Shifting Graphs to the Left

Oct 13, 2025 by miodeya.com 27 views

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Introduction

In mathematics, graph shifts are a fundamental concept in understanding the behavior of functions. When a graph is shifted to the left or right, it affects the function's equation, and understanding this concept is crucial for solving problems in algebra and calculus. In this article, we will explore how to shift a graph to the left and determine the correct function that describes the new graph.

Shifting Graphs to the Left

When a graph is shifted to the left, it means that the graph is moved to the left by a certain number of units. This shift affects the function's equation, and we need to understand how to represent this shift mathematically.

Horizontal Shifts

A horizontal shift to the left by a units can be represented by replacing x with (x + a) in the function's equation. This means that the graph is shifted to the left by a units, and the new function's equation is:

f(x + a)

Applying Horizontal Shifts to the Function f(x) = |x|

Now, let's apply the concept of horizontal shifts to the function f(x) = |x|. If the graph of f(x) = |x| is shifted to the left 2 units, we need to replace x with (x + 2) in the function's equation.

f(x + 2) = |x + 2|

This means that the new function's equation is g(x) = |x + 2|.

Evaluating the Answer Choices

Now that we have determined the correct function that describes the graph of the new function, let's evaluate the answer choices.

A. g(x) = |x| - 2

This option is incorrect because it represents a vertical shift, not a horizontal shift.

B. g(x) = |x - 2|

This option is incorrect because it represents a horizontal shift to the right, not to the left.

C. g(x) = |x + 2|

This option is correct because it represents a horizontal shift to the left by 2 units.

D. g(x) = |x| + 2

This option is incorrect because it represents a vertical shift, not a horizontal shift.

Conclusion

In conclusion, when a graph is shifted to the left, we need to replace x with (x + a) in the function's equation, where a is the number of units the graph is shifted to the left. Applying this concept to the function f(x) = |x|, we find that the new function's equation is g(x) = |x + 2|. This means that option C is the correct answer.

Final Answer

The final answer is C. g(x) = |x + 2|.

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Introduction

In our previous article, we explored how to shift a graph to the left and determine the correct function that describes the new graph. In this article, we will provide a Q&A guide to help you understand graph shifts in functions.

Q: What is a horizontal shift in a function?

A: A horizontal shift in a function is a transformation that moves the graph of the function to the left or right. When a graph is shifted to the left, it means that the graph is moved to the left by a certain number of units.

Q: How do I represent a horizontal shift in a function's equation?

A: To represent a horizontal shift in a function's equation, you need to replace x with (x + a), where a is the number of units the graph is shifted to the left.

Q: What is the difference between a horizontal shift and a vertical shift?

A: A horizontal shift moves the graph of a function to the left or right, while a vertical shift moves the graph of a function up or down.

Q: How do I determine the correct function that describes the graph of a new function after a horizontal shift?

A: To determine the correct function that describes the graph of a new function after a horizontal shift, you need to replace x with (x + a) in the original function's equation, where a is the number of units the graph is shifted to the left.

Q: What is the equation of the new function if the graph of f(x) = |x| is shifted to the left 2 units?

A: The equation of the new function is g(x) = |x + 2|.

Q: What is the equation of the new function if the graph of f(x) = x^2 is shifted to the right 3 units?

A: To shift the graph of f(x) = x^2 to the right 3 units, you need to replace x with (x - 3) in the original function's equation. The new function's equation is g(x) = (x - 3)^2.

Q: What is the equation of the new function if the graph of f(x) = 2x is shifted down 4 units?

A: To shift the graph of f(x) = 2x down 4 units, you need to subtract 4 from the original function's equation. The new function's equation is g(x) = 2x - 4.

Q: What is the equation of the new function if the graph of f(x) = x^3 is shifted to the left 2 units and then up 3 units?

A: To shift the graph of f(x) = x^3 to the left 2 units, you need to replace x with (x + 2) in the original function's equation. The new function's equation is g(x) = (x + 2)^3. To shift the graph up 3 units, you need to add 3 to the new function's equation. The final equation of the new function is g(x) = (x + 2)^3 + 3.

Conclusion

In conclusion, graph shifts in functions are an important concept in mathematics. By understanding how to shift a graph to the left or right, you can determine the correct function that describes the graph of a new function. We hope this Q&A guide has helped you understand graph shifts in functions.

Final Tips

Always replace x with (x + a) in the original function's equation to represent a horizontal shift.
Remember that a horizontal shift moves the graph of a function to the left or right, while a vertical shift moves the graph of a function up or down.
Practice, practice, practice! The more you practice shifting graphs, the more comfortable you will become with the concept.