Math 1050 Lab Activity: Graphing Transformations

Transcription

1 Math 00 Lab Activit: Graphing Transformations Name: We'll focus on quadratic functions to eplore graphing transformations. A quadratic function is a second degree polnomial function. There are two common forms of the equation: standard form: f() = a + b + c (h, k) form: f() = a( - h) + k This lab will focus on the (h, k) form and how to use this form to easil graph the function. PART I: Graphing Basics The graph of a quadratic function will alwas be a parabola with this basic shape verte verte A ke point of interest is the verte, which is the lowest or minimum point on the graph of a parabola that opens up or the highest or maimum point on the graph of a parabola that opens down. Another important propert of these graphs is that the are smmetric about the vertical line that passes through the verte so that each side is a mirror image of the other. This vertical line is called the ais of smmetr. Eample: (-, -) - - In this graph the verte is the point (-, -). The equation of the ais of smmetr is = -. Note: the ais of smmetr is a line. It can onl be correctl described as an equation.

2 For each graph shown below, identif the verte and the equation of the ais of smmetr verte equation of the ais of smmetr verte equation of the ais of smmetr

3 PART II: Graphing f() = The easiest parabola to graph, and the one to which we will compare others, is the function f() = or = which ou will graph b plotting points.. State the domain of this function.. Find at least five eact (, ) ordered pairs that are on the graph and sketch the graph of the function What point is the verte of this graph? What is the equation of the ais of smmetr for this graph?

4 PART III: Graphing Transformations Now ou will eperiment with the graph in the form of f() = a( - h) + k. Since graphing b hand is slow and tedious, we'll use software: or use a graphing calculator. Note that our correct placement of parentheses is important in this process. To raise an epression to a power tpe ^ which is the shift of the 6 ke. For eample to enter ( + ), tpe ( + )^. First click on "graphing calculator" and in equation, graph = b tping =^. Now click on "graph" at the bottom and ou will see our parabola graphed in red. Leave the graph of = in equation for the whole eercise so that ou ma compare all the other graphs to this basic function. Graphing = + k - Vertical Shifts. In equation, graph = +. Click on the graph button and ou will see our original = in red and our new graph = + on the same aes graphed in blue. What happened? Did adding cause the verte to move to a different location? If so where is the verte of = +?. Click on the equation button and in equation, tpe = -. See if ou and our lab budd can predict what this graph will look like before clicking on the graph button. Were ou correct? What is the verte of the graph = -?. Sketch both equations, = + and = - on the plane below. Label which graph is which and label the verte on each graph Tr graphing = + k with various values for k until ou are confident that ou know what the graph will look like.

5 Generalize the concept: (answer with brief phrases) How does the graph of = + k compare to the basic graph of =? What happens when k is a positive number? What happens when k is a negative number? In the equation = + k, where is the verte? Graphing = ( - h) - Horizontal Shifts. Delete all of our equations ecept for = in equation. In equation, graph = ( - ). Click on the graph button and ou will see our original = in red and our new graph = ( - ) on the same aes graphed in blue. Be sure our parenthesis are correct! You will tpe ( - )^ here. What happened? Did replacing with - cause the verte to move? Where is the verte of the graph = ( - )?. Click on the equation button and in equation, tpe = ( + ). What is the verte of the graph = ( + )?

6 6. Sketch both equations, = ( - ) and = ( + ) on the plane below. Label which graph is which and label the verte on each graph Tr graphing = ( - h) with various values for h until ou are confident that ou know what the graph will look like. Generalize the concept: (answer with brief phrases) How does the graph of = ( - h) compare to the basic graph of =? Consider what happens when h is positive and when h is negative. In the equation = ( - h), where is the verte? Do ou understand wh mathematicians put the minus sign in this form? Graphing = a - Stretches and Compressions 7. Delete all of our equations ecept for = in equation. In equation, graph =. Click on the graph button and ou will see our original = in red and our new graph = on the same aes graphed in blue. What happened? Did multipling b cause the verte to move to a different location? Did multipling b change the shape of the parabola?

7 8. Click on the equation button and in equation, tpe =. How does this graph compare with the previous two? 9. Click on the equation button and in equation, tpe = and in equation tpe = 8. Click on the graph button and view these graphs. What do ou see? 0. Sketch all of the equations, =, =, =, and = 8 on the plane below. Label which graph is which What happens if a is a negative number? Delete all of our equations ecept for = in equation. In equation, graph = -. Click on the graph button and ou will see our original = in red and our new graph = - on the same aes graphed in blue. What happened?. Click on the equation button and in equation, tpe = - and in equation, tpe = -. Did our graphs look the wa ou epected them to look?

8 . Sketch all of the equations, = -, = -, and = - on the plane below. Label which graph is which Tr graphing = a with various values for a until ou are confident that ou know what the graph will look like. Generalize the concept: (answer with brief phrases) How does the graph of = a compare to the basic graph of =? Consider what happens when a is positive and when a is negative. What happens when a >? What happens when 0 < a <?

9 PART IV: Putting it All Together, the graph of f() = a( - h) + k Graph and sketch each of the following function. State the verte and the equation of the ais of smmetr. Based on what ou have been learning, tr to predict what the graph will look like before ou click on the graph button.. f() = ( + ) - verte ais of smmetr g() = - ( - ) + verte ais of smmetr

10 . f() = 0. ( + ) + verte ais of smmetr f() = ( - ) verte ais of smmetr Eperiment b graphing f() = a( - h) + k using different values of a, h, and k until ou can consistentl predict what the graph will look like before ou click the graph button. In general: The verte is located at the point If a is positive the graph will open If a is negative the graph will open If a > the graph will appear than the graph of = If 0 < a < the graph will appear than the graph of =