The Graph Of F ( X ) = ∣ X ∣ F(x) = |x| F ( X ) = ∣ X ∣ Is Transformed To G ( X ) = ∣ X + 1 ∣ − 7 G(x) = |x+1| - 7 G ( X ) = ∣ X + 1∣ − 7 . On Which Interval Is The Function Decreasing?A. ( − ∞ , − 7 (-\infty, -7 ( − ∞ , − 7 ] B. ( − ∞ , − 1 (-\infty, -1 ( − ∞ , − 1 ] C. ( − ∞ , 7 (-\infty, 7 ( − ∞ , 7 ] D. ( − ∞ , 1 (-\infty, 1 ( − ∞ , 1 ]

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Introduction

In this article, we will explore the transformation of the graph of the absolute value function f(x)=∣x∣f(x) = |x| to the function g(x)=∣x+1∣−7g(x) = |x+1| - 7. We will analyze the behavior of the function and determine the interval on which the function is decreasing.

Understanding Absolute Value Functions

An absolute value function is a function that takes the absolute value of its input. The graph of an absolute value function is a V-shaped graph with its vertex at the origin. The function f(x)=∣x∣f(x) = |x| is a classic example of an absolute value function.

Transforming the Graph of f(x)=∣x∣f(x) = |x|

To transform the graph of f(x)=∣x∣f(x) = |x|, we need to apply a series of transformations to the function. The function g(x)=∣x+1∣−7g(x) = |x+1| - 7 is obtained by applying the following transformations to the function f(x)=∣x∣f(x) = |x|:

  • Horizontal Shift: The graph of f(x)=∣x∣f(x) = |x| is shifted to the left by 1 unit to obtain the graph of g(x)=∣x+1∣g(x) = |x+1|.
  • Vertical Shift: The graph of g(x)=∣x+1∣g(x) = |x+1| is shifted down by 7 units to obtain the graph of g(x)=∣x+1∣−7g(x) = |x+1| - 7.

Analyzing the Behavior of g(x)=∣x+1∣−7g(x) = |x+1| - 7

To determine the interval on which the function g(x)=∣x+1∣−7g(x) = |x+1| - 7 is decreasing, we need to analyze the behavior of the function. The function g(x)=∣x+1∣−7g(x) = |x+1| - 7 is a piecewise function, and its behavior depends on the value of xx.

Case 1: x<−1x < -1

When x<−1x < -1, the function g(x)=∣x+1∣−7g(x) = |x+1| - 7 can be written as g(x)=−(x+1)−7=−x−8g(x) = -(x+1) - 7 = -x - 8. This is a linear function with a negative slope, and it is decreasing on the interval (−∞,−1)(-\infty, -1).

Case 2: x≥−1x \geq -1

When x≥−1x \geq -1, the function g(x)=∣x+1∣−7g(x) = |x+1| - 7 can be written as g(x)=x+1−7=x−6g(x) = x+1 - 7 = x - 6. This is a linear function with a positive slope, and it is increasing on the interval [−1,∞)[-1, \infty).

Conclusion

Based on the analysis of the behavior of the function g(x)=∣x+1∣−7g(x) = |x+1| - 7, we can conclude that the function is decreasing on the interval (−∞,−1)(-\infty, -1).

The final answer is: B\boxed{B}

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Introduction

In our previous article, we explored the transformation of the graph of the absolute value function f(x)=∣x∣f(x) = |x| to the function g(x)=∣x+1∣−7g(x) = |x+1| - 7. We analyzed the behavior of the function and determined the interval on which the function is decreasing. In this article, we will answer some frequently asked questions related to the graph of a transformed absolute value function.

Q&A

Q1: What is the vertex of the graph of g(x)=∣x+1∣−7g(x) = |x+1| - 7?

A1: The vertex of the graph of g(x)=∣x+1∣−7g(x) = |x+1| - 7 is at the point (−1,−7)(-1, -7).

Q2: What is the x-intercept of the graph of g(x)=∣x+1∣−7g(x) = |x+1| - 7?

A2: The x-intercept of the graph of g(x)=∣x+1∣−7g(x) = |x+1| - 7 is at the point (−8,0)(-8, 0).

Q3: What is the y-intercept of the graph of g(x)=∣x+1∣−7g(x) = |x+1| - 7?

A3: The y-intercept of the graph of g(x)=∣x+1∣−7g(x) = |x+1| - 7 is at the point (0,−7)(0, -7).

Q4: Is the function g(x)=∣x+1∣−7g(x) = |x+1| - 7 increasing or decreasing on the interval [−1,∞)[-1, \infty)?

A4: The function g(x)=∣x+1∣−7g(x) = |x+1| - 7 is increasing on the interval [−1,∞)[-1, \infty).

Q5: Is the function g(x)=∣x+1∣−7g(x) = |x+1| - 7 increasing or decreasing on the interval (−∞,−1)(-\infty, -1)?

A5: The function g(x)=∣x+1∣−7g(x) = |x+1| - 7 is decreasing on the interval (−∞,−1)(-\infty, -1).

Q6: What is the equation of the asymptote of the graph of g(x)=∣x+1∣−7g(x) = |x+1| - 7?

A6: The equation of the asymptote of the graph of g(x)=∣x+1∣−7g(x) = |x+1| - 7 is y=−7y = -7.

Q7: What is the domain of the function g(x)=∣x+1∣−7g(x) = |x+1| - 7?

A7: The domain of the function g(x)=∣x+1∣−7g(x) = |x+1| - 7 is all real numbers, (−∞,∞)(-\infty, \infty).

Q8: What is the range of the function g(x)=∣x+1∣−7g(x) = |x+1| - 7?

A8: The range of the function g(x)=∣x+1∣−7g(x) = |x+1| - 7 is all real numbers greater than or equal to −7-7, [−7,∞)[-7, \infty).

Conclusion

In this article, we answered some frequently asked questions related to the graph of a transformed absolute value function. We hope that this article has been helpful in understanding the behavior of the function g(x)=∣x+1∣−7g(x) = |x+1| - 7. If you have any further questions, please don't hesitate to ask.

Additional Resources

References