If The Parent Function $f(x)=|x|$ Is Shifted To The Left 5 Units And Shifted Up 2 Units To Create $g(x$\], What Is The Equation Of $g(x$\]?A. $g(x)=|x-5|-2$ B. $g(x)=|x+5|$ C. $g(x)=|x+5|-2$ D.
In mathematics, function transformations are essential concepts that help us understand how functions change when they are shifted, stretched, or compressed. In this article, we will explore the concept of shifting a function to the left and up, and how it affects the equation of the function.
Shifting a Function to the Left
When a function is shifted to the left, it means that the graph of the function is moved to the left by a certain number of units. In other words, the x-values of the function are decreased by that number of units. For example, if we have a function f(x) = |x| and we shift it to the left by 5 units, the new function will be g(x) = |x - 5|.
Shifting a Function Up
When a function is shifted up, it means that the graph of the function is moved up by a certain number of units. In other words, the y-values of the function are increased by that number of units. For example, if we have a function f(x) = |x| and we shift it up by 2 units, the new function will be g(x) = |x| + 2.
Combining Shifting Operations
When we combine shifting operations, we need to apply the operations in the correct order. In this case, we need to shift the function f(x) = |x| to the left by 5 units and then shift it up by 2 units. To do this, we first apply the left shift operation to get g(x) = |x - 5|, and then we apply the up shift operation to get g(x) = |x - 5| + 2.
Equation of g(x)
So, what is the equation of g(x)? Based on our previous discussion, we know that g(x) = |x - 5| + 2. However, we can simplify this equation by combining the absolute value and the constant term. We can do this by using the fact that |x - 5| = |-(5 - x)| = |-5 + x| = |5 - x|.
Simplifying the Equation
Using the fact that |x - 5| = |5 - x|, we can rewrite the equation g(x) = |x - 5| + 2 as g(x) = |5 - x| + 2. However, we can simplify this equation further by using the fact that |5 - x| = |-(x - 5)| = |-x + 5| = |x - 5|.
Final Answer
Using the fact that |x - 5| = |5 - x|, we can rewrite the equation g(x) = |x - 5| + 2 as g(x) = |5 - x| + 2. However, we can simplify this equation further by using the fact that |5 - x| = |x - 5|.
Therefore, the equation of g(x) is g(x) = |x - 5| + 2 = |5 - x| + 2 = |x + 5| - 2.
Conclusion
In conclusion, when we shift a function to the left and up, we need to apply the operations in the correct order. In this case, we first shift the function f(x) = |x| to the left by 5 units to get g(x) = |x - 5|, and then we shift it up by 2 units to get g(x) = |x - 5| + 2. Using the fact that |x - 5| = |5 - x|, we can rewrite the equation g(x) = |x - 5| + 2 as g(x) = |5 - x| + 2 = |x - 5| + 2 = |x + 5| - 2.
Answer
The final answer is g(x) = |x + 5| - 2.
References
- [1] "Function Transformations" by Khan Academy
- [2] "Shifting Functions" by Math Open Reference
- [3] "Absolute Value Functions" by Purplemath
Q&A: Function Transformations ==============================
In our previous article, we explored the concept of shifting a function to the left and up, and how it affects the equation of the function. In this article, we will answer some common questions related to function transformations.
Q: What is the difference between shifting a function to the left and shifting it to the right?
A: Shifting a function to the left means that the graph of the function is moved to the left by a certain number of units. In other words, the x-values of the function are decreased by that number of units. Shifting a function to the right means that the graph of the function is moved to the right by a certain number of units. In other words, the x-values of the function are increased by that number of units.
Q: How do I shift a function up or down?
A: To shift a function up, you need to add a certain number of units to the y-values of the function. To shift a function down, you need to subtract a certain number of units from the y-values of the function.
Q: What is the effect of shifting a function on its graph?
A: Shifting a function to the left or right will change the position of the graph, but it will not change the shape of the graph. Shifting a function up or down will change the position of the graph, but it will not change the shape of the graph.
Q: Can I shift a function more than once?
A: Yes, you can shift a function more than once. For example, you can shift a function to the left by 3 units and then shift it up by 2 units.
Q: How do I determine the equation of a shifted function?
A: To determine the equation of a shifted function, you need to apply the shifting operations to the original function. For example, if you have a function f(x) = |x| and you shift it to the left by 5 units, the new function will be g(x) = |x - 5|.
Q: What is the difference between a horizontal shift and a vertical shift?
A: A horizontal shift is a shift to the left or right, while a vertical shift is a shift up or down.
Q: Can I shift a function in both the horizontal and vertical directions?
A: Yes, you can shift a function in both the horizontal and vertical directions. For example, you can shift a function to the left by 3 units and then shift it up by 2 units.
Q: How do I determine the equation of a function that has been shifted in both the horizontal and vertical directions?
A: To determine the equation of a function that has been shifted in both the horizontal and vertical directions, you need to apply the shifting operations to the original function. For example, if you have a function f(x) = |x| and you shift it to the left by 5 units and then shift it up by 2 units, the new function will be g(x) = |x - 5| + 2.
Q: What is the effect of shifting a function on its domain and range?
A: Shifting a function will change its domain and range. For example, if you shift a function to the left by 3 units, its domain will be shifted to the left by 3 units, and its range will remain the same.
Q: Can I shift a function that is not a linear function?
A: Yes, you can shift a function that is not a linear function. For example, you can shift a quadratic function or a polynomial function.
Q: How do I determine the equation of a shifted function that is not a linear function?
A: To determine the equation of a shifted function that is not a linear function, you need to apply the shifting operations to the original function. For example, if you have a function f(x) = x^2 and you shift it to the left by 3 units, the new function will be g(x) = (x + 3)^2.
Conclusion
In conclusion, shifting a function is an important concept in mathematics that can be used to create new functions and to analyze the behavior of functions. By understanding how to shift a function, you can create new functions and analyze the behavior of functions in different ways.
References
- [1] "Function Transformations" by Khan Academy
- [2] "Shifting Functions" by Math Open Reference
- [3] "Absolute Value Functions" by Purplemath