Download Transforming Mathematics with GSP 4: Constructing and Analyzing Functions - Prof. Olive and more Assignments History of Education in PDF only on Docsity!
Chapter 7: Functions
As we saw in Chapter 6, dynamic dilations can be used to construct products and quotients of distances. This capability gives us a way of constructing algebraic relations geometrically with Sketchpad_._ Environments similar to Goldenberg’s Dynagraph (Goldenberg et al., 1992) can be constructed quite simply. The Dynagraph consists of two parallel number lines. The user controls the position of a variable point on one number line (the input variable “x”) and movement of this point causes movement of its image point (y) on the other number line according to a defined functional relation, y=f(x). With Sketchpad that functional relation can be constructed geometrically as well as typed in as an algebraic expression. Constructing algebraic relations geometrically can provide a powerful link between these two branches of mathematics and enhance learning through the dynamic exploration of functions. Dynamic Function Representation on a Number Line In the following section we shall be using our number line as a dynagraph to explore linear and quadratic functions. The input variable will always be a free point on the number line (labeled x) and the output of the function will be a point constructed from this free point using geometric transformations. [Note: To construct a horizontal number line, define a coordinate system and then hide the grid and the vertical axis. Label the origin 0 and the unit point 1. ] Activity 7.1: Constructing a Linear Function In order to construct a linear function on the number line you will need to create a free point on the number line for your “x” variable. Label this point x and find its x - coordinate. The next step is to create a segment that will represent the parameter a (or multiplier) of x in the function f(x)=ax. This could be done on the number line (as with the product A*B in chapter 6) but things start to get crowded and confusing if everything is on the same line. One solution is to create separate segments for each parameter in a function on hidden lines that are parallel to the number line. The following steps demonstrate how to create a segment for parameter a and construct the point ax on the number line:
- Create a free point somewhere below your number line. Label this A. [Note: You could place A on the vertical axis below the origin and then re-hide the vertical axis.]
- Construct a line through this point parallel to your number line.
- Mark the points 0 and 1 on your number line as a vector.
- Translate your new point A by the marked vector. This will create a unit point on your new line. Label it u.
- Place a free point on your new line and label it a.
- Hide your new line and construct the segment Aa.
- Mark the DIRECTED RATIO Aa/Au. You will need to select your three points A, u and a IN THAT ORDER! then choose Mark Ratio under the Transform menu. [Note: Sketchpad always uses the following order when using 3 points to define a directed ratio: Common point, denominator point, numerator point.] 8. Mark the origin ( 0 ) of your number line as the center for dilation and dilate your point x by the marked ratio. Label the dilated image point ax. 9. Measure the abscissa (x-coordinate) of points x, A, a and ax. 10. Calculate the difference of the abscissas of points a and A (xa – xA) and label this measure “a”. Hide the abscissa measures xa and xA and edit the labels for xx and xax as in Figure 7.1. -10 -5 5 10 x = -2.01 (^) ax = -3. a = 1. ax u x^0 A (^) a Figure 7.1: Constructing the Function f(x)=ax. Experiment by sliding your point a back and forth. What happens to x? What happens to ax? Move your variable point x. What happens to ax? Does your point a change when you vary x? Assignment 7. Construct a point on the number line representing the linear function f(x) = ax+b. Where b is represented by another segment constructed similarly to the construction of Aa. What kind of transformation of ax could represent adding a directed measure, b? Figure 7. illustrates one possible representation.
Constructing Powers of x The problem of constructing a point that would be a distance x^2 from the origin is a simple dilation of the point x by the directed ratio 0x/01. Create such a point on your number line. Use the default label x ’ for this new point as Sketchpad has no superscript capability for its labels. Move your x point through the origin. What happens to x ’? Does it behave as you would expect the point x^2 to behave? Now construct a point that will be a distance x^3 from the origin (use the same dilation on the point x ’). Use the default label x ’’ for this point. Move your x point through the origin. What happens to x ’’? Does it behave as you would expect the point x^3 to behave? The above process can be continued to construct any power of x. Activity 7. Construct a point on your number line that can be represented by the quadratic expression ax^2 +bx+c where the parameters a , b , and c are defined by dynamic line segments (see Figure 7.4). -10 -5 5 10 c = 4. b = -1. x = 2. a = 1. bx ax'+bx ax'^ ax'+bx+c u x' u u 0 1 x A (^) a b B C c Figure 7.4. Constructing the Quadratic Expression ax’+bx+c. Move your x point and try and predict the relative motions of each of the components of the quadratic expression ( ax’, bx, ax’+bx, and ax’+bx+c ). What happens to the different components as x passes through the unit point 1? What happens when it passes through 0? Can you find values (lengths) for b and c such that the point ax’+bx+c passes to the left of the origin (keeping a positive)?
