How do you graph a linear function?

Definition

A linear function is any function of the form
$$ f(x) = mx + b $$
where $m$ is the slope and $b$ is the $y$-intercept. The graph of a linear function is always a straight line.

Steps to Graph a Linear Function

1. Identify the slope ($m$) and $y$-intercept ($b$).
2. Plot the $y$-intercept $(0, b)$ on the coordinate plane.
3. Use the slope to find another point. The slope $m = \frac{\text{rise}}{\text{run}}$ tells you how to move from the $y$-intercept.
4. Draw a straight line through the points.

Worked Example

Graph the function: $f(x) = 2x - 3$

Step 1: Identify $m$ and $b$.

• Slope: $m = 2$
• $y$-intercept: $b = -3$

Step 2: Plot the $y$-intercept $(0, -3)$.

Step 3: Use the slope $m = 2 = \frac{2}{1}$.

• From $(0, -3)$, move up $2$ units (rise) and right $1$ unit (run) \to get \to $(1, -1)$. **Step 4:** Plot $(1, -1)$. Draw a straight line through $(0, -3)$ and $(1, -1)$. **Check another point:** - For $x = 2$: %%MATH_DISPLAY_1%% So, $(2, 1)$ is also on the line. ## Takeaways - The graph of a linear function is always a straight line. - The $y$-intercept is where the line crosses the $y$-axis.
• The slope determines the steepness and direction of the line.