Let's analyze and graph each function shown in the image.

Problem 1

1. Function: $f(x) = 3x + 4$

   • This is a linear equation in slope-intercept form: $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
   • Slope (m): 3. This means that for every 1 unit increase in $x$, $y$ increases by 3 units.
   • Y-intercept (b): 4. This is the point where the line crosses the y-axis, at $(0, 4)$.

   Steps to Graph:

   • Start by plotting the y-intercept at $(0, 4)$.
   • Use the slope to find another point. Since the slope is 3, from $(0, 4)$, move 1 unit to the right (positive x-direction) and 3 units up. This gives the point $(1, 7)$.
   • Connect these points with a straight line.

Problem 2

2. Function: $f(x) = -3$

   • This is a constant function, meaning it does not change with $x$; $y$ is always $-3$ regardless of $x$ values.
   • This will produce a horizontal line.

   Steps to Graph:

   • Draw a horizontal line through $y = -3$. This line will pass through points like $(-8, -3)$, $(0, -3)$, and $(8, -3)$.

Let me know if you need further details or explanations.

Relative Questions:

1. How would the graph change if the slope in Problem 1 were negative?
2. What if the y-intercept in Problem 1 was set to 0?
3. How would a vertical line look, and what would its equation be?
4. Can you identify the domain and range of both functions?
5. How would you graph $f(x) = 0$, and what would it represent?

Tip:

For linear functions, always identify the slope and intercept to make graphing easier.