Which Function Is A Horizontal Translation Of The Parent Quadratic Function, $f(x)=x^2$?A. $k(x)=-x^2$ B. $g(x)=(x-4)^2$ C. $j(x)=x^2-4$ D. $h(x)=4x^2$

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Introduction

In mathematics, quadratic functions are a fundamental concept in algebra and are used to model various real-world phenomena. A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants. In this article, we will focus on the horizontal translation of the parent quadratic function, f(x) = x^2.

Understanding Horizontal Translation

Horizontal translation, also known as horizontal shift, is a transformation that moves a function's graph horizontally to the left or right. This type of transformation changes the position of the function's graph without changing its shape or size. In the case of quadratic functions, horizontal translation can be achieved by adding or subtracting a constant value from the variable x.

Parent Quadratic Function

The parent quadratic function, f(x) = x^2, is a basic quadratic function with a leading coefficient of 1 and no linear or constant terms. This function has a minimum point at (0, 0) and opens upwards. The graph of this function is a parabola that is symmetric about the y-axis.

Horizontal Translation of the Parent Quadratic Function

To find a horizontal translation of the parent quadratic function, f(x) = x^2, we need to add or subtract a constant value from the variable x. The general form of a horizontally translated quadratic function is f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.

Analyzing the Options

Let's analyze the given options to determine which one is a horizontal translation of the parent quadratic function, f(x) = x^2.

Option A: k(x) = -x^2

This option is not a horizontal translation of the parent quadratic function. The negative sign in front of x^2 changes the direction of the parabola, but it does not change its position.

Option B: g(x) = (x - 4)^2

This option is a horizontal translation of the parent quadratic function. The value inside the parentheses, x - 4, is a horizontal shift of 4 units to the right. This means that the graph of g(x) is the same as the graph of f(x) = x^2, but shifted 4 units to the right.

Option C: j(x) = x^2 - 4

This option is not a horizontal translation of the parent quadratic function. The value -4 is subtracted from x^2, but it does not change the position of the graph.

Option D: h(x) = 4x^2

This option is not a horizontal translation of the parent quadratic function. The value 4 is multiplied with x^2, but it does not change the position of the graph.

Conclusion

In conclusion, the correct answer is option B: g(x) = (x - 4)^2. This option is a horizontal translation of the parent quadratic function, f(x) = x^2, by 4 units to the right.

Understanding the Concept

Horizontal translation is an important concept in mathematics, and it has many real-world applications. By understanding how to translate quadratic functions horizontally, we can model various phenomena, such as the trajectory of a projectile or the growth of a population.

Real-World Applications

Horizontal translation has many real-world applications, such as:

  • Projectile Motion: The trajectory of a projectile can be modeled using quadratic functions. By applying horizontal translation, we can determine the position of the projectile at any given time.
  • Population Growth: The growth of a population can be modeled using quadratic functions. By applying horizontal translation, we can determine the population size at any given time.
  • Engineering: Horizontal translation is used in engineering to design and optimize systems, such as bridges and buildings.

Final Thoughts

In conclusion, horizontal translation is an important concept in mathematics that has many real-world applications. By understanding how to translate quadratic functions horizontally, we can model various phenomena and solve real-world problems.

Introduction

In our previous article, we discussed the concept of horizontal translation of quadratic functions. We also analyzed the options to determine which one is a horizontal translation of the parent quadratic function, f(x) = x^2. In this article, we will answer some frequently asked questions about quadratic function horizontal translation.

Q&A

Q: What is horizontal translation?

A: Horizontal translation, also known as horizontal shift, is a transformation that moves a function's graph horizontally to the left or right. This type of transformation changes the position of the function's graph without changing its shape or size.

Q: How do I apply horizontal translation to a quadratic function?

A: To apply horizontal translation to a quadratic function, you need to add or subtract a constant value from the variable x. The general form of a horizontally translated quadratic function is f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.

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

A: Horizontal translation moves a function's graph horizontally to the left or right, while vertical translation moves a function's graph vertically up or down. Vertical translation changes the position of the function's graph without changing its shape or size.

Q: Can I apply horizontal translation to any type of function?

A: Yes, you can apply horizontal translation to any type of function, including linear, quadratic, and polynomial functions.

Q: How do I determine the vertex of a horizontally translated quadratic function?

A: To determine the vertex of a horizontally translated quadratic function, you need to find the value of h in the equation f(x) = a(x - h)^2 + k. The vertex is located at the point (h, k).

Q: Can I apply horizontal translation to a function with a negative leading coefficient?

A: Yes, you can apply horizontal translation to a function with a negative leading coefficient. However, the direction of the translation will be opposite to the direction of the translation for a function with a positive leading coefficient.

Q: How do I graph a horizontally translated quadratic function?

A: To graph a horizontally translated quadratic function, you need to graph the original function and then shift it horizontally by the specified amount.

Q: Can I apply horizontal translation to a function with a fractional exponent?

A: Yes, you can apply horizontal translation to a function with a fractional exponent. However, the translation will be applied to the entire function, including the fractional exponent.

Conclusion

In conclusion, horizontal translation is an important concept in mathematics that has many real-world applications. By understanding how to translate quadratic functions horizontally, we can model various phenomena and solve real-world problems. We hope that this article has answered some of the frequently asked questions about quadratic function horizontal translation.

Final Thoughts

Horizontal translation is a powerful tool that can be used to model and solve a wide range of problems. By understanding how to apply horizontal translation to quadratic functions, we can gain a deeper understanding of the underlying mathematics and develop new skills and techniques for solving problems.

Real-World Applications

Horizontal translation has many real-world applications, such as:

  • Projectile Motion: The trajectory of a projectile can be modeled using quadratic functions. By applying horizontal translation, we can determine the position of the projectile at any given time.
  • Population Growth: The growth of a population can be modeled using quadratic functions. By applying horizontal translation, we can determine the population size at any given time.
  • Engineering: Horizontal translation is used in engineering to design and optimize systems, such as bridges and buildings.

Additional Resources

If you want to learn more about quadratic function horizontal translation, we recommend the following resources:

  • Textbooks: There are many textbooks available that cover quadratic function horizontal translation, including "Algebra and Trigonometry" by Michael Sullivan and "Calculus" by Michael Spivak.
  • Online Resources: There are many online resources available that cover quadratic function horizontal translation, including Khan Academy, MIT OpenCourseWare, and Wolfram Alpha.
  • Software: There are many software programs available that can be used to graph and analyze quadratic functions, including Graphing Calculator, Mathematica, and MATLAB.