Quadratic Functions. Teachers Teaching with Technology. Scotland T 3. Symmetry of Graphs. Teachers Teaching with Technology (Scotland)

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1 Teachers Teaching with Technology (Scotland) Teachers Teaching with Technology T 3 Scotland Quadratic Functions Symmetry of Graphs Teachers Teaching with Technology (Scotland)

2 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 T 3 Scotland Quadratic Function: Symmetry of Graphs Page 1 of 9

3 QUADRATIC FUNCTION Calculator Set Up Use the followed by ZOOM ENTER key and come down to 4: as shown to set the initial window range. y = x y= x + y= x + 3 y = x What do you notice about the s and the number in the function? What do you think the s of the point where the graph cuts the y-axis will be? T 3 Scotland Quadratic Function: Symmetry of Graphs Page of 9 y = x 4

4 y= x y = x + y = x + 3 y= x Complete the following statements: 1. All the points on the y-axis have their x equal to. The y-axis is an axis of for all of the above graphs T 3 Scotland Quadratic Function: Symmetry of Graphs Page 3 of 9

5 Turning points The parabola with equation y = x + is shown here. Maximum Turning Point The curve has a maximum turning point at (0,) because the maximum Y value that the equation can have is +. Notice the maximum value occurs on the axis of symmetry. The parabola with equation y = x 3 is shown here. Point 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 Minimum Turning Note Every parabola has an axis of symmetry. For every parabola the axis of symmetry passes through the turning point. T 3 Scotland Quadratic Function: Symmetry of Graphs Page 4 of 9

6 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). y = ( x+ )( x+ 1) y = ( x + 3)( x 1) Multiply out the brackets Multiply out the brackets gives gives y = y = Cuts x-axis at (, ) and (, ) Cuts x-axis at (, ) and (, ) y= ( x+ )( x + 4) y = ( x 1)( x 3) Multiply out the brackets Multiply out the brackets gives gives y = y = Cuts x-axis at (, ) and (, ) Cuts x-axis at (, ) and (, ) y= ( x+ )( x 3) y= x( x 5) Multiply out the brackets Multiply out the brackets gives gives y = y = Cuts x-axis at (, ) and (, ) Cuts x-axis at (, ) and (, ) T 3 Scotland Quadratic Function: Symmetry of Graphs Page 5 of 9

7 Sketch the following graphs and rewrite as bracket equations. You will need to use zoom decimal and zoom standard y= x + 3x+ y= x + x 3 y = ( )( ) y = ( )( ) y= x 5x+ 6 y= x + 5x+ 4 y = ( )( ) y = ( )( ) y= x 4x+ 5 y= x 16 y = ( )( ) y = ( )( ) y= x x y = x + x 6 y = ( )( ) y = ( )( ) T 3 Scotland Quadratic Function: Symmetry of Graphs Page 6 of 9

8 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) y= x + x MAX MIN MAX MIN 6 y = x + x 8 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 (, ) MAX MIN MAX MIN c) d) y= x 4x y= x 4x+ 1 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 (, ) MAX MIN MAX MIN e) f) y= x + x y= x 5x+ 6 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 s of the points where the graph cuts the x-axis? T 3 Scotland Quadratic Function: Symmetry of Graphs Page 7 of 9

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 = e) x = f) x = y = y = y = y = y = y = What do you notice about these s? Given the equations below find the axis of symmetry and the Max / Min turning point. Use your Graphic Calculator to help you. y= x + 8x+ 1 y = x + 4x+ 3 y= x 7x+ 1 y= x + 3x+ 18 y= x + 13x 48 axis of symmetry x = y= x 4x axis of symmetry x = T 3 Scotland Quadratic Function: Symmetry of Graphs Page 8 of 9

10 Given the graphs below find the axis of symmetry and the Max / Min turning point. Use your Graphic Calculator to help you. y= x + x 6 Given the graphs below and the axis of symmetry state the other points that the graph cuts the x- axis. 1 Other point cutting x-axis =(, ) Axis of symmetry x= 55. Other point cutting x- axis =(, ) This graph has a minimum turning point at (,-5) Other point cutting x-axis =(, ) T 3 Scotland Quadratic Function: Symmetry of Graphs Page 9 of 9

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