Select The Correct Answer.Which Function Is A Horizontal Translation Of The Parent Quadratic Function, F ( X ) = X 2 F(x) = X^2 F ( X ) = X 2 ?A. K ( X ) = − X 2 K(x) = -x^2 K ( X ) = − X 2 B. H ( X ) = 4 X 2 H(x) = 4x^2 H ( X ) = 4 X 2 C. G ( X ) = ( X − 4 ) 2 G(x) = (x-4)^2 G ( X ) = ( X − 4 ) 2 D. J ( X ) = X 2 − 4 J(x) = X^2 - 4 J ( X ) = X 2 − 4

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Introduction

In mathematics, quadratic functions are a fundamental concept in algebra and geometry. 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)=ax2+bx+cf(x) = ax^2 + bx + c, where aa, bb, and cc are constants. In this article, we will focus on the horizontal translation of the parent quadratic function, f(x)=x2f(x) = x^2. We will explore the concept of horizontal translation and determine which function is a horizontal translation of the parent quadratic function.

What is Horizontal Translation?

Horizontal translation is a transformation that shifts the graph of a function horizontally. It is a type of transformation that changes the position of the graph along the x-axis. When a function is translated horizontally, the graph moves to the left or right, but the shape of the graph remains the same.

Parent Quadratic Function

The parent quadratic function is f(x)=x2f(x) = x^2. This function has a parabolic shape and opens upwards. The graph of this function is a U-shaped curve that is symmetric about the y-axis.

Horizontal Translation of the Parent Quadratic Function

To find the horizontal translation of the parent quadratic function, we need to shift the graph of f(x)=x2f(x) = x^2 horizontally. This can be done by adding or subtracting a constant value from the variable xx. If we add a constant value to xx, the graph shifts to the right. If we subtract a constant value from xx, the graph shifts to the left.

Option A: k(x)=−x2k(x) = -x^2

Option A is k(x)=−x2k(x) = -x^2. This function is a reflection of the parent quadratic function across the x-axis, but it is not a horizontal translation. The graph of this function is a U-shaped curve that opens downwards.

Option B: h(x)=4x2h(x) = 4x^2

Option B is h(x)=4x2h(x) = 4x^2. This function is a vertical stretch of the parent quadratic function, but it is not a horizontal translation. The graph of this function is a U-shaped curve that is four times as tall as the parent quadratic function.

Option C: g(x)=(x−4)2g(x) = (x-4)^2

Option C is g(x)=(x−4)2g(x) = (x-4)^2. This function is a horizontal translation of the parent quadratic function. The graph of this function is a U-shaped curve that is shifted 4 units to the right.

Option D: j(x)=x2−4j(x) = x^2 - 4

Option D is j(x)=x2−4j(x) = x^2 - 4. This function is a horizontal translation of the parent quadratic function. The graph of this function is a U-shaped curve that is shifted 2 units to the left.

Conclusion

In conclusion, the correct answer is Option C: g(x)=(x−4)2g(x) = (x-4)^2. This function is a horizontal translation of the parent quadratic function, f(x)=x2f(x) = x^2. The graph of this function is a U-shaped curve that is shifted 4 units to the right.

Understanding Horizontal Translations in Quadratic Functions: Key Takeaways

  • Horizontal translation is a transformation that shifts the graph of a function horizontally.
  • The parent quadratic function is f(x)=x2f(x) = x^2.
  • To find the horizontal translation of the parent quadratic function, we need to shift the graph of f(x)=x2f(x) = x^2 horizontally.
  • Option C: g(x)=(x−4)2g(x) = (x-4)^2 is the correct answer. This function is a horizontal translation of the parent quadratic function.

Frequently Asked Questions

Q: What is horizontal translation?

A: Horizontal translation is a transformation that shifts the graph of a function horizontally.

Q: What is the parent quadratic function?

A: The parent quadratic function is f(x)=x2f(x) = x^2.

Q: How do we find the horizontal translation of the parent quadratic function?

A: To find the horizontal translation of the parent quadratic function, we need to shift the graph of f(x)=x2f(x) = x^2 horizontally.

