[list=1][*][size=100]Move point [i]A[/i] along the function graph and make a conjecture about the shape of the path of point [i]S[/i], which corresponds to the slope function.[br][br][/size][/*][*][size=100]Turn on the [img]https://wiki.geogebra.org/uploads/thumb/e/e2/Menu-trace-on.svg/16px-Menu-trace-on.svg.png[/img] trace of point [i]S[/i]. Move point [i]A[/i] to check your conjecture.[br][u]Hint[/u]: Right-click point [i]S[/i] (MacOS: [i]Ctrl[/i]-click, tablet: long tap) and select [img]https://wiki.geogebra.org/uploads/thumb/e/e2/Menu-trace-on.svg/16px-Menu-trace-on.svg.png[/img] [i]Trace on[/i].[br][br][/size][/*][*][size=100]Find the equation of the resulting slope function and enter it into the [i]Input Bar[/i] using [i]g(x)=...[/i] Move point [i]A[/i] along the graph of function [i]f[/i]. If your prediction is correct, the trace of point [i]S[/i] will match the graph of your function [i]g[/i].[br][br][/size][/*][*][size=100]Change the equation of the initial polynomial [i]f[/i] to produce a new problem. For example, enter [code]f(x)= 2 x²[/code] into the [i]Input Bar[/i].[br][u]Hint[/u]: You might want to zoom if point [i]A[/i] lays outside of the visible area after changing the function.[/size][/*][/list]

[table] [tr] [td][size=100]1.[/size][/td] [td][icon]https://wiki.geogebra.org/uploads/thumb/4/40/Menu_view_algebra.svg/120px-Menu_view_algebra.svg.png[/icon][br][/td] [td][size=100]Enter the polynomial [code][/code][font=Courier New]f(x) = x^2/2 + 1[/font].[/size][/td][/tr] [tr] [td][size=100]2.[/size][/td] [td][icon]/images/ggb/toolbar/mode_pointonobject.png[/icon][br][/td] [td][size=100]Create a new point [i]A[/i] on function [i]f[/i].[br][u]Hint[/u]: Point A can only be moved along the function.[/size][/td][/tr] [tr] [td][size=100]3.[/size][/td] [td][size=100][icon]/images/ggb/toolbar/mode_tangent.png[/icon][/size][/td] [td][size=100]Create tangent [i]a[/i] to function [i]f[/i] through point [i]A[/i].[/size][/td][/tr] [tr] [td][size=100]4.[/size][/td] [td][icon]https://wiki.geogebra.org/uploads/thumb/4/40/Menu_view_algebra.svg/120px-Menu_view_algebra.svg.png[/icon][br][/td] [td][size=100]Create the slope of tangent [i]a[/i] using [font=Courier New]m = Slope(a)[/font].[/size][/td][/tr] [tr] [td][size=100]5.[/size][/td] [td][icon]https://wiki.geogebra.org/uploads/thumb/4/40/Menu_view_algebra.svg/120px-Menu_view_algebra.svg.png[/icon][br][/td] [td][size=100]Define point S: [font=Courier New]S = (x(A), m)[/font].[br][u]Hint[/u]: [code]x(A)[/code] gives you the [i]x[/i]-coordinate of point [i]A[/i].[/size][/td][/tr] [tr] [td][size=100]6.[/size][/td] [td][size=100][icon]/images/ggb/toolbar/mode_segment.png[/icon][/size][/td] [td][size=100]Connect points [i]A[/i] and [i]S[/i] using a segment.[/size][/td][/tr][tr] [td][size=100]7.[/size][/td] [td][icon]https://wiki.geogebra.org/uploads/thumb/e/e2/Menu-trace-on.svg/32px-Menu-trace-on.svg.png[/icon][/td] [td][size=100]Turn on the trace of point [i]S[/i]. [br][u]Hint[/u]: Right-click point [i]S[/i] (MacOS: Ctrl-click, tablet: long click) and select [i]Trace on[/i].[br][/size][/td][/tr][/table]