Quadratic Function Transformation Example

GeoGebra Commands & Parameters

1. LaTeX(steps) → counter
2. LaTeX("\text{Move the slider to solve the problem.}") → text0
3. Max({abs(h), abs(k), abs(g(0))}) → base
4. LaTeX("\textcolor{#006633}{y=a(x-h)^2+k}") → text10
5. LaTeX((((((("\text{Substitute }\textcolor{#006633}{(" * h) * ",") * k) * ")}\text{ for }\textcolor{#006633}{(h,k)}\text{ and }\textcolor{#006633}{(0,") * g(0)) * ")}\text{ for }\textcolor{#006633}{(x,y)}.")) → text11
6. LaTeX("\text{Simplify.}") → text13
7. LaTeX((((((("\textcolor{#006633}{" * g(0)) * "=") * Simplify[(0 - h)^(2)]) * "a+(") * k) * ")}")) → text14
8. LaTeX((("\textcolor{#006633}{a=" * a) * "}")) → text16
9. LaTeX("\textcolor{#006633}{y=a(x-h)^2+k}") → text19
10. LaTeX((((((("\text{Let }\textcolor{#006633}{y=g(x)}\text{, }\textcolor{#006633}{a=" * a) * "}\text{, and }\textcolor{#006633}{(h,k)=(") * h) * ",") * k) * ")}.")) → text20
11. LaTeX("\text{Simplify.}") → text22
12. LaTeX((("\textcolor{#006633}{g(x)=" * g) * "}")) → text23
13. LaTeX((((("\text{Subtract }\textcolor{#006633}{" * k) * "}\text{ from both sides and divide by }\textcolor{#006633}{") * ((-h))^(2)) * "}.")) → text15
14. LaTeX((("\textcolor{#006633}{g(x)=" * g) * "}")) → text27
15. If(h > 0, "right", "left") → shiftLR
16. If(k > 0, "up", "down") → shiftUD
17. Extremum(g) → A
18. LaTeX((((("\text{It can be seen from the graph that the vertex of }\textcolor{#006633}{g(x)}\text{ is }\textcolor{#006633}{(" * h) * ", ") * k) * ")}.")) → text5
19. LaTeX("\text{Recall that the vertex of the graph of the quadratic function is also known}\cr\text{as the maximum or minimum point of the function and is represented by the }\cr\text{coordinate }\textcolor{#006633}{(h, k)}\text{. The point }\textcolor{#006633}{(x, y)}\text{ represents some point on the graph}\cr\text{of the function.}") → text3
20. LaTeX("\text{Replace }\textcolor{#006633}{f(x)}\text{ in the equation of the vertex form with }\textcolor{#006633}{y}\text{: }\textcolor{#006633}{y=a(x–h)^2 +k}.") → text2
21. LaTeX("\text{To determine the equation of }\textcolor{#006633}{g(x)}\text{, first identify the vertex of the graph of }\textcolor{#006633}{g(x)}.") → text4
22. LaTeX((("\text{It can be seen from the graph that an additional point on the graph of }\cr\textcolor{#006633}{g(x)}\text{ is }\textcolor{#006633}{(0, " * g(0)) * ")}.")) → text7
23. LaTeX("\text{Next, identify an additional point on the graph of }\textcolor{#006633}{g(x)}.") → text6
24. LaTeX((((((("\text{Use the vertex }\textcolor{#006633}{(" * h) * ", ") * k) * ")}\text{ and the additional point }\textcolor{#006633}{(0, ") * g(0)) * ")}\text{ to find the value of }\textcolor{#006633}{a}\cr\text{in the standard form of the equation.}")) → text8
25. LaTeX("\text{Standard form of a quadratic equation}") → text9
26. LaTeX("\text{Standard form of a quadratic equation}") → text18
27. LaTeX((("\text{The equation of the transformed function }\textcolor{#006633}{g(x)}\text{ in vertex form is }\cr\textcolor{#006633}{g(x)=" * g) * "}.")) → text24
28. LaTeX("\text{Describe the transformation(s) of the parent function that resulted }\cr\text{in the function }\textcolor{#006633}{g(x)}.") → text25
29. LaTeX("\text{To describe the transformation(s) of the parent function }\textcolor{#000099}{f(x) = x^2}\text{, }\cr\text{start inside the parentheses and work outward.}") → text26
30. LaTeX((((("\text{Then, }\textcolor{#006633}{" * a) * "}\text{ was multiplied by the “new” squared term, }\textcolor{#006633}{(") * Simplify[x - h]) * ")^2}.")) → text29
31. If(h ≠ 0, LaTeX[(((("\text{Finally, }\textcolor{#006633}{" * k) * "}\text{ was added to the new squared term with its new }\cr\text{coefficient to give the final result, }\textcolor{#006633}{g(x) = ") * g) * "}.")], LaTeX["\text{No addition was made so the vertex is still on the x-axis.}"]) → text30
32. LaTeX((((((("\textcolor{#006633}{(" * g(0)) * ")=a \left[(0)-(") * h) * ")\right]^2+(") * k) * ")}")) → text12
33. LaTeX("\text{Use the value of }\textcolor{#006633}{a}\text{ and the coordinate of the vertex to write the equation of }\cr\text{the transformed function }\textcolor{#006633}{g(x)}.") → text17
34. LaTeX((((((("\textcolor{#006633}{\left[g(x)\right]=(" * a) * ")\left[x-(") * h) * ")\right]^2+(") * k) * ")}")) → text21
35. If(h ≠ 0, LaTeX[(((((("\text{The value }\textcolor{#006633}{" * h) * "}\text{ was subtracted from }\textcolor{#000099}{x}\text{ to} \text{ shift the vertex ") * shiftLR) * ", which }\cr\text{gives }\textcolor{#006633}{") * Simplify[x - h]) * "}\text{ in place of }\textcolor{#000099}{x}.")], LaTeX["\text{ No change was made in }\textcolor{#000099}{x}\text{ so the function is still centered on the y-axis.}"]) → text28
36. LaTeX("\text{Use the vertex form of a quadratic function, }\textcolor{#006633}{f(x) = a(x – h)^2 + k}\text{, }\cr\text{to determine the equation of the transformed function }\textcolor{#006633}{g(x)}.") → text1
37. LaTeX("\text{Select the Original Problem A button to return to the graph of g(x).}\cr\text{Select the Original Problem B button to explore the graph of h(x).}\cr\text{(Note: Due to functionality, this applet refers to both g(x) and h(x) as g(x).)}\cr\text{Select the New Random Problem button to randomly generate a}\cr\text{similar practice problem.}") → RandomizationText