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Let's break this problem down step-by-step.

We are given two functions:

1. The function in the graph appears to be \( f(x) \), and we are tasked with identifying its slope and y-intercept.
2. The second function provided in the equation form is \( g(x) = -2x + 3 \).

### Comparing the slopes:

1. **For the graph function \( f(x) \)**:
   - By observing the graph, the line has a negative slope. To calculate the slope, we look at two points on the graph: one at \( (0, 3) \) (the y-intercept) and another at \( (-5, 5) \). 
   - Slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 3}{-5 - 0} = \frac{2}{-5} = -\frac{2}{5} \).
   - So, the slope of \( f(x) \) is \( -\frac{2}{5} \).

2. **For the function \( g(x) = -2x + 3 \)**:
   - The slope is clearly \( -2 \) because the equation is in slope-intercept form, \( y = mx + b \), where \( m \) is the slope.

### Comparing the y-intercepts:

1. **For \( f(x) \)**:
   - From the graph, the y-intercept occurs at \( (0, 3) \), meaning the y-intercept is 3.

2. **For \( g(x) = -2x + 3 \)**:
   - The y-intercept is also 3 because the equation \( g(x) = -2x + 3 \) shows that when \( x = 0 \), \( g(0) = 3 \).

### Conclusion:

- The slopes are different: \( -\frac{2}{5} \) for \( f(x) \) and \( -2 \) for \( g(x) \).
- The y-intercepts are the same: 3 for both functions.

Therefore, the correct answer is **D. The slopes are different but the y-intercepts are the same.**

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Would you like more details on the slope calculation or how to interpret the graph?

Here are 5 follow-up questions that you might find helpful:
1. What does the slope of a line tell us about the graph?
2. How do you find the slope of a line from two points on the graph?
3. What is the significance of the y-intercept in a linear equation?
4. How would the graph of \( g(x) = -2x + 3 \) look compared to \( f(x) \)?
5. Can you explain the difference between positive and negative slopes?

**Tip**: Always remember that the slope of a line measures its steepness, and a negative slope indicates a line that falls as you move from left to right!

Compare the slopes and y-intercepts of two functions: the graph of f(x) and the equation g(x) = -2x + 3.

Learn how to compare the slopes and y-intercepts of two functions: the graph of f(x) and the linear equation g(x) = -2x + 3. This problem involves key algebraic concepts such as slope calculation and interpreting y-intercepts. Suitable for students in Grades 8-10.

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