Applying the Horizontal Shift

Oct 13, 2025 by windailys.com 50 views

Introduction

In mathematics, functions are used to describe the relationship between variables. When a function is translated, it means that its graph is shifted or moved in a specific direction. In this article, we will explore the transformation of the function $$f(x) = x^2$$ to $$g(x) = (x-5)^2 + 1$$. We will analyze the effect of this transformation on the graph of the function and determine which graph represents $$g(x)$$.

Understanding the Original Function $$f(x) = x^2$$

The original function $$f(x) = x^2$$ is a quadratic function that represents a parabola. The graph of this function is a U-shaped curve that opens upwards. The vertex of the parabola is at the origin (0, 0). The function is symmetric about the y-axis, and its domain and range are all real numbers.

Understanding the Translated Function $$g(x) = (x-5)^2 + 1$$

The translated function $$g(x) = (x-5)^2 + 1$$ is obtained by shifting the graph of $$f(x) = x^2$$ horizontally and vertically. The horizontal shift is 5 units to the right, and the vertical shift is 1 unit upwards.

Horizontal Shift

The horizontal shift of 5 units to the right means that the graph of $$g(x)$$ is shifted 5 units to the right of the graph of $$f(x)$$. This means that the vertex of the parabola is now at the point (5, 1).

Vertical Shift

The vertical shift of 1 unit upwards means that the graph of $$g(x)$$ is shifted 1 unit upwards of the graph of $$f(x)$$. This means that the vertex of the parabola is now at the point (5, 1).

Combining the Shifts

When we combine the horizontal and vertical shifts, we get the graph of $$g(x) = (x-5)^2 + 1$$. This graph is a parabola that opens upwards, with its vertex at the point (5, 1).

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

To determine which graph represents $$g(x)$$, we need to analyze the graph of $$g(x)$$ and compare it with the graph of $$f(x)$$. We can see that the graph of $$g(x)$$ is a parabola that opens upwards, with its vertex at the point (5, 1). This is consistent with the graph of $$f(x)$$, which is also a parabola that opens upwards.

Conclusion

In conclusion, the graph of $$g(x) = (x-5)^2 + 1$$ is a translated version of the graph of $$f(x) = x^2$$. The horizontal shift of 5 units to the right and the vertical shift of 1 unit upwards result in a parabola that opens upwards, with its vertex at the point (5, 1). Therefore, the graph that represents $$g(x)$$ is the graph that is shifted 5 units to the right and 1 unit upwards of the graph of $$f(x)$$.

Example

Let's consider an example to illustrate the transformation of the function $$f(x) = x^2$$ to $$g(x) = (x-5)^2 + 1$$. Suppose we have a point (2, 4) on the graph of $$f(x)$$. To find the corresponding point on the graph of $$g(x)$$, we need to apply the horizontal and vertical shifts.

Applying the Horizontal Shift

The horizontal shift of 5 units to the right means that the x-coordinate of the point is increased by 5. Therefore, the new x-coordinate is 2 + 5 = 7.

Applying the Vertical Shift

The vertical shift of 1 unit upwards means that the y-coordinate of the point is increased by 1. Therefore, the new y-coordinate is 4 + 1 = 5.

Conclusion

Therefore, the corresponding point on the graph of $$g(x)$$ is (7, 5).

Graphical Representation

The graph of $$g(x) = (x-5)^2 + 1$$ can be represented graphically as follows:

The graph of $$f(x) = x^2$$ is a parabola that opens upwards, with its vertex at the origin (0, 0).
The graph of $$g(x) = (x-5)^2 + 1$$ is a parabola that opens upwards, with its vertex at the point (5, 1).
The graph of $$g(x)$$ is shifted 5 units to the right and 1 unit upwards of the graph of $$f(x)$$.

