[b][size=150]Cubic Functions: Transformations and Parameters[br][/size][/b][size=150][br][b]Cubic functions[/b] are third-degree polynomials, often represented in the transformation form [br][color=#ff0000][i][b]f(x) = a a(bx - h))3 + k. [/b][/i][/color]Unlike linear functions, these form a characteristic "[i][b]S[/b][/i]" shape or "[i][b]wiggle[/b][/i]," showing a changing curvature and featuring exactly one inflection point.[br][br][/size][b]The Impact of Setting Parameters[/b][size=150][br]By setting the specific values for each parameter, we can precisely manipulate the graph's orientation and position on the coordinate plane:[br][br][size=100]•[b] Vertical Stretch and Reflection ([i]a[/i])[/b]: By setting the value of [b][i]a[/i][/b], we determine the vertical steepness and whether the graph is reflected. In the example provided, setting [color=#ff0000][b][i]a = —0.6 [/i][/b][/color]results in a vertical compression and a reflection, causing the graph to decrease from left to right.[br]•[b] Horizontal Scaling ([i]b[/i]):[/b] Setting the parameter [b][i]b [/i][/b]controls the horizontal stretch or compression. Here,[br]setting [color=#ff0000][i][b]b = 1.3[/b][/i][/color] slightly compresses the curve horizontally toward the inflection point.[br]•[b] Horizontal and Vertical Shifts ([i]h and k)[/i]:[/b] By setting the values for [b][i]h [/i][/b]and [b][i]k[/i][/b], we define the exact location[br]of the inflection point. For this specific graph, setting [b][i][color=#ff0000]h = 1[/color][/i][/b] and[color=#ff0000][i][b] k = 2.6[/b][/i][/color] repositions the center of the[br]"[b][i]S"[/i][/b] curve to the coordinates [b][i](1, 2.6).[/i][/b][br][/size][br][/size][b]Summary of the Transformation[/b][size=150][size=100][br]Through the intentional setting of these parameters, the parent function [color=#ff0000][b]y = x[sup]3[/sup][/b][/color] (shown in green) is[br]transformed into the red curve. This illustrates how every coefficient in a cubic equation directly dictates a physical movement or reshaping of the graph, resulting in a smooth, repositioned, and reflected S-shaped curve.[b][br][/b][/size][b][br]The cubic function is [/b][color=#ff0000]y = x[/color][color=#ff0000][sup]3 [/sup][/color][b] , denoted by function g. [br][br][/b][/size]The transformed basic function is [b][color=#ff0000]y = a(bx - h)[/color][/b][b][size=150][color=#ff0000][sup]3[/sup][/color][/size][/b][b][color=#ff0000] +k[/color][/b][br][br][b][color=#ff0000][size=150]Note[/size][/color][/b]: The 'slider' feature on the x-y coordinate plane can be used to change the [color=#ff0000][b]a, b, h, and k[/b][/color] values [br] for the following exercises. To do so, place the cursor and hold it on the dot of the slider and [br] slide it to the desired m and b values.[br] To move the slider to a different location on the x-y plane, place the cursor and hold it on the line [br] of the slider and move it to the desired location.[br][br][b][color=#ff00ff]Note: You can zoom in or out with the mouse.[/color][/b]

[b][size=150]Perform the following cubic function transformation:[br][/size][/b][br][b]Vertical shift of 3 units up. [br][br][/b] The new function is [b][color=#ff0000]y=[/color][/b][b][size=150][color=#ff0000]x[/color][color=#ff0000][sup]3[/sup][/color][/size][/b][b][color=#ff0000] +3[/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=0 since there is no horizontal shift [br] Set k=3 which represents the vertical shift of 3 units up.[br][/color][br][color=#ff00ff][b] Observe the transformation of the cubic function.[/b][/color]

[b][size=150]Perform the following cubic function transformation:[br][/size][/b][br][b]Vertical shift of 3 units down. [br][br][/b] The new function is [b][color=#ff0000]y=[/color][/b][b][size=150][color=#ff0000]x[/color][color=#ff0000][sup]3[/sup][/color][/size][/b][b][color=#ff0000] - 3[/color][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=0 since there is no horizontal shift [br] Set k= - 3 which represents the vertical shift of 3 units down.[br][/color][br][b][color=#ff00ff] Observe the transformation of [/color][/b][b][color=#ff00ff]the cubic function[/color][/b][b][color=#ff00ff].[/color][/b]

