To graph the function $f(x) = 5x - 7$, let's break down its components.

1. Identify the Slope and Y-Intercept:

   • The function $f(x) = 5x - 7$ is in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
   • Here, $m = 5$ and $b = -7$.
   • This slope, $m = 5$, indicates that for every unit increase in $x$, $f(x)$ increases by 5 units.
   • The y-intercept $b = -7$ is the point where the line crosses the y-axis, at $(0, -7)$.

2. Plotting Points:

   • Start by plotting the y-intercept at $(0, -7)$.
   • Use the slope $5$ (rise over run, or $\frac{5}{1}$) to find another point. From $(0, -7)$, move up 5 units and right 1 unit to reach $(1, -2)$.

3. Drawing the Line:

   • Connect these points to form a straight line.

Let me know if you'd like a visual plot of this, or have any further questions!

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Further Related Questions

1. How do we interpret the slope in real-world applications?
2. What is the derivative of $f(x) = 5x - 7$ and what does it represent?
3. How would the graph change if the slope was negative?
4. Can we calculate the x-intercept of this function, and how?
5. What would happen if we added a constant to the function, like $f(x) = 5x - 7 + c$?

Tip

For any linear function $f(x) = mx + b$, the slope $m$ directly controls the line's steepness, while $b$ sets its position along the y-axis.