Patricia graphed *f*(*x*) = *x* and *g*(*x*) = *x* – 3 on the same coordinate grid.

Which statement best describes the relationships between the two functions?

The two functions, \( f(x) = x \) and \( g(x) = x - 3 \), are linear functions. They have the same slope but different y-intercepts.

The relationship between these functions can be described as follows:

- **\( g(x) \) is a vertical translation of \( f(x) \) downward by 3 units.** This means that every point on \( f(x) \) is shifted 3 units downward to obtain \( g(x) \).

Mathematically, \( g(x) \) can be rewritten as \( f(x) - 3 \), showing that \( g(x) \) is the same as \( f(x) \) but with all the y-values decreased by 3.

Would you like more details or have any questions? Here are five related questions to deepen understanding:

1. What is the slope of both functions \( f(x) \) and \( g(x) \)?
2. How would you graph both functions on the same coordinate plane?
3. What would happen if the transformation applied was \( g(x) = x + 3 \) instead of \( g(x) = x - 3 \)?
4. How would you describe the relationship between \( f(x) \) and \( g(x) \) if the transformation was \( g(x) = 2x - 3 \)?
5. Can you describe how the equation \( g(x) = x - 3 \) affects the intercepts on the graph?

**Tip:** When comparing two linear functions, pay close attention to the changes in their slopes and y-intercepts to understand the type of transformation.

Patricia graphed f(x) = x and g(x) = x – 3 on the same coordinate grid. Which statement best describes the relationships between the two functions?

Explore the relationship between the linear functions f(x) = x and g(x) = x - 3, understanding how vertical translations affect function graphs. This math problem covers key concepts such as linear function transformations, with a focus on grade-appropriate algebra skills.

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