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### Extracted Questions and Solutions:

1. **Problem 1:**
   *The graph on the xy-plane of the linear function f contains the point (0, 4). For every increase of 3 units in x, f(x) decreases by 5 units. Which of the following equations defines the function f?*

   **Solution:**
   - The point (0, 4) indicates that the y-intercept of the function is 4.
   - For every increase of 3 units in \(x\), \(f(x)\) decreases by 5 units. This is the slope (\(m\)) of the line, given by:
     \[
     m = \frac{\text{change in } y}{\text{change in } x} = \frac{-5}{3}.
     \]
   - The equation of the line in slope-intercept form \(y = mx + b\) becomes:
     \[
     f(x) = -\frac{5}{3}x + 4.
     \]
   - The correct option is **(D) \(-\frac{5}{3}x + 4\).**

2. **Problem 2:**
   *The graph of the function \(f\) shown is the xy-plane. Which of the following equations defines \(f\)?*

   **Solution:**
   - The graph intercepts the y-axis at (0, 2), so the y-intercept \(b = 2\).
   - The slope can be calculated by finding another point on the graph. Assuming a rise of 3 units and a run of 2 units, the slope is:
     \[
     m = \frac{3}{2}.
     \]
   - Thus, the equation of the line is:
     \[
     f(x) = \frac{3}{2}x + 2.
     \]
   - The correct option is **(C) \(y = \frac{3}{2}x + 2\).**

3. **Problem 3:**
   *The graph shows the positions of Tom and Mallika during a race... which of the following is correct?*

   **Solution:**
   - Tom’s and Mallika’s lines intersect at \(x = 30\), indicating that they both reached the same distance at 30 seconds.
   - Before the intersection point, Tom's line is below Mallika's, indicating that Mallika is ahead.
   - After 30 seconds, Tom's line is above Mallika's, indicating Tom has overtaken Mallika.
   - Therefore, the correct answer is **(B) Tom overtook Mallika after 30 seconds.**

4. **Problem 4:**
   *The equation, the points (−4, 4) and (2, 1) are on the line... find the slope.*

   **Solution:**
   - The slope \(m\) is given by the formula:
     \[
     m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 4}{2 - (-4)} = \frac{-3}{6} = -\frac{1}{2}.
     \]
   - So, the correct slope is **(A) -\frac{1}{2}.**

Would you like more details on any of these solutions, or do you have any other questions related to these problems?

### Related Questions:

1. How do you convert a graph's visual data into a mathematical equation?
2. What are some techniques to quickly identify the slope and intercept from a linear graph?
3. How can you interpret the intersection of two lines in terms of real-world problems, such as races or economics?
4. What is the significance of the slope in a linear equation, and how does it relate to the rate of change?
5. What are other forms of linear equations besides the slope-intercept form?

**Tip:** When working with graph problems, always identify key points (like intercepts) and calculate slopes using clear coordinate pairs to build your equations accurately.

Explore solutions to linear function problems using graph interpretation. Learn how to determine slopes, intercepts, and equations from graphs. Suitable for high school students.

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