Key Features of Linear Functions and Their Graphs

Linear functions are a type of math function shown by this equation:

$$f(x) = mx + b$$

Here’s what those letters mean:

• $m$ is the slope (how steep the line is),
• $b$ is the y-intercept (the spot where the line crosses the y-axis).

Features

1. Constant Rate of Change:
   The slope $m$ shows a steady rate of change. For example, if $m = 2$, then every time $x$ increases by 1, $f(x)$ goes up by 2.

2. Graph Shape:
   The graph of a linear function is always a straight line.

3. Intercepts:

   • Y-Intercept: This happens when $x = 0$; the point is $(0, b)$.
   • X-Intercept: This happens when $f(x) = 0$. You can find it by setting $mx + b = 0$ and solving for $x$.

4. Domain and Range:
   For linear functions, both the domain (possible $x$ values) and range (possible $f(x)$ values) are all real numbers.

5. Parallel Lines:
   Two linear functions will run parallel to each other if they have the same slope ($m_1 = m_2$) but different y-intercepts ($b_1 \neq b_2$).

6. Graphing:
   To draw a linear function, start at the y-intercept. Then, use the slope to find more points. If $m = \frac{rise}{run}$, you move up or down for the rise and left or right for the run.

Conclusion

Knowing these features is really important for understanding and drawing linear functions in algebra and pre-calculus.