I can see that the image is clearer now. Let me analyze the content and identify the math problems for you.

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.