Conclusion

References

[1] "Graphing Functions" by Khan Academy
[2] "Functions" by Math Open Reference
[3] "Graphing Quadratic Functions" by Purplemath

Final Thoughts

In this article, we have explored the transformation of the function $$f(x) = x^2$$ to $$g(x) = (x-5)^2 + 1$$. We have analyzed the effect of the horizontal and vertical shifts on the graph of the function and determined which graph represents $$g(x)$$. The graph of $$g(x)$$ is a parabola that opens upwards, with its vertex at the point (5, 1). Therefore, the graph that represents $$g(x)$$ is the graph that is shifted 5 units to the right and 1 unit upwards of the graph of $$f(x)$$.

Introduction

In our previous article, we explored the transformation of the function $$f(x) = x^2$$ to $$g(x) = (x-5)^2 + 1$$. We analyzed the effect of the horizontal and vertical shifts on the graph of the function and determined which graph represents $$g(x)$$. In this article, we will answer some frequently asked questions about the graph of a translated function.

Q1: What is the effect of the horizontal shift on the graph of $$f(x) = x^2$$?

A1: The horizontal shift of 5 units to the right means that the graph of $$g(x)$$ is shifted 5 units to the right of the graph of $$f(x)$$. This means that the vertex of the parabola is now at the point (5, 1).

Q2: What is the effect of the vertical shift on the graph of $$f(x) = x^2$$?

A2: The vertical shift of 1 unit upwards means that the graph of $$g(x)$$ is shifted 1 unit upwards of the graph of $$f(x)$$. This means that the vertex of the parabola is now at the point (5, 1).

Q3: How do I determine which graph represents $$g(x)$$?

A3: To determine which graph represents $$g(x)$$, you need to analyze the graph of $$g(x)$$ and compare it with the graph of $$f(x)$$. You can see that the graph of $$g(x)$$ is a parabola that opens upwards, with its vertex at the point (5, 1). This is consistent with the graph of $$f(x)$$, which is also a parabola that opens upwards.

Q4: What is the domain and range of the graph of $$g(x) = (x-5)^2 + 1$$?

A4: The domain of the graph of $$g(x)$$ is all real numbers, and the range is also all real numbers. This is because the graph of $$g(x)$$ is a parabola that opens upwards, and it has no restrictions on its domain or range.

Q5: How do I apply the horizontal and vertical shifts to a point on the graph of $$f(x) = x^2$$?

A5: To apply the horizontal and vertical shifts to a point on the graph of $$f(x)$$, you need to increase the x-coordinate of the point by 5 (for the horizontal shift) and increase the y-coordinate of the point by 1 (for the vertical shift).

Q6: What is the vertex of the parabola of the graph of $$g(x) = (x-5)^2 + 1$$?

A6: The vertex of the parabola of the graph of $$g(x)$$ is at the point (5, 1).

Q7: How do I graph the function $$g(x) = (x-5)^2 + 1$$?

A7: To graph the function $$g(x)$$, you need to graph the function $$f(x) = x^2$$ and then apply the horizontal and vertical shifts to the graph of $$f(x)$$. This will give you the graph of $$g(x)$$.

Q8: What is the difference between the graph of $$f(x) = x^2$$ and the graph of $$g(x) = (x-5)^2 + 1$$?

A8: The graph of $$g(x)$$ is a translated version of the graph of $$f(x)$$. The horizontal shift of 5 units to the right and the vertical shift of 1 unit upwards result in a parabola that opens upwards, with its vertex at the point (5, 1).

Q9: How do I determine the equation of the graph of $$g(x) = (x-5)^2 + 1$$?

A9: To determine the equation of the graph of $$g(x)$$, you need to analyze the graph of $$g(x)$$ and compare it with the graph of $$f(x)$$. You can see that the graph of $$g(x)$$ is a parabola that opens upwards, with its vertex at the point (5, 1). This is consistent with the graph of $$f(x)$$, which is also a parabola that opens upwards.

Q10: What is the significance of the graph of $$g(x) = (x-5)^2 + 1$$?

A10: The graph of $$g(x)$$ is a translated version of the graph of $$f(x)$$. The horizontal shift of 5 units to the right and the vertical shift of 1 unit upwards result in a parabola that opens upwards, with its vertex at the point (5, 1). This graph represents a real-world situation where a function is translated in a specific direction.