Comparing Functions on Parallel Number Lines Crowding all of the above points on one number line can become confusing. Keeping track of the relative motions becomes troublesome. Creating parallel number lines for each function or components of a function can help simplify the picture. One quick and easy way to create a parallel number line that maintains all of the functional relations is to translate your existing number line vertically.
- Create a free point somewhere above the origin point. Label this point 0’. [Note: You can show the hidden vertical axis and place 0’ on the vertical axis.]
- Mark the vector 00’.
- Select your number line and all the points on it.
- Translate your number line by Marked Vector.
- Translate this new number line by the same vector.
- Repeat step 5 until you have one number line for each point.
- Label one point on each number line and hide all of the other points on the line except your labeled point, the origin and unit point.
- For each of your new number lines do the following in order to obtain a numbered axis: a. Select the origin and unit point and choose construct circle from the Construct menu. b. With this circle still selected, choose Define Unit Circle from the Graph menu. A dialog box will appear asking if you really want to create a new coordinate system. Click the Yes button. c. Choose Hide Grid from the Graph menu. d. Hide the vertical axis and the unit circle. The above steps should leave you with something like Figure 7.5. Move the point x on the bottom number line and observe the relative motions of each of the points on the other number lines. Move x through the origin. Write down any conjectures you may have.
Roots (or zeros) of quadratic functions can also be investigated on your parallel number lines. Simply move your x point and observe if or when the ax’+bx+c point passes through the origin for various values of a, b and c. The position(s) of your x point will be the zeros (or roots) of the quadratic function. Assignment 7. Explore the roots for various values of a, b and c. Find values for which there are no roots, only one root, or two roots. Could there ever be more than two zeros for a quadratic function? Fix a and b and adjust c to create a function with just one root. Why does this work? Try adjusting a, b and c to find a function with a specific root. Figure 7.7 shows an apparent root at x = -. The multiple parallel number lines can also be used to investigate composition of functions. For instance, try creating points for the composition f(g(x)), where f(x)=ax^2 and g(x)=x-b. Explore this form of a quadratic. Where are the roots? Compare this form to the standard form. Describe the composition f(g(x)) in terms of geometric transformations. -10 -5 5 10 -10 -5 5 10 -10 -5 5 10 -10 -5 5 10 -10 -5 5 10 -10 -5 5 10 c = -8. b = -1. x = -2. a = 1. ax'+bx+c (^0) ax'+bx 0 ax' 0 x' 0 bx u u u x 0 1 A a b B c C 0 Figure 7.7: Finding a Quadratic Function with a Root at -2.
Creating Dynagraphs Algebraically in GSP 4 The secret to constructing Dynagraphs algebraically using GSP 4 is to create two horizontal number lines (an input axis and an output axis) and to use the function calculator to calculate the value of your function for some variable point on the input axis. You then use this calculated value to plot a point on the output axis. As you move your variable point on the input axis the plotted point on the output axis moves appropriately. The major concern in using this method is to make sure the appropriate coordinate system is marked when you are calculating or plotting coordinates. Use the following steps as a guide:
- Open a new sketch and choose Define Coordinate System from the Graph menu.
- Place a point on the y-axis about an inch below the x-axis.
- Hide the y-axis and hide the grid.
- Select your point below the visible x-axis and choose Define Origin from the Graph menu. A warning dialog will ask you if you really want to define a new coordinate system. Click on the Yes button.
- Hide the grid (choose Hide Grid from the Graph menu) and hide the new y-axis.
