The graph of f(x) = 5x is shown to the right. Determine the graph of f(x) = 5x +3. Choose the correct graph below. O A. 11 O B. G O C. O D. 2

Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
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### Graph Equations and Transformations

#### Problem Statement:
The graph of \( f(x) = 5x \) is shown to the right. Determine the graph of \( f(x) = 5x + 3 \).

![Graph of f(x) = 5x](attachment)

#### Options for the Correct Graph:
Choose the correct graph below.

**Option A:**
![Option A](attachment)

**Option B:**
![Option B](attachment)

**Option C:**
![Option C](attachment)

**Option D:**
![Option D](attachment)

### Detailed Explanation of Graphs:
1. **Original Graph \( f(x) = 5x \)**
   - The given graph shows a linear function \( f(x) = 5x \).
   - The line passes through the origin (0,0) with a slope of 5, indicating that for every unit increase in \( x \), \( y \) increases by 5 units.

2. **Transforming to \( f(x) = 5x + 3 \)**
   - To determine the graph of \( f(x) = 5x + 3 \), we need to apply a vertical shift of the original line.
   - Adding 3 to the function causes the entire graph to shift up by 3 units.
   - The slope remains the same (5), but the line will now intersect the y-axis at (0, 3) instead of (0, 0).

### Graph Options Analysis:
- **Option A**: Shows a graph passing through (0, 0), not shifted up.
- **Option B**: This graph intersects the y-axis at (0, 3) and maintains the same slope as the original.
- **Option C**: This graph shows an incorrect slope alteration.
- **Option D**: This graph shows a vertical shift but not by the correct amount.

**Correct Answer: Option B**

Graph **Option B** accurately represents the solution \( f(x) = 5x + 3 \), which is the vertical shift of the original function \( f(x) = 5x \) by 3 units upward.
Transcribed Image Text:### Graph Equations and Transformations #### Problem Statement: The graph of \( f(x) = 5x \) is shown to the right. Determine the graph of \( f(x) = 5x + 3 \). ![Graph of f(x) = 5x](attachment) #### Options for the Correct Graph: Choose the correct graph below. **Option A:** ![Option A](attachment) **Option B:** ![Option B](attachment) **Option C:** ![Option C](attachment) **Option D:** ![Option D](attachment) ### Detailed Explanation of Graphs: 1. **Original Graph \( f(x) = 5x \)** - The given graph shows a linear function \( f(x) = 5x \). - The line passes through the origin (0,0) with a slope of 5, indicating that for every unit increase in \( x \), \( y \) increases by 5 units. 2. **Transforming to \( f(x) = 5x + 3 \)** - To determine the graph of \( f(x) = 5x + 3 \), we need to apply a vertical shift of the original line. - Adding 3 to the function causes the entire graph to shift up by 3 units. - The slope remains the same (5), but the line will now intersect the y-axis at (0, 3) instead of (0, 0). ### Graph Options Analysis: - **Option A**: Shows a graph passing through (0, 0), not shifted up. - **Option B**: This graph intersects the y-axis at (0, 3) and maintains the same slope as the original. - **Option C**: This graph shows an incorrect slope alteration. - **Option D**: This graph shows a vertical shift but not by the correct amount. **Correct Answer: Option B** Graph **Option B** accurately represents the solution \( f(x) = 5x + 3 \), which is the vertical shift of the original function \( f(x) = 5x \) by 3 units upward.
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