If The Parent Function F ( X ) = ∣ X ∣ F(x) = |x| F ( X ) = ∣ X ∣ Is Shifted To The Left 5 Units And Shifted Up 2 Units To Create G ( X G(x G ( X ], What Is The Equation Of G ( X G(x G ( X ]?

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 $g(x)$?

Understanding the Parent Function

The parent function $f(x) = |x|$ is a fundamental function in mathematics, representing the absolute value of a variable $x$. This function has a V-shaped graph with its vertex at the origin (0, 0). The absolute value function is symmetric about the y-axis and has a minimum value of 0.

Shifting the Parent Function

When a function is shifted to the left or right, it means that the graph of the function is moved horizontally. Similarly, when a function is shifted up or down, it means that the graph of the function is moved vertically. In this problem, we are asked to shift the parent function $f(x) = |x|$ to the left by 5 units and up by 2 units to create a new function $g(x)$.

Shifting to the Left by 5 Units

To shift the parent function $f(x) = |x|$ to the left by 5 units, we need to replace $x$ with $(x + 5)$. This is because when we shift the graph of a function to the left, we are essentially moving the graph to the left by a certain number of units. In this case, we are moving the graph 5 units to the left.

Shifting Up by 2 Units

To shift the parent function $f(x) = |x|$ up by 2 units, we need to add 2 to the function. This is because when we shift the graph of a function up, we are essentially moving the graph up by a certain number of units. In this case, we are moving the graph 2 units up.

Combining the Shifts

Now that we have shifted the parent function $f(x) = |x|$ to the left by 5 units and up by 2 units, we can combine the two shifts to create the new function $g(x)$. To do this, we replace $x$ with $(x + 5)$ and add 2 to the function.

The Equation of $g(x)$

The equation of $g(x)$ is given by:

$$g(x) = |x + 5| + 2$$

This equation represents the new function $g(x)$, which is obtained by shifting the parent function $f(x) = |x|$ to the left by 5 units and up by 2 units.

Graphical Representation

To visualize the graph of $g(x)$, we can plot the graph of the parent function $f(x) = |x|$ and then shift it to the left by 5 units and up by 2 units. The resulting graph will be a V-shaped graph with its vertex at the point $(-5, 2)$.

Properties of $g(x)$

The function $g(x)$ has several properties that are worth noting. First, the function is still symmetric about the y-axis, just like the parent function $f(x) = |x|$. Second, the function has a minimum value of 2, which is obtained when $x = -5$. Finally, the function has a maximum value of 2, which is obtained when $x = 5$.

Conclusion

In this problem, we were asked to shift the parent function $f(x) = |x|$ to the left by 5 units and up by 2 units to create a new function $g(x)$. We found that the equation of $g(x)$ is given by $g(x) = |x + 5| + 2$. We also discussed the graphical representation and properties of $g(x)$.

Step-by-Step Solution

1. Shift the parent function $f(x) = |x|$ to the left by 5 units: Replace $x$ with $(x + 5)$.
2. Shift the parent function $f(x) = |x|$ up by 2 units: Add 2 to the function.
3. Combine the shifts: Replace $x$ with $(x + 5)$ and add 2 to the function.
4. Write the equation of $g(x)$: The equation of $g(x)$ is given by $g(x) = |x + 5| + 2$.

Key Concepts

• Parent function: The parent function $f(x) = |x|$ is a fundamental function in mathematics, representing the absolute value of a variable $x$.
• Shifting: Shifting a function to the left or right means moving the graph of the function horizontally, while shifting a function up or down means moving the graph of the function vertically.
• Equation of $g(x)$: The equation of $g(x)$ is given by $g(x) = |x + 5| + 2$.

Real-World Applications

• Signal processing: The absolute value function is used in signal processing to represent the amplitude of a signal.
• Optimization: The absolute value function is used in optimization problems to represent the distance between two points.
• Machine learning: The absolute value function is used in machine learning to represent the error between predicted and actual values.

Further Reading

• Absolute value function: The absolute value function is a fundamental function in mathematics, representing the absolute value of a variable $x$.
• Shifting functions: Shifting functions is a technique used in mathematics to create new functions by moving the graph of a function horizontally or vertically.
• Equations of functions: The equation of a function is a mathematical expression that represents the function.
  Q&A: 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 $g(x)$?

Q: What is the parent function $f(x) = |x|$?

A: The parent function $f(x) = |x|$ is a fundamental function in mathematics, representing the absolute value of a variable $x$. This function has a V-shaped graph with its vertex at the origin (0, 0).

Q: What happens when we shift the parent function $f(x) = |x|$ to the left by 5 units?

A: When we shift the parent function $f(x) = |x|$ to the left by 5 units, we replace $x$ with $(x + 5)$. This means that the graph of the function is moved 5 units to the left.

Q: What happens when we shift the parent function $f(x) = |x|$ up by 2 units?

A: When we shift the parent function $f(x) = |x|$ up by 2 units, we add 2 to the function. This means that the graph of the function is moved 2 units up.

Q: How do we combine the shifts to create the new function $g(x)$?

A: To combine the shifts, we replace $x$ with $(x + 5)$ and add 2 to the function. This gives us the equation of $g(x)$ as $g(x) = |x + 5| + 2$.

Q: What is the equation of $g(x)$?

A: The equation of $g(x)$ is given by $g(x) = |x + 5| + 2$.

Q: What are the properties of $g(x)$?

A: The function $g(x)$ has several properties that are worth noting. First, the function is still symmetric about the y-axis, just like the parent function $f(x) = |x|$. Second, the function has a minimum value of 2, which is obtained when $x = -5$. Finally, the function has a maximum value of 2, which is obtained when $x = 5$.

Q: How do we visualize the graph of $g(x)$?

A: To visualize the graph of $g(x)$, we can plot the graph of the parent function $f(x) = |x|$ and then shift it to the left by 5 units and up by 2 units. The resulting graph will be a V-shaped graph with its vertex at the point $(-5, 2)$.

Q: What are some real-world applications of the absolute value function?

A: The absolute value function has several real-world applications, including signal processing, optimization, and machine learning.

Q: What are some further reading resources for learning more about the absolute value function and shifting functions?

A: Some further reading resources for learning more about the absolute value function and shifting functions include textbooks on mathematics and online resources such as Khan Academy and Wolfram Alpha.

Frequently Asked Questions

• Q: What is the parent function $f(x) = |x|$? A: The parent function $f(x) = |x|$ is a fundamental function in mathematics, representing the absolute value of a variable $x$.
• Q: How do we shift the parent function $f(x) = |x|$ to the left by 5 units? A: We replace $x$ with $(x + 5)$.
• Q: How do we shift the parent function $f(x) = |x|$ up by 2 units? A: We add 2 to the function.
• Q: What is the equation of $g(x)$? A: The equation of $g(x)$ is given by $g(x) = |x + 5| + 2$.

Common Mistakes

• Mistake 1: Shifting the parent function $f(x) = |x|$ to the left by 5 units and up by 2 units, but forgetting to replace $x$ with $(x + 5)$.
• Mistake 2: Shifting the parent function $f(x) = |x|$ up by 2 units, but forgetting to add 2 to the function.
• Mistake 3: Not combining the shifts to create the new function $g(x)$.

Conclusion

In this Q&A article, we have discussed the parent function $f(x) = |x|$, shifting the parent function to the left by 5 units and up by 2 units, and combining the shifts to create the new function $g(x)$. We have also answered some frequently asked questions and discussed common mistakes to avoid.