- Label the top axis input and the bottom axis output (click on each axis with the label tool and edit each label). At this point you should have two horizontal axes as in figure 7.8 below. -10 -5 5 10 output -10 -5 5 10 input Figure 7.8: Two horizontal axes
- Place a free point on the input axis and label it “x”.
- VERY IMPORTANT STEP: Select the origin point of your input axis and choose Mark Coordinate System from the Graph menu (this step makes the input axis the active coordinate system for coordinate measurements).
- Label this new point f(x). Construct a segment between points x and f(x). At this point your sketch should look something like figure 7.10 below. This completes your construction of a Dynagraph. You can make a custom tool of this construction for creating more Dynagraphs. The givens for your dynagraph tool should be the origin point of a new input axis and a new function. (For tips on creating custom tools see your GSP manual.) -10 -5 5 10 output -10 -5 5 10 input
f ( input x )= 7.
f ( x) = e x^ input (^) x = 2. f(x) x Figure 7.10: Completed Dynagraph in GSP 4 Challenge a classmate to try and discover your function by hiding the function expression and then ask your class mate to experiment by changing the position of the input point, x and observing the effect this has on the output point, f(x). Exloring Range and Domain with Dynagraphs Open the file Dynagraphs.gsp inside the Algebra folder. This sketch is from the CD accompanying Exploring Algebra with the Geometer’s Sketchpad (2003). You will find two pages of mystery dynagraphs. Explore each of them in turn and try and deduce the functions for each dynagraph. There are buttons to provide more information (such as the scale point – i.e. the unit point for each axis, or the actual numbers on the axes). There is also a button to show the functions but resist using this until you have thoroughly explored all of the dynagraphs on that page. Comparing the range and domain of these different functions may help in your exploration. The domain of a function is the set of input values for which the function produces an output; the range of a function is the set of output values that the function can produce. In order to visually record the range of one of the dynagraphs in the Dynagraphs.gsp sketch, first select the triangular region attached to the output point and choose Trace Locus from the Display menu. Move the input point slowly to see the trace of the output triangle along the bottom of the output axis. Figure 7.11 shows the trace for functions h and i from page one of the sketch.
Figure 7.11: Two Dynagrpaphs with Ranges Traced Note that the range for function h appears to be discrete rather than continuous, taking only certain values along the number line, whereas the range for function i appears to be only positive real numbers, but does appear to be continuous. What possibilities do these traces of the ranges of these functions suggest for the type of function in each case? Explore the range and domain of the functions on page 2 of this sketch. Are there functions that have a limited domain (not all real numbers)? Are there functions that have a bounded range? Are there functions that appear to have “holes” in their range (a value or values that the function can not achieve)? Activity 7.3: Composition of Functions using Dynagraphs When we form the composition of two functions, such as g(f(x)), the output of the inner function (f(x)) becomes the input for the outer function. We can make this connection explicit using two dynagraphs. For instance, with the two dynagraphs in figure 7.11, in order to form the composed function i(h(x) we would want the output of h (the point h(c) ) to be the input point for function i. The input point for function i is D. We need to make point D become point h(C). We can do this by splitting point D from its axis and then merging it with point h(C). The following steps achieve this process:
1. Deselect everything by clicking on a blank part of the sketch. 2. Select point D and then choose Split point from axis under the Edit menu. Point D (with its attached pentagon) will move away from the axis. 3. Leave point D selected and also select point h(C). 4. Choose Merge Points from the Edit menu. Your two dynagraphs should now look like figure 7.
for students in a Mathematics Education course at the University of Georgia) ask you to explore how asymptotic behavior appears within a dynagraph:
- How would you characterize vertical asymptotes using the dynagraph representation? (see function w on page. 2 of the Dynagraph.gsp sketch). Come up with other rational functions that behave differently about the vertical asymptote.
- How would you characterize horizontal asymptotes using this representation? Start by analyzing the dynagraph of: y=(2x–3)/(x+1) (look at its “behavior at infinity”) then come up with other examples of functions with horizontal asymptotes.
- How would you characterize slant asymptotes using this representation? Start by analyzing the dynagraph of: y=(x^2 +3x-5)/(x+1) (look at its “behavior at infinity”) then come up with other examples of functions with slant asymptotes.