[b][size=150]Perform the following cubic function transformation:[br][/size][/b][b][br]Horizontal shift of 3 units to the right. [br][br][/b] The new function is [b][color=#ff0000]y=(x-3)[/color][/b][b][size=150][color=#ff0000][sup]3[/sup][/color][/size][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=3 which represents the horizontal shift of 3 units to the right. [br] Set k=0 since there is not vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of [/color][/b][b][color=#ff00ff]the cubic function[/color][/b][b][color=#ff00ff]. [/color][/b]

[b][size=150]Perform the following cubic function transformation:[br][/size][/b][br][b]Horizontal shift of 3 units to the left. [br][br][/b] The new function is [b][color=#ff0000]y=(x+3)[/color][/b][b][size=150][color=#ff0000][sup]3[/sup][/color][/size][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=- 3 which represents the horizontal shift of 3 units to the left. [br] Set k=0 since there is not vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]cubic[/color] [/b][b][color=#ff00ff]function.[/color][/b]

[b][size=150]Perform the following cubic function transformation:[br][/size][/b][br][b]Vertical shift of 3 units up plus a horizontal shift of 3 units to the right. [br][br][/b] New function: [b]y = [color=#ff0000](x-3)[/color][/b][b][size=150][color=#ff0000][sup]3[/sup][/color][/size][/b][b][color=#ff0000] +3[/color][/b] , denoted by function f. [br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=3 which represents the horizontal shift of 3 units to the right. [br] Set k=3 which represents the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]cubic [/color][/b][b][color=#ff00ff]function.[/color][/b]

[b][size=150]Perform the following cubic function transformation:[br][/size][/b][b][br]Vertical shift of 3 units down plus a horizontal shift of 3 units to the left. [br] [br][/b] New function: [b][color=#ff0000]y = (x+3)[/color][/b][b][size=150][color=#ff0000][sup]3[/sup][/color][/size][/b][b][color=#ff0000] - 3 [/color][/b], denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h=- 3 which represents the horizontal shift of 3 units to the left. [br] Set k=- 3 which represents the vertical shift of 3 units down.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]cubic [/color][/b][b][color=#ff00ff]function.[/color][/b]

[b][size=150]Perform the following cubic function transformation:[br][/size][/b][b][br]Vertical shift of 3 units down plus a horizontal shift of 3 units to the right. [br][br][/b] New function: [b][color=#ff0000]y = (x - 3)[/color][/b][b][size=150][color=#ff0000][sup]3[/sup][/color][/size][/b][b][color=#ff0000] - 3 [/color][/b], denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h= 3 which represents the horizontal shift of 3 units to the right. [br] Set k=- 3 which represents the vertical shift of 3 units down.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]cubic [/color][/b][b][color=#ff00ff]function.[/color][/b]

[b][size=150]Perform the following cubic function transformation:[br][/size][/b][b][br]Vertical shift of 3 units up plus a horizontal shift of 3 units to the left. [br][br][/b] New function: [b][color=#ff0000]y = (x + 3)[/color][/b][b][size=150][color=#ff0000][sup]3[/sup][/color][/size][/b][b][color=#ff0000] + 3 [/color][/b],denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b=1.[br] Set h= - 3 which represents the horizontal shift of 3 units to the left. [br] Set k= 3 which represents the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]cubic [/color][/b][b][color=#ff00ff]function.[/color][/b]

[b][size=150]Perform the following cubic function transformation:[br][/size][/b][br][b]Vertical stretch by a factor of 3. [br][br][/b] New function: [color=#ff0000] [b]y = 3 [/b][/color][b][size=150][color=#ff0000]x[/color][color=#ff0000][sup]3[/sup][/color][/size][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=3. Set b=1.[br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]cubic[/color] [/b][b][color=#ff00ff]function.[/color][/b]

