Finding the Equation of the New Function

Oct 13, 2025 by talynt.org 84 views

Introduction

In mathematics, functions are used to describe the relationship between variables. A linear function is a type of function that can be represented in the form of f(x) = mx + b, where m is the slope and b is the y-intercept. The parent function f(x) = x is a linear function with a slope of 1 and a y-intercept of 0. Shifting this function down by 6 units results in a new function g(x). In this article, we will explore how to find the equation of the new function g(x) after shifting the parent function f(x) = x down by 6 units.

What is a Function Shift?

A function shift is a transformation that changes the position of a function on the coordinate plane. There are four types of function shifts: horizontal shifts, vertical shifts, horizontal stretches, and vertical stretches. In this case, we are dealing with a vertical shift, which changes the position of the function up or down.

Vertical Shifts

A vertical shift is a transformation that changes the position of a function up or down. When a function is shifted down by a certain number of units, the new function is obtained by subtracting that number from the original function. In this case, we are shifting the parent function f(x) = x down by 6 units.

Finding the Equation of the New Function

To find the equation of the new function g(x), we need to subtract 6 from the original function f(x) = x. This can be represented as:

g(x) = f(x) - 6 g(x) = x - 6

Therefore, the equation of the new function g(x) is g(x) = x - 6.

Conclusion

In conclusion, shifting the parent function f(x) = x down by 6 units results in a new function g(x). The equation of the new function g(x) is obtained by subtracting 6 from the original function f(x) = x. This can be represented as g(x) = x - 6.

Answer

The correct answer is B. g(x) = x - 6.

Additional Examples

Here are some additional examples of function shifts:

Shifting the parent function f(x) = x up by 3 units results in a new function g(x). The equation of the new function g(x) is g(x) = x + 3.
Shifting the parent function f(x) = x down by 2 units results in a new function g(x). The equation of the new function g(x) is g(x) = x - 2.
Shifting the parent function f(x) = x right by 4 units results in a new function g(x). The equation of the new function g(x) is g(x) = x - 4.
Shifting the parent function f(x) = x left by 1 unit results in a new function g(x). The equation of the new function g(x) is g(x) = x + 1.

Tips and Tricks

Here are some tips and tricks for working with function shifts:

When shifting a function up or down, subtract the number of units from the original function.
When shifting a function right or left, add or subtract the number of units from the original function.
Use the equation g(x) = f(x) + c to represent a vertical shift, where c is the number of units shifted.
Use the equation g(x) = f(x + d) to represent a horizontal shift, where d is the number of units shifted.

Common Mistakes

Here are some common mistakes to avoid when working with function shifts:

Not subtracting the correct number of units when shifting a function up or down.
Not adding or subtracting the correct number of units when shifting a function right or left.
Using the wrong equation to represent a function shift.
Not considering the direction of the shift when working with function shifts.

Conclusion

Q: What is a function shift?

A: A function shift is a transformation that changes the position of a function on the coordinate plane. There are four types of function shifts: horizontal shifts, vertical shifts, horizontal stretches, and vertical stretches.

Q: What is a vertical shift?

A: A vertical shift is a transformation that changes the position of a function up or down. When a function is shifted down by a certain number of units, the new function is obtained by subtracting that number from the original function.

Q: How do I find the equation of a new function after a vertical shift?

A: To find the equation of a new function after a vertical shift, you need to subtract the number of units shifted from the original function. For example, if the original function is f(x) = x and it is shifted down by 6 units, the new function is g(x) = x - 6.

Q: What is a horizontal shift?

A: A horizontal shift is a transformation that changes the position of a function right or left. When a function is shifted right by a certain number of units, the new function is obtained by adding that number to the original function.

Q: How do I find the equation of a new function after a horizontal shift?

A: To find the equation of a new function after a horizontal shift, you need to add the number of units shifted to the original function. For example, if the original function is f(x) = x and it is shifted right by 4 units, the new function is g(x) = x + 4.

Q: What is a horizontal stretch?

A: A horizontal stretch is a transformation that changes the width of a function. When a function is stretched horizontally by a certain factor, the new function is obtained by multiplying the original function by that factor.

Q: How do I find the equation of a new function after a horizontal stretch?

A: To find the equation of a new function after a horizontal stretch, you need to multiply the original function by the stretch factor. For example, if the original function is f(x) = x and it is stretched horizontally by a factor of 2, the new function is g(x) = 2x.

Q: What is a vertical stretch?

A: A vertical stretch is a transformation that changes the height of a function. When a function is stretched vertically by a certain factor, the new function is obtained by multiplying the original function by that factor.

Q: How do I find the equation of a new function after a vertical stretch?

A: To find the equation of a new function after a vertical stretch, you need to multiply the original function by the stretch factor. For example, if the original function is f(x) = x and it is stretched vertically by a factor of 3, the new function is g(x) = 3x.

Q: Can I combine multiple function shifts?

A: Yes, you can combine multiple function shifts. For example, if a function is shifted down by 2 units and then shifted right by 3 units, the new function is obtained by subtracting 2 from the original function and then adding 3 to the result.

Q: How do I determine the order of function shifts?

A: To determine the order of function shifts, you need to consider the direction of the shift. If a function is shifted down and then shifted right, the down shift should be performed first. If a function is shifted right and then shifted down, the right shift should be performed first.

Q: Can I use function shifts to solve real-world problems?

A: Yes, you can use function shifts to solve real-world problems. For example, if you are modeling the growth of a population, you can use a function shift to represent a change in the population size over time.

Q: What are some common mistakes to avoid when working with function shifts?

A: Some common mistakes to avoid when working with function shifts include:

Not subtracting the correct number of units when shifting a function up or down.
Not adding or subtracting the correct number of units when shifting a function right or left.
Using the wrong equation to represent a function shift.
Not considering the direction of the shift when working with function shifts.

Conclusion

In conclusion, function shifts are an important concept in mathematics that can be used to describe the relationship between variables. By understanding how to shift functions up, down, right, and left, we can create new functions and solve problems in a variety of contexts. Remember to use the correct equations and techniques when working with function shifts, and avoid common mistakes to ensure accurate results.