## Functional Transformations on $ f(x) $Given:- Points $(-2, 3)$ and $(5, -1)$ are on the graph of $ f(x) $.### Problem StatementState the coordinates of these points after the following functional transformation was performed on $ f(x) $:\[ g(x) = -\frac{1}{2} f(x-1) - 2 \]### Transformation StepsLet's determine the new coordinates of the points after the transformation in the form of a table.#### Original Points| Point $ (x, y) $ on $ f(x) $ | $ x $ | $ y $ ||-------------------------------|--------|--------|| Original Point 1 | $-2$ | $ 3 $ || Original Point 2 | $ 5 $ | $-1$ |### Steps for Transformation1. **Horizontal Shift**: $ f(x-1) $2. **Vertical Scaling and Reflection**: $ -\frac{1}{2} $3. **Vertical Shift**: $ -2 $#### Transformed PointsFor each point, apply the transformations in order:1. **Horizontal Shift** ($ x $ coordinate shift by $ +1 $): - New $ x $ coordinate = $ x + 1 $2. **Vertical Scaling and Reflection** ($ y $ multiply by $ -\frac{1}{2}$): - New $ y $ coordinate = $ y \times -\frac{1}{2} $3. **Vertical Shift** (subtract 2 from new $ y $ value): - New $ y $ coordinate = New $ y $ coordinate - 2#### Computation for each point:1. Point $(-2, 3)$: - Horizontal Shift: $(-2+1, 3)$ = $(-1, 3)$ - Vertical Scaling and Reflection: $(-1, 3 \times -\frac{1}{2})$ = $(-1, -1.5)$ - Vertical Shift: $(-1, -1.5 - 2)$ = $(-1, -3.5)$2. Point $(5, -1)$: - Horizontal Shift:

If points (-2,3) and (5, -1) are on the graph of f(x) To state the coordinates of these points after…

If points (-2, 3) and (5, –1) are on the graph of f (x), state the coordinates of these points after the following functional transformations were performed on f. Show all steps in the form of a table. 1 b) g(x) = f(x – 1) – 2

If points (-2, 3) and (5, –1) are on the graph of f (x), state the coordinates of these points after the following functional transformations were performed on f. Show all steps in the form of a table. 1 b) g(x) = f(x – 1) – 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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Question

## Functional Transformations on $ f(x) $

Given:
- Points $(-2, 3)$ and $(5, -1)$ are on the graph of $ f(x) $.

### Problem Statement
State the coordinates of these points after the following functional transformation was performed on $ f(x) $:

\[ g(x) = -\frac{1}{2} f(x-1) - 2 \]

### Transformation Steps

Let's determine the new coordinates of the points after the transformation in the form of a table.

#### Original Points
| Point $ (x, y) $ on $ f(x) $ | $ x $ | $ y $ |
|-------------------------------|--------|--------|
| Original Point 1 | $-2$ | $ 3 $ |
| Original Point 2 | $ 5 $ | $-1$ |

### Steps for Transformation
1. **Horizontal Shift**: $ f(x-1) $
2. **Vertical Scaling and Reflection**: $ -\frac{1}{2} $
3. **Vertical Shift**: $ -2 $

#### Transformed Points

For each point, apply the transformations in order:

1. **Horizontal Shift** ($ x $ coordinate shift by $ +1 $):
- New $ x $ coordinate = $ x + 1 $

2. **Vertical Scaling and Reflection** ($ y $ multiply by $ -\frac{1}{2}$):
- New $ y $ coordinate = $ y \times -\frac{1}{2} $

3. **Vertical Shift** (subtract 2 from new $ y $ value):
- New $ y $ coordinate = New $ y $ coordinate - 2

#### Computation for each point:
1. Point $(-2, 3)$:
- Horizontal Shift: $(-2+1, 3)$ = $(-1, 3)$
- Vertical Scaling and Reflection: $(-1, 3 \times -\frac{1}{2})$ = $(-1, -1.5)$
- Vertical Shift: $(-1, -1.5 - 2)$ = $(-1, -3.5)$

2. Point $(5, -1)$:
- Horizontal Shift:

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