TRANSFORMATIONS OF QUADRATIC FUNCTIONS Quadratic functions may be expressed in what is known as the Turning Point form: π¦ = π(π₯ β β)2 + π In this activity you will explore the effect that these variables a, h and k have on the graph of a quadratic function. Open the GeoGebra link: https://www.geogebra.org/m/mk3yXd7C Set the initial settings as follows: a = dilation in y-axis h = horizontal translation k = vertical translation Use the sliders to change the values. Only change one variable at a time. Leave the other two in these initial settings. DILATION AND REFLECTION β The effect of changing βaβ a Equation Sketch Shape compared to π = ππ Axis of symmetry equation π₯=0 Turning point (x, y) max / min (0, 0) min 1 π¦ = (π₯)2 Original basis of comparison 2 π¦ = 2(π₯)2 Stretch in y-axis Makes the graph skinnier For a given value of x, the y-value has increased by a factor of 2 π₯=0 (0, 0) min Compress in y-axis Makes the graph fatter For a given value of x, the y-value has decreased by a factor of 0.8 π₯=0 (0, 0) min Reflection across y = 0 (x-axis) π₯=0 (0, 0) max (x, y)β(x, 2y) 0.8 2 π¦ = 0.8(π₯) (x, y)β(x, 0.8y) -1 2 π¦ = β1(π₯) (x, y)β(x, -y) QUESTION 1 (ANSWER IN EXERCISE BOOK) Use the above information to describe what happens to the graph of π¦ = (π₯)2 (in terms of axis of symmetry and turning point) when: a>1 0<a<1 a<0 TRANSLATION β The effect of changing βkβ k Equation Sketch Shape compared to π = ππ 0 π¦ = (π₯)2 Original basis of comparison 1 π¦ = (π₯)2 + 1 Vertical translation, 1 unit up the y-axis For a given value of x, the yvalue has increase by 1 unit Axis of symmetry equation π₯=0 Turning point (x, y) max / min (0, 0) min π₯=0 (0, 1) min π₯=0 (0, 3) min (x, y)β(x, y+1) 3 π¦ = (π₯)2 + 3 Vertical translation, 1 unit up the y-axis For a given value of x, the yvalue has increase by 3 units (x, y)β(x, y+3) -1 π¦ = (π₯)2 β 1 Vertical translation, 1 unit down the y-axis For a given value of x, the yvalue has decrease by 1 unit π₯=0 (0, -1) min -2 π¦ = (π₯)2 β 2 Vertical translation, 2 units down the y-axis For a given value of x, the yvalue has decrease by 2 units π₯=0 (0, -2) min (x, y)β(x, y-1) (x, y)β(x, y-2) QUESTION 2 (ANSWER IN EXERCISE BOOK) Use the above information to describe what happens to the graph of π¦ = (π₯)2 (in terms of axis of symmetry and turning point) when: k>0 k<0 TRANSLATION β The effect of changing βhβ h Equation Sketch Shape compared to π = ππ Axis of symmetry equation π₯=0 Turning point (x, y) max / min (0, 0) min 0 π¦ = (π₯)2 Original basis of comparison 1 π¦ = (π₯ β 1)2 Horizontal translation, 1 unit to the right in the xaxis π₯=1 (1, 0) min (x, y)β(x+1, y) 3 π¦ = (π₯ β 3)2 + 3 Horizontal translation, 3 units to the right in the xaxis π₯=3 (3, 0) min -1 π¦ = (π₯ + 1)2 β 1 Horizontal translation, 1 unit left in the x-axis π₯ = β1 (-1, 0) min Horizontal translation, 2 units left in the x-axis π₯ = β2 (-2, 0) min (x, y)β(x+3, y) (x, y)β(x-1, y) -2 π¦ = (π₯ + 2)2 β 2 (x, y)β(x-2, y) QUESTION 3 (ANSWER IN EXERCISE BOOK) Use the above information to describe what happens to the graph of π¦ = (π₯)2 (in terms of axis of symmetry and turning point) when: h>0 h<0 SUMMARY The transformations of a quadratic are easily seen when it is expressed in turning point form π¦ = π(π₯ β β)2 + π Where: a = dilation in y-axis h = horizontal translation k = vertical translation Key features of a graph include: ο· The coordinate of the turning point, (h, k) and whether it is a maximum or a minimum ο· The y-intercept (0, y) β a quadratic will only have one y-intercept ο· The x-intercepts (x, 0) β a quadratic may have 0, 1, or 2 x-intercepts. ο· Axis of symmetry PRACTICE Use the GeoGebra to complete the following table. When copying the graph into the grid, use a screen shot and then adjust the picture size in the PICTURE TOOLS > FORMAT tab to be 2.9 cm wide. Donβt worry about the height, just set the width and the height will automatically adjust. >> Equation π¦ = 0.5(π₯ β 3)2 β 2 π¦ = β2(π₯ + 1.5)2 + 1.2 π¦ = βπ₯ 2 + 3 π¦= 1 2 π₯ +2 2 Sketch a h k 2 -1 1.5 -0.3 0 4 -1 -4 0 Axis of symmetry equation Turning Point (x, y) max/min x-int (when y = 0) y-int (when x = 0)