[b][size=150]Perform the following cubic function transformation:[br][/size][/b][br][b]Vertical shrink by a factor of 1/3.[br][br][/b] New function: [b][color=#ff0000]y = 1/3 [/color][/b][b][size=150][color=#ff0000]x[/color][color=#ff0000][sup]3[/sup][/color][/size][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=1/3. Set b=1.[br] Set h= - 3 which represents the horizontal shift of 3 units to the left. [br] Set k= 3 which represents the vertical shift of 3 units up.[br][br][/color][color=#ff00ff][b] Observe the transformation of the [/b][/color][b][color=#ff00ff]cubic [/color][/b][color=#ff00ff][b]function.[/b][/color]

[b][size=150]Perform the following cubic function transformation:[br][/size][/b][b][br]Horizontal stretch by a factor of 1/3.[br][br][/b] New function: [color=#ff0000][b]y = (1/3x)[/b][/color][b][size=150][color=#ff0000][sup]3[/sup][/color][/size][/b] , denoted by function f.[br][br] [color=#0000ff]Set a =1. Set b=1/3.[br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]cubic [/color][/b][b][color=#ff00ff]function.[/color][/b]

[b][size=150]Perform the following cubic function transformation:[br][/size][/b][br][b]Horizontal shrink by a factor of 3[br][/b] New function: [b][color=#ff0000]y = (3x)[/color][/b] [b][size=150][color=#ff0000][sup]3[/sup][/color][/size][/b] , denoted by function f.[br][br] [color=#0000ff]Set a =1. Set b=3.[br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]cubic [/color][/b][b][color=#ff00ff]function.[/color][/b]

[b][size=150]Perform the following cubic function transformation:[br][/size][/b][b][br]Vertical shift of 3 units up plus, horizontal shift of 3 units to the left[br] and a vertical stretch by a factor of 2. [br][br][/b] New function: [b][color=#ff0000]y = 2(x + 3)[/color][/b][b][size=150][color=#ff0000][sup]3[/sup][/color][/size][/b][b][color=#ff0000] + 3 [/color][/b], denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b = 1.[br] Set h= - 3 which represents the horizontal shift of 3 units to the left. [br] Set k= 3 which represents the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]cubic [/color][/b][b][color=#ff00ff]function.[/color][/b]

[b][size=150]Perform the following cubic function transformation:[br][/size][/b][b][br]Vertical shift of 3 units up plus, horizontal shift of 3 units to the left[br] and a vertical shrink by a factor of 1/2. [br][br][/b] New function: [b][color=#ff0000]y = 1/2(x + 3)[/color][/b][b][size=150][color=#ff0000][sup]3[/sup][/color][/size][/b][b][color=#ff0000] + 3 [/color][/b], denoted by function f.[br][br] [color=#0000ff]Set a=1. Set b = 1.[br] Set h= - 3 which represents the horizontal shift of 3 units to the left. [br] Set k= 3 which represents the vertical shift of 3 units up.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]cubic [/color][/b][b][color=#ff00ff]function.[/color][/b]

[b][size=150]Perform the following cubic function transformation:[br][/size][/b][br][b]Vertical reflection over the x-axis. [br][br][/b] New function: [b][color=#ff0000]y = - [/color][/b][b][size=150][color=#ff0000]x[/color][color=#ff0000][sup]3[/sup][/color][/size][/b] , denoted by function f.[br][br] [color=#0000ff]Set a=-1. Set b = 1.[br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][br][/color][color=#ff00ff][b] Observe the transformation of the [/b][/color][b][color=#ff00ff]cubic [/color][/b][color=#ff00ff][b]function.[/b][/color]

[b][size=150]Perform the following cubic function transformation:[br][/size][/b][br][b]Reflection over the y-axis. [br][br][/b] New function: [b] [color=#ff0000]y = (-x)[/color][/b][b][size=150][color=#ff0000][sup]3 [/sup][/color][/size][/b], denoted by function f.[br][br] [color=#0000ff] Set a=1. Set b = -1. [br] Set h= 0 since there is no horizontal shift.[br] Set k= 0 since there is no vertical shift.[br][/color][br][b][color=#ff00ff] Observe the transformation of the [/color][/b][b][color=#ff00ff]cubic [/color][/b][b][color=#ff00ff]function.[/color][/b]

[b]Repeat this exercise as many times as desired until concept is mastered. [/b] [br][br]Use different values of [color=#ff0000][b]a, b, h and k[/b][/color].