With explanation of the answer please. ### Problem 5: Graphing Functions Based On f(x) = |x|#### InstructionsSketch the graph of the following functions based on your knowledge of the graph of $ f(x) = |x| $.#### (a) $ g(x) = |x - 1| + 1 $#### (b) $ h(x) = -|x + 3| - 2 $### Explanation1. **Understanding the Basic Graph $ f(x) = |x| $**: - The graph of $ f(x) = |x| $ is a V-shaped graph that opens upwards. - The vertex of the graph is at the origin (0,0). - The graph is symmetric about the y-axis, with the left side sloping downward from the vertex and the right side sloping upward from the vertex.2. **Transformation of $ g(x) = |x - 1| + 1 $**: - **Horizontal Shift**: The term $ (x - 1) $ inside the absolute value function causes the graph to shift horizontally to the right by 1 unit. - **Vertical Shift**: The term $ +1 $ outside the absolute value function causes the graph to shift vertically upwards by 1 unit. - **Resulting Graph**: The vertex of $ g(x) $ will be at (1,1), and the graph will maintain its V-shape, opening upwards.3. **Transformation of $ h(x) = -|x + 3| - 2 $**: - **Horizontal Shift**: The term $ (x + 3) $ inside the absolute value function causes the graph to shift horizontally to the left by 3 units. - **Vertical Shift**: The term $ -2 $ outside the absolute value function causes the graph to shift vertically downwards by 2 units. - **Vertical Reflection**: The negative sign in front of the absolute value causes the graph to reflect over the x-axis, meaning it will open downwards. - **Resulting Graph**: The vertex of $h(x) $ will be at (-3, -2), and the graph will maintain its V-shape but open downwards.Graph these transformations to visualize the respective functions.

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Answered: 5. Sketch the graph of the following…

5. Sketch the graph of the following functions based on your knowledge of the graph of f(x) = |x|. (a) g(x) = |x – 1|+1 (b) h(x) = -|x + 3| – 2

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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With explanation of the answer please.

### Problem 5: Graphing Functions Based On f(x) = |x|

#### Instructions
Sketch the graph of the following functions based on your knowledge of the graph of $ f(x) = |x| $.

#### (a) $ g(x) = |x - 1| + 1 $

#### (b) $ h(x) = -|x + 3| - 2 $

### Explanation

1. **Understanding the Basic Graph $ f(x) = |x| $**:
- The graph of $ f(x) = |x| $ is a V-shaped graph that opens upwards.
- The vertex of the graph is at the origin (0,0).
- The graph is symmetric about the y-axis, with the left side sloping downward from the vertex and the right side sloping upward from the vertex.

2. **Transformation of $ g(x) = |x - 1| + 1 $**:
- **Horizontal Shift**: The term $ (x - 1) $ inside the absolute value function causes the graph to shift horizontally to the right by 1 unit.
- **Vertical Shift**: The term $ +1 $ outside the absolute value function causes the graph to shift vertically upwards by 1 unit.
- **Resulting Graph**: The vertex of $ g(x) $ will be at (1,1), and the graph will maintain its V-shape, opening upwards.

3. **Transformation of $ h(x) = -|x + 3| - 2 $**:
- **Horizontal Shift**: The term $ (x + 3) $ inside the absolute value function causes the graph to shift horizontally to the left by 3 units.
- **Vertical Shift**: The term $ -2 $ outside the absolute value function causes the graph to shift vertically downwards by 2 units.
- **Vertical Reflection**: The negative sign in front of the absolute value causes the graph to reflect over the x-axis, meaning it will open downwards.
- **Resulting Graph**: The vertex of $h(x) $ will be at (-3, -2), and the graph will maintain its V-shape but open downwards.

Graph these transformations to visualize the respective functions.

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