- How would you characterize the asymptotes of: y=(x^2 +3x-5)/(x^2 +1)? What do you notice about the range of this function? From Dynagraphs to Cartesian Coordinates The transition from the parallel number lines representation of functions to the more traditional Cartesian Coordinate representation can be made dynamically using your GSP dynagraph. Even though both GSP 3 and GSP 4 have built-in coordinate systems, it can be enlightening and interesting to create your own two-dimensional coordinate system from your one-dimensional number line or dynagraph. What you will be doing, in fact, is transforming a mathematical mapping from R1 to R1 (the real numbers) into a mapping from R1 to R2 (2-space). A pair of parallel number lines can be transformed into non-parallel, intersecting number lines to form coordinate axes in 2-space. The axes can be oblique as well as rectangular, and do not have to share a common origin. One simple way to construct such a flexible coordinate system is to construct a rotated image of the second (output) number line about its origin. The following steps are provided as a guide: 1. Move the output axis of your dynagraph above the input axis. 2. Mark the origin of the output axis as a center of rotation. 3. Place a free point somewhere above the unit point of the output axis and label this point tilt.
4. Select (in this order) the unit point and origin point of your output axis AND the point tilt and choose Mark Angle from the Transform menu. 5. Select the output axis and the output point (f(x)) and choose Rotate from the Transform menu. The output axis and point should rotate about its origin by the marked angle. Your dynagraph will look something like figure 7.13. 6. Label the rotated f(x) point and hide the original output axis and point. -10 -5 5 10 15 output -10 -5 5 10 15 input
inputx = 4.80 f( x ) = 2⋅(x-3 )^2^ f (^ input x ) =^ 6.
f(x) f(x) 0 tilt 1 0' 1' x Figure 7.13: A Rotated Dynagraph Output Axis By moving the two origin points together you will have what is called an Oblique Cartesian Coordinate System , named after the French philosopher and mathematician, René Descartes (1596-1650) who is considered one of the fathers of analytic geometry. When the two axes are perpendicular to one another, the system is called a Rectangular Cartesian Coordinate System. Research the web to find out about René Descartes and his coordinate system. Did Descartes ever use oblique coordinates? If so, for what purpose? A point is plotted in these systems by constructing lines through the input and output points that are parallel to the other axis. That is, a line through x parallel to the rotated output axis and a line through f(x) parallel to the horizontal input axis. Where these two lines intersect will be the point in the coordinate plane corresponding to the ordered pair (x, f(x)) or (x, y) using the conventional label of y for the output axis. It is important to realize that the intersection of these two lines is what coordinates the motion of the input and output points
Edit your function by double clicking on the f(x) expression and test your conjectures with different kinds of functions. Sketchpad has a built-in rectangular coordinate graphing system that automatically constructs the locus of the ordered pair (x, f(x)) when a function is plotted using the Plot Function option under the Graph menu. Select the function expression in your sketch and plot its graph using this feature of Sketchpad. You should have a figure that looks something like figure 7.15. Figure 7.15: Oblique and Rectangular Graphs of a Quadratic Function Rotate the tilt point until both graphs coincide. Move your input point x and observe how the plotted point (x,y) moves along the plotted function graph. For the next part of this chapter will shall use Sketchpad’s function plotting feature to investigate transformations of the quadratic function. Activity 7.4: Transformations of the Quadratic Function In a new sketch, plot the function f(x)=x^2 using Plot Function under the Graph menu. A parabola should have been plotted. Place a free point on this parabola using the Point tool (just click on the parabola with the Point tool). Label this point A. Measure the coordinates of point A (select point A and then select Coordinates from the Measure menu). Observe how the coordinates change as you move point A along the parabola. We shall now use this point on the graph of f(x) to translate the parabola (function plots themselves cannot be transformed directly). -10 -5 5 10 15 input input (^) x = 4.76 f^ (^ x)^ = 2^ ⋅^ (^ x-3 )^2^ f^ ( input^^ x )^ =^ 6. Move 0' -> 0 f(x)^ (x,y)y)) 0 tilt 0' 1 x
In order to translate the parabola, we need to define a vector of translation. We shall then use this vector to translate point A on the parabola Create a free point somewhere in your sketch and label it V. Select the origin point and point V IN THAT ORDER and choose Mark Vector from the Transform menu. Select point A and choose Translate under the Transform menu. Display the label of the translated point A’. Select A’ and choose Trace Point under the Display menu. Move point A. What shape does the translated point, A’ trace out? What do you notice about your free point V relative to the trace of A’? Measure the coordinates of points V and A’. Move point A. What do you notice about the coordinates of A, V and A’? Move point V. What do you notice about the coordinates of A, V and A’? You may want to measure the abscissa and ordinates of points A and V and do some calculations with these to check your conjecture concerning relations among these coordinates. In order to more closely observe and keep track of the changes in the parabolic graph of the quadratic as point V is moved we can construct the locus of point A’ (instead of tracing its path) as A is moved. First choose Erase Traces under the Display menu, then select A’ and turn off its trace by choosing the checked Trace Point under the Display menu. Select points A’ and A then choose Locus from the Construct menu. Your sketch should look something like figure 7.16. Move point V around your sketch. What do you notice about point V and the translated parabola?