Q: Which function is a horizontal translation of the parent quadratic function?

A: Option C: g(x)=(x−4)2g(x) = (x-4)^2 is the correct answer. This function is a horizontal translation of the parent quadratic function.

References

Introduction

Quadratic functions are a fundamental concept in algebra and geometry. A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. In this article, we will provide a comprehensive Q&A guide on quadratic functions, covering topics such as the parent quadratic function, horizontal translation, and more.

Q&A Guide

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)=ax2+bx+cf(x) = ax^2 + bx + c, where aa, bb, and cc are constants.

Q: What is the parent quadratic function?

A: The parent quadratic function is f(x)=x2f(x) = x^2. This function has a parabolic shape and opens upwards. The graph of this function is a U-shaped curve that is symmetric about the y-axis.

Q: What is horizontal translation?

A: Horizontal translation is a transformation that shifts the graph of a function horizontally. It is a type of transformation that changes the position of the graph along the x-axis. When a function is translated horizontally, the graph moves to the left or right, but the shape of the graph remains the same.

Q: How do we find the horizontal translation of the parent quadratic function?

A: To find the horizontal translation of the parent quadratic function, we need to shift the graph of f(x)=x2f(x) = x^2 horizontally. This can be done by adding or subtracting a constant value from the variable xx. If we add a constant value to xx, the graph shifts to the right. If we subtract a constant value from xx, the graph shifts to the left.

Q: Which function is a horizontal translation of the parent quadratic function?

A: Option C: g(x)=(x−4)2g(x) = (x-4)^2 is the correct answer. This function is a horizontal translation of the parent quadratic function.

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. The general form of a linear function is f(x)=mx+bf(x) = mx + b, where mm and bb are constants.

Q: How do we graph a quadratic function?

A: To graph a quadratic function, we can use the following steps:

  1. Find the vertex of the parabola.
  2. Determine the direction of the parabola (upward or downward).
  3. Plot the vertex and the points on the parabola.
  4. Draw a smooth curve through the points.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point on the parabola that is the lowest or highest point. It is the point where the parabola changes direction.

Q: How do we find the vertex of a parabola?

A: To find the vertex of a parabola, we can use the following formula:

x=−b2ax = -\frac{b}{2a}

This formula gives us the x-coordinate of the vertex. To find the y-coordinate, we can plug the x-coordinate into the equation of the parabola.

Q: What is the axis of symmetry of a parabola?

A: The axis of symmetry of a parabola is a vertical line that passes through the vertex of the parabola. It is a line of symmetry that divides the parabola into two equal parts.

Q: How do we find the axis of symmetry of a parabola?

A: To find the axis of symmetry of a parabola, we can use the following formula:

x=−b2ax = -\frac{b}{2a}

This formula gives us the x-coordinate of the axis of symmetry. The axis of symmetry is a vertical line that passes through the point (x,y)(x, y).

Conclusion

In conclusion, quadratic functions are a fundamental concept in algebra and geometry. A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. In this article, we have provided a comprehensive Q&A guide on quadratic functions, covering topics such as the parent quadratic function, horizontal translation, and more.

Frequently Asked Questions

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)=ax2+bx+cf(x) = ax^2 + bx + c, where aa, bb, and cc are constants.

Q: What is the parent quadratic function?

A: The parent quadratic function is f(x)=x2f(x) = x^2. This function has a parabolic shape and opens upwards. The graph of this function is a U-shaped curve that is symmetric about the y-axis.

Q: What is horizontal translation?

A: Horizontal translation is a transformation that shifts the graph of a function horizontally. It is a type of transformation that changes the position of the graph along the x-axis. When a function is translated horizontally, the graph moves to the left or right, but the shape of the graph remains the same.

Q: How do we find the horizontal translation of the parent quadratic function?

A: To find the horizontal translation of the parent quadratic function, we need to shift the graph of f(x)=x2f(x) = x^2 horizontally. This can be done by adding or subtracting a constant value from the variable xx. If we add a constant value to xx, the graph shifts to the right. If we subtract a constant value from xx, the graph shifts to the left.

References