Select the correct answer from each drop-down menu.
Consider the function f(x) = 3x + 1 and the graph of the function g(x) shown below.

A coordinate plane linear graph function shows a line intersecting Y-axis at minus 5 and X-axis at 1.5.

The graph g(x) is the graph of f(x) translated 
 units 
, and g(x) = 
.

Let's break down the problem:

1. **Function \( f(x) = 3x + 1 \)**:
   This is a linear function with a slope of 3 and a y-intercept of 1.

2. **Graph of \( g(x) \)**:
   From the description, the graph of \( g(x) \) intersects the y-axis at -5 and the x-axis at 1.5 (or \( \frac{3}{2} \)).

Now, let's analyze what this implies:

- **Translation of the y-intercept**: The function \( f(x) \) has a y-intercept of 1, while \( g(x) \) has a y-intercept of -5. This suggests a vertical translation. The shift in the y-direction is from \( 1 \) to \( -5 \), which is a shift downward by 6 units.

- **Slope of \( g(x) \)**: The slope remains the same because the description does not mention any change in the steepness of the graph. Thus, the slope of \( g(x) \) is still 3.

### Translation in terms of units:
The function \( f(x) = 3x + 1 \) is translated **6 units down** to get the function \( g(x) \).

### Function \( g(x) \):
Since the slope of the graph is unchanged, and the y-intercept has shifted to -5, the equation for \( g(x) \) is \( g(x) = 3x - 5 \).

### Final answers:
1. The graph \( g(x) \) is the graph of \( f(x) \) translated **6** units **down**.
2. \( g(x) = 3x - 5 \).

Would you like further details or have any questions?

Here are some related questions to expand on this:

1. How can we determine the slope of a line from its graph?
2. What is the general form of a linear function, and how can we modify it for translations?
3. How do vertical and horizontal translations affect the graph of a function?
4. What are the differences between vertical and horizontal translations in linear functions?
5. How do you solve for x-intercepts and y-intercepts algebraically?

**Tip**: When translating graphs, remember that vertical shifts affect only the y-intercept, while horizontal shifts affect the x-values.

Select the correct answer from each drop-down menu. Consider the function f(x) = 3x + 1 and the graph of the function g(x) shown below. A coordinate plane linear graph function shows a line intersecting Y-axis at minus 5 and X-axis at 1.5. The graph g(x) is the graph of f(x) translated __ units __, and g(x) = __.

Understand how to translate the linear function f(x) = 3x + 1 into g(x) = 3x - 5 through a vertical shift of 6 units downward. Learn key algebraic concepts like graph translations and linear functions. Ideal for students in Grades 8-10.

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