The Function $g(x$\] Is A Transformation Of The Quadratic Parent Function, $f(x)=x^2$. What Function Is $g(x$\]?A. $g(x)=-3x^2$ B. $g(x)=-\frac{1}{3}x^2$ C. $g(x)=3x^2$ D.

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Understanding the Quadratic Parent Function

The quadratic parent function, denoted as f(x)=x2f(x)=x^2, is a fundamental function in mathematics that represents a parabola opening upwards with its vertex at the origin (0,0). This function is a quadratic function, which means it can be written in the form f(x)=ax2+bx+cf(x)=ax^2+bx+c, where aa, bb, and cc are constants. In the case of the quadratic parent function, a=1a=1, b=0b=0, and c=0c=0.

Transformations of the Quadratic Parent Function

Transformations of the quadratic parent function involve changing the function in some way to create a new function. These transformations can include shifting, scaling, reflecting, and rotating the function. In this case, we are interested in finding a function g(x)g(x) that is a transformation of the quadratic parent function f(x)=x2f(x)=x^2.

Identifying the Transformation

To identify the transformation, we need to examine the given options and determine which one represents a transformation of the quadratic parent function. Let's analyze each option:

Option A: g(x)=−3x2g(x)=-3x^2

This option represents a transformation of the quadratic parent function by multiplying the function by −3-3. This means that the function is being scaled by a factor of −3-3, which will result in a parabola that opens downwards and is wider than the original function.

Option B: g(x)=−13x2g(x)=-\frac{1}{3}x^2

This option represents a transformation of the quadratic parent function by multiplying the function by −13-\frac{1}{3}. This means that the function is being scaled by a factor of −13-\frac{1}{3}, which will result in a parabola that opens downwards and is narrower than the original function.

Option C: g(x)=3x2g(x)=3x^2

This option represents a transformation of the quadratic parent function by multiplying the function by 33. This means that the function is being scaled by a factor of 33, which will result in a parabola that opens upwards and is wider than the original function.

Option D:

This option is not provided, so we will not consider it.

Conclusion

Based on the analysis of each option, we can conclude that the function g(x)g(x) that is a transformation of the quadratic parent function f(x)=x2f(x)=x^2 is:

  • Option A: g(x)=−3x2g(x)=-3x^2, which represents a transformation of the quadratic parent function by scaling it by a factor of −3-3.
  • Option B: g(x)=−13x2g(x)=-\frac{1}{3}x^2, which represents a transformation of the quadratic parent function by scaling it by a factor of −13-\frac{1}{3}.
  • Option C: g(x)=3x2g(x)=3x^2, which represents a transformation of the quadratic parent function by scaling it by a factor of 33.

Therefore, the correct answer is:

  • Option A: g(x)=−3x2g(x)=-3x^2 is the correct answer, as it represents a transformation of the quadratic parent function by scaling it by a factor of −3-3.

Final Answer

The final answer is: A\boxed{A}

Understanding the Quadratic Parent Function

The quadratic parent function, denoted as f(x)=x2f(x)=x^2, is a fundamental function in mathematics that represents a parabola opening upwards with its vertex at the origin (0,0). This function is a quadratic function, which means it can be written in the form f(x)=ax2+bx+cf(x)=ax^2+bx+c, where aa, bb, and cc are constants. In the case of the quadratic parent function, a=1a=1, b=0b=0, and c=0c=0.

Transformations of the Quadratic Parent Function

Transformations of the quadratic parent function involve changing the function in some way to create a new function. These transformations can include shifting, scaling, reflecting, and rotating the function. In this case, we are interested in finding a function g(x)g(x) that is a transformation of the quadratic parent function f(x)=x2f(x)=x^2.

Q&A

Q: What is the quadratic parent function?

A: The quadratic parent function is a fundamental function in mathematics that represents a parabola opening upwards with its vertex at the origin (0,0). It is denoted as f(x)=x2f(x)=x^2.

Q: What are transformations of the quadratic parent function?

A: Transformations of the quadratic parent function involve changing the function in some way to create a new function. These transformations can include shifting, scaling, reflecting, and rotating the function.

Q: How do you identify a transformation of the quadratic parent function?

A: To identify a transformation of the quadratic parent function, you need to examine the given function and determine how it differs from the original function. This can involve looking for changes in the coefficient of the x2x^2 term, the xx term, or the constant term.

Q: What is the difference between scaling and shifting a function?

A: Scaling a function involves changing the size of the function, while shifting a function involves moving it to a new location. For example, if you scale a function by a factor of 2, it will be twice as large as the original function. If you shift a function by 3 units to the right, it will be moved 3 units to the right of the original function.

Q: How do you determine the type of transformation that has occurred?

A: To determine the type of transformation that has occurred, you need to examine the function and look for changes in the coefficient of the x2x^2 term, the xx term, or the constant term. You can also use graphing software or a graphing calculator to visualize the function and determine the type of transformation that has occurred.

Q: What is the significance of the quadratic parent function?

A: The quadratic parent function is a fundamental function in mathematics that represents a parabola opening upwards with its vertex at the origin (0,0). It is used as a building block for more complex functions and is an important concept in algebra and calculus.

Conclusion

In conclusion, the function g(x)g(x) that is a transformation of the quadratic parent function f(x)=x2f(x)=x^2 is:

  • Option A: g(x)=−3x2g(x)=-3x^2, which represents a transformation of the quadratic parent function by scaling it by a factor of −3-3.
  • Option B: g(x)=−13x2g(x)=-\frac{1}{3}x^2, which represents a transformation of the quadratic parent function by scaling it by a factor of −13-\frac{1}{3}.
  • Option C: g(x)=3x2g(x)=3x^2, which represents a transformation of the quadratic parent function by scaling it by a factor of 33.

Therefore, the correct answer is:

  • Option A: g(x)=−3x2g(x)=-3x^2 is the correct answer, as it represents a transformation of the quadratic parent function by scaling it by a factor of −3-3.

Final Answer

The final answer is: A\boxed{A}