Select Ad and A, then choose Locus under the Construct menu to construct the locus of your dilated point as A moves on the original parabola. Your sketch should look like figure 7.17 at this point. Select the parameter d , and then change its value by using the + and – keys on your keyboard. What happens to the dilated parabola? [Note: The default step for changing a parameter with the +/- keys is one unit. You can change this step. Select parameter d and choose Properties from the Edit menu. Select the Parameter tab on the pop-up window. At the bottom of this dialog box change the “Change by:” value to 0.1 units for Keyboard (+/-) adjustments. Click on the OK button.] Now press the + key several times. What do you notice? Figure 7.17: Dilated and Translated Quadratic Functions Assignment 7. Challenges:
- Create a new function using parameter d whose graph will coincide with the dilated parabola of f(x). Call your new function h(x).
- Create a new quadratic function based on the free point V as the vertex of your parabola and the parameter d as the dilation factor, that will be the translation of your function h(x) by the vector 0V. Call your new function j(x).
- How does j(x) compare to the general quadratic function in vertex form: y=a(x- h)^2 +k, where (h, k) are the coordinates of the vertex of the parabola? What is the 12 10 8 6 4 2 - - - -
-10 -5 5 10 15 d = 2. xA+xV = 5.21 yA+yV = 1. yV = -2. yA = 4. xV = 3. xA = 2. V: (3.14, -2.54) At: (5.21, 1.73) A: (2.07, 4.27) f ( x ) = x^2 A'd A't 0 V A'
relationship between the parameter a in this general vertex form and your parameter d? Extra Challenge: Starting with a free point V as the vertex of your parabola, create a general quadratic function that can be moved around the screen simply by moving it’s vertex, V, and can change its “openness” from very narrow opening up to very wide and eventually to opening down by moving a control point on the parabola. This control point should move with the parabola when you move point V (see figures 7.18a and 7.18b for two possible states of this completely dynamic quadratic). Figure 7.18a and 7.18b: A dynamic quadratic graph controlled by points V and A Need to add: From vertex-form of quadratic function to Completing the Square to find roots and eventually the quadratic formula Draft: for f(x)=a(x-h)^2 + k -- where (h, k) are the coordinates of the vertex of the parabola, We can solve for roots by putting f(x)=0: a(x-h)^2 + k=0 a(x-h)^2 =-k (x-h)^2 = € − k a x-h=
± − ak , thus x=h
± − ak. This can be thought of as the quadratic formula in
vertex form. Converting from vertex form to standard form: f(x)=ax^2 +bx+c by expanding the binomial a(x-h)^2 and equating like terms, gives h= € − b 2 a , k=c- € b^2 4 a
. Substituting for h and k in 12 10 8 6 4 2 -5 5 y (^) A -k = 0. g ( x ) = a ⋅( x-h ) 2 +kk a = 0. y) (^) A' = 3. f ( x) = ( x-h ) 2 +kk k = 2. h = 1. V A' 12 10 8 6 4 2 -5 5 y (^) A -k = -0. g ( x ) = a ⋅( x-h ) 2 +kk a = -0. y) (^) A' = 10. f ( x) = ( x-h ) 2 +kk k = 10. h = 1. V A'
