© Teachers Teaching with Technology (Scotland)
Teachers Teaching with Technology
T3 Scotland
Quadratic Functions
Symmetry of Graphs
©Teachers Teaching with Technology (Scotland)
QUADRATIC FUNCTION
Aim
To demonstrate how the TI-83 can be used to explore quadratic graphs and their axis of
symmetry.
Objectives
Mathematical objectives
By 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 objectives
By 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 ZOOM key and come down to 4: as shown
followed by ENTER to set the initial window range.
Plot the function y = x
and sketch it on the blank screen shown
Graph cuts the y-axis at
coordinate
Plot the function y = x + 2
and sketch it on the blank screen shown
Graph cuts the y-axis at
coordinate
Plot the function y = x + 3
and sketch it on the blank screen shown
Graph cuts the y-axis at
coordinate
Plot the function y = x − 2
and sketch it on the blank screen shown
Graph cuts the y-axis at
coordinate
2
2
2
2
What do you notice about the coordinates and the number in the function?
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
What do you think the coordinates of the point where the graph
y = x 2 − 4 cuts the y-axis
will be?
______________________________________________________________________________
T3 Scotland
Quadratic Function: Symmetry of Graphs
Page 2 of 9
Plot the function y = − x
and sketch it on the blank screen shown
Graph cuts the y-axis at
coordinate
Plot the function y = − x + 2
and sketch it on the blank screen shown
Graph cuts the y-axis at
coordinate
Plot the function y = − x + 3
and sketch it on the blank screen shown
Graph cuts the y-axis at
coordinate
Plot the function y = − x − 2
and sketch it on the blank screen shown
Graph cuts the y-axis at
coordinate
2
2
2
2
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
Turning points
The parabola with equation y = − x + 2 is shown here.
The 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.
2
Maximum Turning Point
Minimum Turning
The parabola with equation y = x − 3 is shown here.
The 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
2
Point
Note
Every 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
Calculator Set Up
Use the
ZOOM
followed by
key and come down to 6: as shown
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).
y = ( x + 3)( x − 1)
y = ( x + 2)( x + 1)
Multiply out the brackets
gives
Multiply out the brackets
gives
y=
y=
Cuts x-axis at (
,
) and (
,
)
Cuts x-axis at (
,
) and (
Multiply out the brackets
gives
Multiply out the brackets
gives
y=
y=
,
) and (
,
)
Cuts x-axis at (
y = − ( x + 2)( x − 3)
,
) and (
Multiply out the brackets
gives
y=
y=
T3 Scotland
,
,
)
,
)
y = x ( x − 5)
Multiply out the brackets
gives
Cuts x-axis at (
)
y = ( x − 1)( x − 3)
y = ( x + 2)( x + 4)
Cuts x-axis at (
,
) and (
,
)
Cuts x-axis at (
Quadratic Function: Symmetry of Graphs
,
) and (
Page 5 of 9
Sketch the following graphs and rewrite as “bracket” equations.
You will need to use zoom decimal and zoom standard
y = x2 + 2x − 3
y = x 2 + 3x + 2
y= (
)(
y = x 2 − 5x + 6
y= (
y= (
)
)(
y= (
)
T3 Scotland
)(
)
)(
)
)(
)
y = x 2 − 16
)(
y= (
)
y = x2 + x − 6
y = − x2 − 2x
y= (
)
y = x 2 + 5x + 4
y = − x2 − 4 x + 5
y= (
)(
)(
)
Quadratic Function: Symmetry of Graphs
y= (
Page 6 of 9
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.
MAX
MIN
MAX
a)
MIN
b)
y = x2 + 2 x − 8
y = x2 + x − 6
Cuts x-axis at (
,
),(
,
)
Cuts x-axis at (
Axis of symmetry cuts x-axis at (
,
)
,
,
)
,
)
)Coordinates of max / min turning point (
MIN
MAX
c)
),(
Axis of symmetry cuts x-axis at (
Coordinates of max / min turning point (
MAX
,
,
)
,
)
,
)
MIN
d)
y = x2 − 4 x
Cuts x-axis at (
y = − x 2 − 4 x + 12
,
),(
,
Cuts x-axis at (
)
Axis of symmetry cuts x-axis at (
,
,
,
)
,
)
) Coordinates of max / min turning point (
MIN
MAX
e)
),(
Axis of symmetry cuts x-axis at (
)
Coordinates of max / min turning point (
MAX
,
MIN
f)
y = − x2 + 2x
Cuts x-axis at (
y = − x 2 − 5x + 6
,
),(
,
Cuts x-axis at (
)
Axis of symmetry cuts x-axis at (
,
),(
,
)
Axis of symmetry cuts X axis at (
)
Coordinates of max / min turning point (
,
,
, )
) Coordinates of max / min turning point (
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
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 =
y=
y=
y=
y=
e) x =
f) x =
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.
y = x2 + 4x + 3
y = − x 2 + 8 x + 12
Axis of symmetry x =
Coordinates of the
Max / Min turning point (
,
)
,
)
,
)
,
)
y = − x 2 + 3x + 18
y = x 2 − 7 x + 12
Axis of symmetry x =
Coordinates of the
Max / Min turning point (
,
)
Axis of symmetry x =
Coordinates of the
Max / Min turning point (
y = − x2 − 4 x
y = x 2 + 13x − 48
axis of symmetry x =
Coordinates of the
Max / Min turning point (
T3 Scotland
Axis of symmetry x =
Coordinates of the
Max / Min turning point (
,
)
axis of symmetry x =
Coordinates of the
Max / Min turning point (
Quadratic Function: Symmetry of Graphs
Page 8 of 9
Given the graphs below find the axis of symmetry and the Max / Min turning point.
Use your Graphic Calculator to help you.
y = x2 + x − 6
Given the graphs below and the axis of symmetry state the other points that the graph cuts the xaxis.
Axis of symmetry x = −
1
2
Other point cutting x-axis =(
− 55
.
Axis of symmetry x=
,
)
Other point cutting x- axis =(
This graph has a minimum
turning point at (2,-25)
Axis of symmetry x =
Other point cutting x-axis =(
T3 Scotland
Quadratic Function: Symmetry of Graphs
,
)
Page 9 of 9
,
)