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The question asks to determine which relationship represents a function with the same rate of change as the graphed function.

### Step 1: Identify the slope (rate of change) from the graph.
The graph shows a straight line, indicating a linear function. To find the rate of change (slope), use the slope formula:

\[
\text{slope} = \frac{\Delta y}{\Delta x}
\]

From the graph, we can observe two points:
- (0, 0) and (4, 3)

Now calculate the slope:

\[
\text{slope} = \frac{3 - 0}{4 - 0} = \frac{3}{4}
\]

So, the rate of change (slope) for the graphed function is \( \frac{3}{4} \).

### Step 2: Examine each answer choice.

- **A**: \( y = x + 5 \)  
  The slope here is \( 1 \), since it's in the form \( y = mx + b \), where \( m = 1 \).
  **Not the correct choice** because the slope is different.

- **B**: Table with points (0, -5), (4, -12), (8, -19), (12, -26)  
  Find the rate of change between points:
  \[
  \text{slope} = \frac{-12 - (-5)}{4 - 0} = \frac{-7}{4}
  \]
  This slope is \( -\frac{7}{4} \).  
  **Not the correct choice** because the slope is negative.

- **C**: \( y = 5x - 3 \)  
  The slope here is \( 5 \), since it's in the form \( y = mx + b \), where \( m = 5 \).
  **Not the correct choice** because the slope is different.

- **D**: Table with points (4, -6), (8, -11), (12, -16), (16, -21)  
  Find the rate of change between points:
  \[
  \text{slope} = \frac{-11 - (-6)}{8 - 4} = \frac{-5}{4}
  \]
  This slope is \( -\frac{5}{4} \), but upon checking all intervals, it's consistently the same.  
  **Not the correct choice** because the slope is negative.

### Conclusion:
None of the choices given directly match the slope \( \frac{3}{4} \) from the graph. There might be a mismatch between the slope of the function in the graph and the available options.

Would you like to go over more details or have further questions? Here are 5 relative questions that could expand the discussion:

1. How can we derive the equation of a line given two points?
2. What other ways can we calculate the rate of change from a graph?
3. What is the significance of the slope in real-world applications?
4. Can we determine the slope from just a table of values without graphing?
5. How does a positive slope differ from a negative slope in interpreting functions?

**Tip**: Always look for two clear points on a graph to accurately determine the slope.

The graph of a function is shown on the coordinate plane. Which relationship represents a function with the same rate of change as the function graphed?

Determine which function has the same rate of change as the graphed function by calculating the slope using the formula (y2 - y1) / (x2 - x1). Suitable for Grades 7-9 students, this problem reinforces key concepts in linear algebra and graph interpretation.

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