Identifying the Translation

Oct 13, 2025 by miodeya.com 100 views

Understanding the Problem

When dealing with translations of functions, it's essential to understand the concept of shifting and scaling. The given function f(x) = x^2 is a quadratic function that opens upwards. A translation of this function results in a new function g(x), which may be shifted horizontally, vertically, or both. In this case, we need to determine the function rule for g(x) in the form a(x - h)^2 + k, where a, h, and k are integers or simplified fractions.

Identifying the Translation

To write the function rule for g(x), we need to identify the translation of f(x) = x^2. Let's assume the graph of g(x) is a translation of f(x) = x^2 by (h, k) units. This means the graph of g(x) is shifted h units to the left or right and k units up or down from the graph of f(x) = x^2.

Writing the Function Rule

The general form of a quadratic function is f(x) = a(x - h)^2 + k. To write the function rule for g(x), we need to determine the values of a, h, and k. Since the graph of g(x) is a translation of f(x) = x^2, we can start by comparing the two functions.

Comparing the Functions

Let's compare the functions f(x) = x^2 and g(x) = a(x - h)^2 + k. We know that g(x) is a translation of f(x) = x^2, so we can write:

g(x) = f(x - h) + k

Substituting f(x) = x^2 into the equation, we get:

g(x) = (x - h)^2 + k

Determining the Values of a, h, and k

To determine the values of a, h, and k, we need to analyze the graph of g(x). Let's assume the graph of g(x) is a translation of f(x) = x^2 by (h, k) units. This means the graph of g(x) is shifted h units to the left or right and k units up or down from the graph of f(x) = x^2.

Horizontal Translation

If the graph of g(x) is shifted h units to the left, then h is negative. If the graph of g(x) is shifted h units to the right, then h is positive.

Vertical Translation

If the graph of g(x) is shifted k units up, then k is positive. If the graph of g(x) is shifted k units down, then k is negative.

Determining the Value of a

The value of a determines the direction and magnitude of the vertical stretch or compression of the graph of g(x). If a is positive, the graph of g(x) is stretched vertically. If a is negative, the graph of g(x) is compressed vertically.

Writing the Function Rule

Now that we have determined the values of a, h, and k, we can write the function rule for g(x) in the form a(x - h)^2 + k.

Example 1: Horizontal Translation

Suppose the graph of g(x) is a translation of f(x) = x^2 by 2 units to the left. In this case, h = -2. The function rule for g(x) is:

g(x) = (x + 2)^2 + k

Example 2: Vertical Translation

Suppose the graph of g(x) is a translation of f(x) = x^2 by 3 units up. In this case, k = 3. The function rule for g(x) is:

g(x) = (x - h)^2 + 3

Example 3: Horizontal and Vertical Translation

Suppose the graph of g(x) is a translation of f(x) = x^2 by 2 units to the left and 3 units up. In this case, h = -2 and k = 3. The function rule for g(x) is:

g(x) = (x + 2)^2 + 3

Conclusion

In conclusion, the function rule for g(x) can be written in the form a(x - h)^2 + k, where a, h, and k are integers or simplified fractions. To determine the values of a, h, and k, we need to analyze the graph of g(x) and compare it to the graph of f(x) = x^2. By understanding the concept of shifting and scaling, we can write the function rule for g(x) and determine the values of a, h, and k.

Frequently Asked Questions

Q: What is the function rule for g(x) if the graph of g(x) is a translation of f(x) = x^2 by 1 unit to the right?

A: The function rule for g(x) is g(x) = (x - 1)^2 + k.

Q: What is the function rule for g(x) if the graph of g(x) is a translation of f(x) = x^2 by 2 units down?

A: The function rule for g(x) is g(x) = (x - h)^2 - 2.

Q: What is the function rule for g(x) if the graph of g(x) is a translation of f(x) = x^2 by 3 units to the left and 2 units up?

A: The function rule for g(x) is g(x) = (x + 3)^2 + 2.

References

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

Q&A: Writing the Function Rule for g(x)

Q: What is the function rule for g(x) if the graph of g(x) is a translation of f(x) = x^2 by 1 unit to the right?

A: The function rule for g(x) is g(x) = (x - 1)^2 + k.

Q: What is the function rule for g(x) if the graph of g(x) is a translation of f(x) = x^2 by 2 units down?

A: The function rule for g(x) is g(x) = (x - h)^2 - 2.

Q: What is the function rule for g(x) if the graph of g(x) is a translation of f(x) = x^2 by 3 units to the left and 2 units up?

A: The function rule for g(x) is g(x) = (x + 3)^2 + 2.

Q: How do I determine the value of a in the function rule for g(x)?

A: The value of a determines the direction and magnitude of the vertical stretch or compression of the graph of g(x). If a is positive, the graph of g(x) is stretched vertically. If a is negative, the graph of g(x) is compressed vertically.

Q: What is the difference between a horizontal translation and a vertical translation?

A: A horizontal translation is a shift of the graph of g(x) to the left or right, while a vertical translation is a shift of the graph of g(x) up or down.

Q: How do I write the function rule for g(x) if the graph of g(x) is a translation of f(x) = x^2 by (h, k) units?

A: To write the function rule for g(x), you need to determine the values of a, h, and k. The function rule for g(x) is g(x) = a(x - h)^2 + k.

Q: What is the significance of the value of k in the function rule for g(x)?

A: The value of k determines the vertical translation of the graph of g(x). If k is positive, the graph of g(x) is shifted up. If k is negative, the graph of g(x) is shifted down.

Q: How do I determine the value of h in the function rule for g(x)?

A: The value of h determines the horizontal translation of the graph of g(x). If h is positive, the graph of g(x) is shifted to the right. If h is negative, the graph of g(x) is shifted to the left.

Q: What is the difference between a quadratic function and a quadratic expression?

A: A quadratic function is an equation of the form f(x) = ax^2 + bx + c, while a quadratic expression is an expression of the form ax^2 + bx + c.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you need to determine the vertex of the parabola, the x-intercepts, and the y-intercept.

Q: What is the significance of the vertex of a quadratic function?

A: The vertex of a quadratic function is the maximum or minimum point of the parabola.

Q: How do I find the x-intercepts of a quadratic function?

A: To find the x-intercepts of a quadratic function, you need to set the function equal to zero and solve for x.

Q: What is the significance of the y-intercept of a quadratic function?

A: The y-intercept of a quadratic function is the point where the parabola intersects the y-axis.

Conclusion

In conclusion, the function rule for g(x) can be written in the form a(x - h)^2 + k, where a, h, and k are integers or simplified fractions. To determine the values of a, h, and k, you need to analyze the graph of g(x) and compare it to the graph of f(x) = x^2. By understanding the concept of shifting and scaling, you can write the function rule for g(x) and determine the values of a, h, and k.

Frequently Asked Questions

Q: What is the function rule for g(x) if the graph of g(x) is a translation of f(x) = x^2 by 1 unit to the right?

A: The function rule for g(x) is g(x) = (x - 1)^2 + k.

Q: What is the function rule for g(x) if the graph of g(x) is a translation of f(x) = x^2 by 2 units down?

A: The function rule for g(x) is g(x) = (x - h)^2 - 2.

Q: What is the function rule for g(x) if the graph of g(x) is a translation of f(x) = x^2 by 3 units to the left and 2 units up?

A: The function rule for g(x) is g(x) = (x + 3)^2 + 2.

References

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