Explore how a parabola can be intersected with a linear function by determining the roots of their difference function.

Change the values of the sliders to explore how the parameters of the linear equation affect the line and the intersection point(s) with the parabola.

[table][tr][td][size=100]1.[/size][/td][td][size=100][icon]https://wiki.geogebra.org/uploads/thumb/4/47/Menu_view_cas.svg/32px-Menu_view_cas.svg.png[/icon][/size][/td][td][size=100]In the [i]CAS View[/i], create a quadratic polynomial by entering [code]f(x):= x^2 – 3/2 * x + 2[/code] into the first row and hitting the [i]Enter[/i] key.[br][/size][/td][/tr][tr][td][size=100]2.[/size][/td][td][size=100][icon]https://www.geogebra.org/images/ggb/toolbar/mode_slider.png[/icon][br][/size][/td][td][size=100]Select the [i]Slider[/i] tool from the [i]Graphics View Toolbar[/i] and create two sliders [i]a [/i]and [i]b [/i]by clicking inside the [img]https://wiki.geogebra.org/uploads/thumb/c/c8/Menu_view_graphics.svg/16px-Menu_view_graphics.svg.png[/img] [i]Graphics View [/i] and using the default settings of sliders.[br][/size][/td][/tr][tr][td][br][/td][td][br][/td][td][size=100][b]Hint:[/b] After clicking on the [img]https://wiki.geogebra.org/uploads/thumb/c/c8/Menu_view_graphics.svg/16px-Menu_view_graphics.svg.png[/img] [i]Graphics View[/i], a window appears allowing you to specify the parameters of your slider. Click [i]Apply[/i] to close the window and create a slider.[/size][br][/td][/tr][tr][td][size=100]3.[/size][/td][td][size=100][icon]https://wiki.geogebra.org/uploads/thumb/4/47/Menu_view_cas.svg/32px-Menu_view_cas.svg.png[/icon][/size][/td][td][size=100]In the [i]CAS View[/i], create a linear function by entering g(x):= a * x + b into the next row and hitting the [i]Enter[/i] key.[/size][/td][/tr][tr][td][size=100]4.[/size][/td][td][size=100][icon]https://www.geogebra.org/images/ggb/toolbar/mode_move.png[/icon][/size][/td][td][size=100]In the [img]https://wiki.geogebra.org/uploads/thumb/c/c8/Menu_view_graphics.svg/16px-Menu_view_graphics.svg.png[/img] [i]Graphics View[/i], use the [i]Move[/i] tool to change the value of slider [i]a[/i] to 0.5 and the value of slider [i]b[/i] to 2.[/size][/td][/tr][tr][td][size=100]5.[/size][/td][td][size=100][icon]https://wiki.geogebra.org/uploads/thumb/4/47/Menu_view_cas.svg/32px-Menu_view_cas.svg.png[/icon][/size][/td][td][size=100]In the [i]CAS View[/i], enter h(x):= f(x) – g(x) to determine the difference between those functions.[/size][/td][/tr][tr][td][size=100]6.[/size][/td][td][size=100][icon]https://www.geogebra.org/images/ggb/toolbar/mode_factor.png[/icon][/size][/td][td][size=100]Enter h(x), then select the tool [i]Factor[/i] from the [i]CAS View Toolbar[/i] to factorize the polynomial.[br][/size][/td][/tr][tr][td][br][/td][td][br][/td][td][size=100][b]Hint:[/b] You can now use these factors to determine the roots of [i]h(x)[/i].[/size][br][/td][/tr][tr][td][size=100]7.[/size][/td][td][size=100][icon]https://wiki.geogebra.org/uploads/thumb/4/47/Menu_view_cas.svg/32px-Menu_view_cas.svg.png[/icon][/size][/td][td][size=100]Enter Solve(h(x)) to confirm the roots.[/size][/td][/tr][tr][td][size=100]8.[/size][/td][td][size=100][icon]https://wiki.geogebra.org/uploads/thumb/4/47/Menu_view_cas.svg/32px-Menu_view_cas.svg.png[/icon][/size][/td][td][size=100]Enter Intersect(f(x), g(x)) to create the intersection points of functions [i]f(x)[/i] and [i]g(x)[/i].[br][/size][/td][/tr][tr][td][size=100][/size][/td][td][size=100][icon]https://www.geogebra.org/images/ggb/toolbar/mode_showhideobject.png[/icon][/size][/td][td][size=100][b]Hint: [/b]You can display the intersection points in the [img]https://wiki.geogebra.org/uploads/thumb/c/c8/Menu_view_graphics.svg/16px-Menu_view_graphics.svg.png[/img] [i]Graphics View[/i] by clicking on the disabled [i]Visibility [/i]button below the corresponding row number in the [img]https://wiki.geogebra.org/uploads/thumb/4/47/Menu_view_cas.svg/16px-Menu_view_cas.svg.png[/img] [i]CAS View[/i].[/size][/td][/tr][tr][td][size=100]9.[/size][/td][td][size=100][br][/size][/td][td][size=100][b]Exploration:[/b] Try to find out what the intersection points of [i]f(x)[/i] and [i]g(x)[/i] have in common with the roots of the difference function [i]h(x)[/i]. [br]Change the parameters of the linear function to find out for which values of [i]a [/i]and [i]b [/i]there are two, one, or no intersection point(s).[br][/size][/td][/tr][tr][td][br][/td][td][size=100][icon]https://www.geogebra.org/images/ggb/toolbar/mode_move.png[/icon][/size][br][/td][td][size=100][b]Hint:[/b] Use the [i]Move[/i] tool to change the values of the sliders and create new functions to explore.[/size][br][/td][/tr][/table]