Explore the effects of the four function transformations [math]a[/math], [math]b[/math], [math]h[/math], and [math]k[/math].

In order to use mathematical functions in real-life applications, we usually have to [i]transform[/i] the function in one or more ways. Often this is necessary because of the units we are using, or perhaps to account for changes in starting position, or maybe we need to re-orient the graph to match our application. For example, the path of a baseball thrown from the outfield to second base follows a parabola, whose parent function is [math]f(x)=x^2[/math]. But if we graph this function, we get a tiny parabola at the origin, opening upward. So first we have to turn the parabola upside-down and stretch it vertically to the height of the throw. We'll also have to shift it up so that the vertex is above the ground. Now, we will have to stretch it out horizontally to match the distance the ball is thrown. And last we'll have to shift it horizontally to line up with the outfield and with second base (let's assume the origin is home plate).[br][br]In math, these four transformations are called, respectively, vertical stretch [math]a[/math] (which can include vertical reflection when we stretch by a negative number), vertical shift [math]k[/math], horizontal stretch [math]b[/math] (which can include horizontal reflection when we stretch by a negative number), and horizontal shift [math]h[/math]. This app illustrates which of these transformations affect which part of the function equation, and shows the corresponding graph changes.[br][br]Be aware that in Precalculus, we usually will not change [math]b[/math] and [math]h[/math] at the same time - either we'll have [math]b=1[/math] or [math]h=0[/math]. But this app does not restrict you on this point, so feel free to explore their interaction. (Basically, [math]b[/math] stretches/compresses the graph horizontally around the vertical line [color=#009900][math]x=h[/math][/color], and [math]a[/math] stretches/compresses the graph vertically around the horizontal line [color=#ff0000][math]y=k[/math][/color]).[br][br]You can change [math]f(x)[/math] to any function you like by entering a new formula in the "[color=#000000]f(x)=[/color]" box. Click the "[color=#000000]Reset All[/color]" button to set all four transformation values back to their "non-transforming" values.

Choose a graph to trace: Sine, Cosine, or both[br]Click on Start Animation to begin or stop the trace.[br]You may also drag the orange point around the circle to manually trace the curves.

Given y = sin(x), each point on the curve is given by (x, sin(x)).[br]Each point on the curve y = cos(x) is given by (x,cos(x)).[br][br]Investigation: Graph both y = sin(x) and y = cos(x) simultaneously. How is the graph of y = sin(x) related to the graph of y = cos(x)?[br][br][br][br]This is a modification of the GeoGebra applet created by Winzeler_BASIS.

Ellipse - The set of all points such that the sum of the distance from two points is a constant. [br][br]Position Vertice V, Foci F1, F2, and Center C. [br][br]Drag the point J along the slider segment to draw the ellipse. [br][br]Use Ctrl-F to clear the trace.

Jerel L. Welker[br]Lincoln Southwest High School Math Department[br]Lincoln, Nebraska[br]jwelker@lps.org