If G ( X ) = 4 X 2 − 16 G(x)=4x^2-16 G ( X ) = 4 X 2 − 16 Were Shifted 5 Units To The Right And 2 Units Down, What Would The New Equation Be?A. H ( X ) = 4 ( X + 5 ) 2 − 17 H(x)=4(x+5)^2-17 H ( X ) = 4 ( X + 5 ) 2 − 17 B. H ( X ) = 4 ( X − 5 ) 2 − 18 H(x)=4(x-5)^2-18 H ( X ) = 4 ( X − 5 ) 2 − 18 C. H ( X ) = 4 ( X − 5 ) 2 + 14 H(x)=4(x-5)^2+14 H ( X ) = 4 ( X − 5 ) 2 + 14 D. H ( X ) = 4 ( X − 18 ) 2 − 5 H(x)=4(x-18)^2-5 H ( X ) = 4 ( X − 18 ) 2 − 5

Introduction

In mathematics, shifting functions is a fundamental concept that involves changing the position of a function on the coordinate plane. This can be achieved by shifting the function horizontally or vertically. In this article, we will explore the concept of shifting functions and how it can be applied to a given quadratic function.

What is Shifting a Function?

Shifting a function involves changing its position on the coordinate plane. This can be done by shifting the function horizontally or vertically. When a function is shifted horizontally, its x-coordinate changes, while its y-coordinate remains the same. On the other hand, when a function is shifted vertically, its y-coordinate changes, while its x-coordinate remains the same.

Horizontal Shifting

Horizontal shifting involves changing the x-coordinate of a function. This can be achieved by adding or subtracting a constant value to the x-variable of the function. If a function is shifted to the right, the x-variable is increased by a constant value. Conversely, if a function is shifted to the left, the x-variable is decreased by a constant value.

Vertical Shifting

Vertical shifting involves changing the y-coordinate of a function. This can be achieved by adding or subtracting a constant value to the function. If a function is shifted up, the y-variable is increased by a constant value. Conversely, if a function is shifted down, the y-variable is decreased by a constant value.

Shifting the Given Function

The given function is $g(x)=4x^2-16$. To shift this function 5 units to the right and 2 units down, we need to apply both horizontal and vertical shifting.

Horizontal Shifting

To shift the function 5 units to the right, we need to add 5 to the x-variable of the function. This can be achieved by replacing $x$ with $(x-5)$ in the function.

g(x-5) = 4(x-5)^2 - 16

Vertical Shifting

To shift the function 2 units down, we need to subtract 2 from the function.

h(x) = g(x-5) - 2

Substituting the expression for $g(x-5)$ into the equation above, we get:

h(x) = 4(x-5)^2 - 16 - 2

Simplifying the equation above, we get:

h(x) = 4(x-5)^2 - 18

Conclusion

In conclusion, shifting a function involves changing its position on the coordinate plane. This can be achieved by shifting the function horizontally or vertically. By applying both horizontal and vertical shifting to the given function $g(x)=4x^2-16$, we get the new equation $h(x)=4(x-5)^2-18$.

Answer

The correct answer is B. $h(x)=4(x-5)^2-18$.

Discussion

This problem requires a good understanding of shifting functions and how it can be applied to a given quadratic function. The concept of shifting functions is a fundamental concept in mathematics and is used extensively in various fields such as physics, engineering, and economics.

Related Topics

• Graphing Functions: Graphing functions is an essential concept in mathematics that involves representing functions on a coordinate plane.
• Function Operations: Function operations involve performing arithmetic operations on functions.
• Inverse Functions: Inverse functions involve finding the inverse of a function.

References

• Mathematics for Dummies: This book provides a comprehensive introduction to mathematics and covers various topics such as algebra, geometry, and calculus.
• Calculus for Dummies: This book provides a comprehensive introduction to calculus and covers various topics such as limits, derivatives, and integrals.

Further Reading

• Shifting Functions: This article provides a comprehensive introduction to shifting functions and covers various topics such as horizontal and vertical shifting.
• Graphing Functions: This article provides a comprehensive introduction to graphing functions and covers various topics such as graphing quadratic functions and graphing polynomial functions.
  Shifting Functions in Mathematics: Q&A =====================================

Introduction

In our previous article, we explored the concept of shifting functions and how it can be applied to a given quadratic function. In this article, we will answer some frequently asked questions related to shifting functions.

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

A: Horizontal shifting involves changing the x-coordinate of a function, while vertical shifting involves changing the y-coordinate of a function.

Q: How do I shift a function to the right?

A: To shift a function to the right, you need to add a constant value to the x-variable of the function. For example, if you want to shift the function $f(x)$ 3 units to the right, you need to replace $x$ with $(x-3)$ in the function.

Q: How do I shift a function down?

A: To shift a function down, you need to subtract a constant value from the function. For example, if you want to shift the function $f(x)$ 2 units down, you need to subtract 2 from the function.

Q: Can I shift a function both horizontally and vertically?

A: Yes, you can shift a function both horizontally and vertically. For example, if you want to shift the function $f(x)$ 3 units to the right and 2 units down, you need to replace $x$ with $(x-3)$ in the function and subtract 2 from the function.

Q: How do I determine the new equation of a shifted function?

A: To determine the new equation of a shifted function, you need to apply the horizontal and vertical shifting rules. For example, if you want to shift the function $f(x)=x^2+2x-3$ 2 units to the right and 3 units down, you need to replace $x$ with $(x-2)$ in the function and subtract 3 from the function.

Q: Can I shift a function with a negative exponent?

A: Yes, you can shift a function with a negative exponent. However, you need to be careful when applying the horizontal and vertical shifting rules. For example, if you want to shift the function $f(x)=\frac{1}{x}$ 2 units to the right, you need to replace $x$ with $(x-2)$ in the function.

Q: How do I graph a shifted function?

A: To graph a shifted function, you need to apply the horizontal and vertical shifting rules to the original graph of the function. For example, if you want to graph the function $f(x)=x^2+2x-3$ shifted 2 units to the right and 3 units down, you need to shift the original graph of the function 2 units to the right and 3 units down.

Q: Can I shift a function with a radical?

A: Yes, you can shift a function with a radical. However, you need to be careful when applying the horizontal and vertical shifting rules. For example, if you want to shift the function $f(x)=\sqrt{x}$ 2 units to the right, you need to replace $x$ with $(x-2)$ in the function.

Conclusion

In conclusion, shifting functions is a fundamental concept in mathematics that involves changing the position of a function on the coordinate plane. By understanding the horizontal and vertical shifting rules, you can apply these rules to a given function to determine the new equation of the shifted function.

Related Topics

• Graphing Functions: Graphing functions is an essential concept in mathematics that involves representing functions on a coordinate plane.
• Function Operations: Function operations involve performing arithmetic operations on functions.
• Inverse Functions: Inverse functions involve finding the inverse of a function.

References

• Mathematics for Dummies: This book provides a comprehensive introduction to mathematics and covers various topics such as algebra, geometry, and calculus.
• Calculus for Dummies: This book provides a comprehensive introduction to calculus and covers various topics such as limits, derivatives, and integrals.

Further Reading

• Shifting Functions: This article provides a comprehensive introduction to shifting functions and covers various topics such as horizontal and vertical shifting.
• Graphing Functions: This article provides a comprehensive introduction to graphing functions and covers various topics such as graphing quadratic functions and graphing polynomial functions.