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

Feb 28, 2025 by ADMIN 181 views

**Horizontal Translation of Quadratic Functions: Understanding the Correct Answer**

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, and determine which function is a horizontal translation of the parent quadratic function.

Understanding Horizontal Translation

Horizontal translation, also known as horizontal shift, is a transformation that moves a function to the left or right. In the case of a quadratic function, a horizontal translation will change the position of the vertex of the parabola. The parent quadratic function, f(x) = x^2, has a vertex at (0, 0). A horizontal translation will move the vertex to a new position, either to the left or to the right.

Analyzing the Options

Let's analyze each option to determine which function is a horizontal translation of the parent quadratic function.

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

This function can be rewritten as g(x) = (x - 4)(x - 4) = x^2 - 8x + 16. This function is a horizontal translation of the parent quadratic function, f(x) = x^2, by 4 units to the left. The vertex of the parabola has been moved from (0, 0) to (-4, 0).

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

This function is not a horizontal translation of the parent quadratic function, f(x) = x^2. Instead, it is a vertical stretch of the parent function by a factor of 4. The vertex of the parabola remains at (0, 0).

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

This function is not a horizontal translation of the parent quadratic function, f(x) = x^2. Instead, it is a reflection of the parent function across the x-axis. The vertex of the parabola remains at (0, 0).

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

This function can be rewritten as j(x) = (x - 0)(x - 0) - 4 = x^2 - 4. This function is a horizontal translation of the parent quadratic function, f(x) = x^2, by 0 units to the left. However, the constant term -4 has been added, which will shift the parabola down by 4 units. The vertex of the parabola has been moved from (0, 0) to (0, -4).

Conclusion

Based on the analysis of each option, we can conclude that Option A: g(x) = (x - 4)^2 is the correct answer. This function is a horizontal translation of the parent quadratic function, f(x) = x^2, by 4 units to the left.

Understanding the Concept

Horizontal translation is an important concept in mathematics, particularly in algebra and geometry. It is used to model various real-world phenomena, such as the motion of objects under the influence of gravity or the growth of populations over time. By understanding the concept of horizontal translation, we can better analyze and solve problems involving quadratic functions.

Real-World Applications

Horizontal translation has numerous real-world applications, including:

Physics: Horizontal translation is used to model the motion of objects under the influence of gravity, such as the trajectory of a projectile or the motion of a pendulum.
Biology: Horizontal translation is used to model the growth of populations over time, such as the growth of a bacterial culture or the spread of a disease.
Economics: Horizontal translation is used to model the behavior of economic systems, such as the movement of prices over time or the growth of a company's revenue.

Conclusion

In conclusion, horizontal translation is an important concept in mathematics, particularly in algebra and geometry. By understanding the concept of horizontal translation, we can better analyze and solve problems involving quadratic functions. The correct answer to the question is Option A: g(x) = (x - 4)^2, which is a horizontal translation of the parent quadratic function, f(x) = x^2, by 4 units to the left.

Introduction

Quadratic functions are a fundamental concept in mathematics, particularly in algebra and geometry. They are used to model various real-world phenomena, such as the motion of objects under the influence of gravity or the growth of populations over time. In this article, we will provide a comprehensive Q&A guide to quadratic functions, covering topics such as the definition of quadratic functions, the standard form of quadratic functions, and the properties of quadratic functions.

Q&A

Q: What is a quadratic function?

A: 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.

Q: What is the standard form of a quadratic function?

A: The standard form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants. This form is also known as the general form of a quadratic function.

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the point on the graph of the function where the function changes from decreasing to increasing or vice versa. The vertex is also known as the minimum or maximum point of the function.

Q: How do you find the vertex of a quadratic function?

A: To find the vertex of a quadratic function, you can use the formula x = -b/2a, where a and b are the coefficients of the quadratic function. This formula gives you the x-coordinate of the vertex.

Q: What is the axis of symmetry of a quadratic function?

A: The axis of symmetry of a quadratic function is a vertical line that passes through the vertex of the function. The axis of symmetry is also known as the line of symmetry.

Q: How do you find the axis of symmetry of a quadratic function?

A: To find the axis of symmetry of a quadratic function, you can use the formula x = -b/2a, where a and b are the coefficients of the quadratic function. This formula gives you the x-coordinate of the axis of symmetry.

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

A: A quadratic function is a polynomial function of degree two, while a linear function is a polynomial function of degree one. Quadratic functions have a parabolic shape, while linear functions have a straight line shape.

Q: Can you give an example of a quadratic function?

A: Yes, an example of a quadratic function is f(x) = x^2 + 3x + 2. This function is a quadratic function because it has a degree of two.

Q: Can you give an example of a linear function?

A: Yes, an example of a linear function is f(x) = 2x + 3. This function is a linear function because it has a degree of one.

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

A: The vertex form of a quadratic function is f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the function.

Q: How do you convert a quadratic function from standard form to vertex form?

A: To convert a quadratic function from standard form to vertex form, you can complete the square. This involves rewriting the quadratic function in the form f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the function.

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

A: A quadratic function is a polynomial function of degree two, while a polynomial function is a function that can be written in the form f(x) = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0, where a_n, a_(n-1), ..., a_1, a_0 are constants and n is a positive integer.

Q: Can you give an example of a polynomial function?

A: Yes, an example of a polynomial function is f(x) = 2x^3 + 3x^2 - 4x + 1. This function is a polynomial function because it can be written in the form f(x) = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0.

Conclusion

In conclusion, quadratic functions are a fundamental concept in mathematics, particularly in algebra and geometry. They are used to model various real-world phenomena, such as the motion of objects under the influence of gravity or the growth of populations over time. By understanding the properties and behavior of quadratic functions, we can better analyze and solve problems involving quadratic functions.