a student graphed f(x)=x and g(x)=f(x)+3 on the same grid. how are the graphs of f and g related

The graphs of \( f(x) = x \) and \( g(x) = f(x) + 3 \) are related through a vertical translation.

- **Graph of \( f(x) = x \)**: This is a straight line that passes through the origin with a slope of 1, forming a 45-degree angle with the x-axis.

- **Graph of \( g(x) = f(x) + 3 = x + 3 \)**: This is also a straight line with the same slope of 1, but the entire graph is shifted **3 units upward** compared to the graph of \( f(x) \).

In other words, every point on the graph of \( f(x) = x \) is translated vertically by 3 units to form the graph of \( g(x) = x + 3 \). The shape and slope of the two lines are the same; only their vertical positions differ.

### Would you like further clarification or more details on graph transformations?

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Here are five follow-up questions to deepen your understanding:

1. What would happen if we added a negative constant to \( f(x) \) instead of a positive one?
2. How would the graph change if \( g(x) = f(x) - 3 \)?
3. Can you describe how the slope affects the orientation of the graph?
4. How would the graphs change if we multiplied \( f(x) \) by a constant instead of adding one?
5. What is the y-intercept of the graph of \( g(x) \)?

**Tip**: When shifting graphs, adding a constant moves the graph vertically, while adding inside the function argument affects horizontal shifts.

A student graphed f(x)=x and g(x)=f(x)+3 on the same grid. How are the graphs of f and g related?

Explore how the graph of f(x) = x and g(x) = x + 3 are related through vertical translation. Learn key algebraic concepts and graph transformations with a clear and detailed explanation. Suitable for students in Grades 8-10.

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