The graph in the image shows a linear function with a negative slope, and it's sloping down from left to right. We need to determine which of the provided functions best matches the graph.

The given options are:

• $f(x) = 3x - 11$
• $f(x) = -4x + 12$
• $f(x) = 4x + 13$
• $f(x) = -5x - 19$

Steps to find the correct function:

1. Identify the slope: The graph shows a downward-sloping line, which means the slope must be negative. This eliminates the functions with positive slopes: $f(x) = 3x - 11$ and $f(x) = 4x + 13$.

2. Examine the remaining options:

   • $f(x) = -4x + 12$
   • $f(x) = -5x - 19$

   Both functions have negative slopes. Now we will compare the steepness and y-intercepts to the graph.

3. Y-intercept:

   • The graph crosses the y-axis slightly above $y = 10$.
   • For $f(x) = -4x + 12$, the y-intercept is $12$, which is close to the graph.
   • For $f(x) = -5x - 19$, the y-intercept is $-19$, which is far off from what we observe on the graph.

Thus, the correct function that most likely matches the graph is: $f(x) = -4x + 12$

Would you like further details on any part of the explanation?

Here are 5 related questions you could explore:

1. What is the importance of slope and y-intercept in determining the equation of a line?
2. How do we calculate the slope of a line given two points?
3. Can the slope of a line ever be zero or undefined?
4. How does the steepness of a line relate to the magnitude of the slope?
5. How can we graph a line if we are only given its equation?

Tip: To determine the equation of a line from a graph, always first check its slope and y-intercept.