The Graph Of F ( X ) = X 2 F(x) = X^2 F ( X ) = X 2 Is Translated To Form G ( X ) = ( X − 2 ) 2 − 3 G(x) = (x-2)^2 - 3 G ( X ) = ( X − 2 ) 2 − 3 .Which Graph Represents G ( X G(x G ( X ]?

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Understanding the Concept of Translation in Graphs

Translation is a fundamental concept in mathematics, particularly in graph theory. It involves shifting a graph horizontally or vertically to form a new graph. In this article, we will explore how the graph of f(x)=x2f(x) = x^2 is translated to form g(x)=(x−2)2−3g(x) = (x-2)^2 - 3. We will analyze the process of translation and determine which graph represents g(x)g(x).

The Original Graph of f(x)=x2f(x) = x^2

The graph of f(x)=x2f(x) = x^2 is a parabola that opens upwards. It has a vertex at the origin (0,0)(0, 0) and is symmetric about the y-axis. The equation f(x)=x2f(x) = x^2 represents a quadratic function that is always positive.

The Translated Graph of g(x)=(x−2)2−3g(x) = (x-2)^2 - 3

To form the graph of g(x)=(x−2)2−3g(x) = (x-2)^2 - 3, we need to apply two types of translations to the graph of f(x)=x2f(x) = x^2. The first translation is a horizontal shift of 2 units to the right, and the second translation is a vertical shift of 3 units downwards.

Horizontal Translation

The horizontal translation of 2 units to the right is achieved by replacing xx with (x−2)(x-2) in the equation f(x)=x2f(x) = x^2. This results in the equation (x−2)2(x-2)^2. The graph of (x−2)2(x-2)^2 is a parabola that opens upwards and has a vertex at (2,0)(2, 0).

Vertical Translation

The vertical translation of 3 units downwards is achieved by subtracting 3 from the equation (x−2)2(x-2)^2. This results in the equation (x−2)2−3(x-2)^2 - 3. The graph of (x−2)2−3(x-2)^2 - 3 is a parabola that opens upwards and has a vertex at (2,−3)(2, -3).

Determining Which Graph Represents g(x)g(x)

To determine which graph represents g(x)g(x), we need to analyze the two types of translations applied to the graph of f(x)=x2f(x) = x^2. The first translation is a horizontal shift of 2 units to the right, and the second translation is a vertical shift of 3 units downwards.

Graph A

Graph A represents the equation f(x)=(x−2)2f(x) = (x-2)^2. This graph is a parabola that opens upwards and has a vertex at (2,0)(2, 0).

Graph B

Graph B represents the equation g(x)=(x−2)2−3g(x) = (x-2)^2 - 3. This graph is a parabola that opens upwards and has a vertex at (2,−3)(2, -3).

Graph C

Graph C represents the equation h(x)=(x+2)2+3h(x) = (x+2)^2 + 3. This graph is a parabola that opens upwards and has a vertex at (−2,3)(-2, 3).

Conclusion

Based on the analysis of the two types of translations applied to the graph of f(x)=x2f(x) = x^2, we can conclude that Graph B represents g(x)=(x−2)2−3g(x) = (x-2)^2 - 3. The horizontal translation of 2 units to the right and the vertical translation of 3 units downwards result in a parabola that opens upwards and has a vertex at (2,−3)(2, -3).

Final Answer

The graph that represents g(x)g(x) is Graph B.

Discussion

The concept of translation in graphs is a fundamental idea in mathematics. It involves shifting a graph horizontally or vertically to form a new graph. In this article, we analyzed how the graph of f(x)=x2f(x) = x^2 is translated to form g(x)=(x−2)2−3g(x) = (x-2)^2 - 3. We determined that Graph B represents g(x)g(x) based on the two types of translations applied to the graph of f(x)=x2f(x) = x^2.

Key Takeaways

  • The graph of f(x)=x2f(x) = x^2 is a parabola that opens upwards and has a vertex at the origin (0,0)(0, 0).
  • The graph of g(x)=(x−2)2−3g(x) = (x-2)^2 - 3 is a parabola that opens upwards and has a vertex at (2,−3)(2, -3).
  • The horizontal translation of 2 units to the right and the vertical translation of 3 units downwards result in a parabola that opens upwards and has a vertex at (2,−3)(2, -3).

References

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Frequently Asked Questions

Q1: What is the concept of translation in graphs?

A1: Translation is a fundamental concept in mathematics, particularly in graph theory. It involves shifting a graph horizontally or vertically to form a new graph.

Q2: How is the graph of f(x)=x2f(x) = x^2 translated to form g(x)=(x−2)2−3g(x) = (x-2)^2 - 3?

A2: To form the graph of g(x)=(x−2)2−3g(x) = (x-2)^2 - 3, we need to apply two types of translations to the graph of f(x)=x2f(x) = x^2. The first translation is a horizontal shift of 2 units to the right, and the second translation is a vertical shift of 3 units downwards.

Q3: What is the effect of the horizontal translation of 2 units to the right on the graph of f(x)=x2f(x) = x^2?

A3: The horizontal translation of 2 units to the right results in a parabola that opens upwards and has a vertex at (2,0)(2, 0).

Q4: What is the effect of the vertical translation of 3 units downwards on the graph of f(x)=x2f(x) = x^2?

A4: The vertical translation of 3 units downwards results in a parabola that opens upwards and has a vertex at (2,−3)(2, -3).

Q5: Which graph represents g(x)=(x−2)2−3g(x) = (x-2)^2 - 3?

A5: Graph B represents g(x)=(x−2)2−3g(x) = (x-2)^2 - 3. The horizontal translation of 2 units to the right and the vertical translation of 3 units downwards result in a parabola that opens upwards and has a vertex at (2,−3)(2, -3).

Q6: What is the significance of the vertex of the graph of g(x)=(x−2)2−3g(x) = (x-2)^2 - 3?

A6: The vertex of the graph of g(x)=(x−2)2−3g(x) = (x-2)^2 - 3 is at (2,−3)(2, -3). This indicates that the graph is a parabola that opens upwards and has a minimum value of -3 at the point (2, -3).

Q7: How can we determine which graph represents g(x)g(x)?

A7: To determine which graph represents g(x)g(x), we need to analyze the two types of translations applied to the graph of f(x)=x2f(x) = x^2. The first translation is a horizontal shift of 2 units to the right, and the second translation is a vertical shift of 3 units downwards.

Q8: What are the key takeaways from this article?

A8: The key takeaways from this article are:

  • The graph of f(x)=x2f(x) = x^2 is a parabola that opens upwards and has a vertex at the origin (0,0)(0, 0).
  • The graph of g(x)=(x−2)2−3g(x) = (x-2)^2 - 3 is a parabola that opens upwards and has a vertex at (2,−3)(2, -3).
  • The horizontal translation of 2 units to the right and the vertical translation of 3 units downwards result in a parabola that opens upwards and has a vertex at (2,−3)(2, -3).

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