8

y x x= −2 4

y x x= − +2 2

y x x= − − +2 4 12

y x x= − − +2 5 6

MAX MIN MAX MIN

MAX MIN MAX MIN

MAX MIN MAX MIN

Evaluate the functions used on the previous page at the point where the axis of symmetry cuts the x-axis.

a) x = b) x = c) x = d) x = e) x = f) x = y = y = y = y = y = y =

What do you notice about these coordinates?______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Given the equations below find the axis of symmetry and the Max / Min turning point.Use your Graphic Calculator to help you.

T3 Scotland Quadratic Function: Symmetry of Graphs Page 8 of 9

Axis of symmetry x =Coordinates of the Max / Min turning point ( , )

Axis of symmetry x =Coordinates of the Max / Min turning point ( , )

Axis of symmetry x =Coordinates of the Max / Min turning point ( , )

Axis of symmetry x =Coordinates of the Max / Min turning point ( , )

axis of symmetry x =Coordinates of the Max / Min turning point ( , )

axis of symmetry x =Coordinates of the Max / Min turning point ( , )

y x x= + +2 4 3

y x x= − +2 7 12

y x x= + −2 13 48

y x x= − + +2 3 18

y x x= − + +2 8 12

y x x= − −2 4

Given the graphs below find the axis of symmetry and the Max / Min turning point.Use your Graphic Calculator to help you.

Given the graphs below and the axis of symmetry state the other points that the graph cuts the x- axis.

T3 Scotland Quadratic Function: Symmetry of Graphs Page 9 of 9

y x x= + −2 6

This graph has a minimumturning point at (2,-25)

Axis of symmetry x =

Other point cutting x-axis =( , )

Axis of symmetry x =

Other point cutting x-axis =( , )

− 12

Axis of symmetry x=

Other point cutting x- axis =( , )

−55.