Answered

Which function is a horizontal translation of the parent quadratic function, $f(x)=x^2$?


A$k(x)=-x^2$
B$h(x)=4 x^2$
C$g(x)=(x-4)^2$
D$j(x)=x^2-4$

Verified Answer :

  • The parent quadratic function is \(f(x) = x^2\).
  • Horizontal translation occurs when the input \(x\) is replaced by \(x-h\), resulting in the form \(f(x-h) = (x-h)^2\).
  • Comparing the given options, \(g(x) = (x-4)^2\) fits the form of a horizontal translation where \(h=4\).
  • The final answer is \(\boxed{g(x)=(x-4)^2}\).

Explanation

  1. Analysis of Function Transformations

The parent quadratic function is defined as \(f(x) = x^2\). Horizontal and vertical transformations alter the position or shape of the graph. A horizontal translation of a function \(f(x)\) is represented by \(f(x - h)\), where \(h\) is the horizontal shift. A vertical translation is represented by \(f(x) + k\), a vertical reflection by \(-f(x)\), and a vertical stretch or compression by \(a \cdot f(x)\). We examine each of the provided options to identify the horizontal translation.

  1. Evaluating the Options

We evaluate each function:

  • \(k(x) = -x^2\): This is the negative of the parent function, representing a reflection across the x-axis.
  • \(h(x) = 4x^2\): This is the parent function multiplied by a constant factor of 4, representing a vertical stretch by a factor of 4.
  • \(g(x) = (x-4)^2\): This is in the form of \(f(x - 4)\), where \(h = 4\). This indicates a horizontal translation of the parent function \(f(x) = x^2\) to the right by 4 units.
  • \(j(x) = x^2 - 4\): This is the parent function with a constant subtracted, representing a vertical translation down by 4 units.
  1. Conclusion

Comparing the given options to the definition of a horizontal translation, we find that the function \(g(x) = (x-4)^2\) is the only one that shifts the parent function horizontally. Thus, it is the correct answer.

Examples

Horizontal translations are frequently used in engineering and design when mapping the trajectory of objects or aligning components. For instance, if you have a software program that simulates the parabolic path of a ball thrown from the origin, \(\(f(x) = x^2\)\), and you need to adjust the starting point 4 units to the right, you apply the horizontal translation \(\(g(x) = (x-4)^2\)\). Understanding these transformations allows for precise control over the positioning and movement of shapes and functions in various modeling applications.

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