To graph the linear function $f(x) = -\frac{3}{4}x - 6$, follow these steps:

1. Identify the Slope and Y-intercept

The equation is in slope-intercept form: $f(x) = mx + b$, where:

• $m = -\frac{3}{4}$ is the slope.
• $b = -6$ is the y-intercept.

This tells us:

• The graph crosses the y-axis at $(0, -6)$.
• The slope $-\frac{3}{4}$ means that for every increase of $4$ units in $x$, $y$ decreases by $3$ units.

2. Plot the Y-intercept

Start by marking the point $(0, -6)$ on the graph.

3. Use the Slope to Find Another Point

From $(0, -6)$:

• Move 4 units to the right (positive $x$-direction).
• Move 3 units down (negative $y$-direction) because the slope is negative. This gives the second point at $(4, -9)$.

4. Draw the Line

Using a ruler, draw a straight line passing through the points $(0, -6)$ and $(4, -9)$. Extend the line in both directions and add arrowheads to indicate it continues infinitely.

5. Verify with Additional Points (Optional)

You can substitute other $x$-values to calculate $y$-values for more points, e.g.:

• $x = -4$: $f(-4) = -\frac{3}{4}(-4) - 6 = 3 - 6 = -3$. Plot $(-4, -3)$.

Summary of Key Points

• Y-intercept: $(0, -6)$
• Second point: $(4, -9)$
• Line has a downward slope ($-3/4$).

Would you like me to provide a sketch of this graph or clarify any step?

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Related Questions:

1. How does the slope affect the steepness and direction of the line?
2. What happens to the graph if the slope becomes $\frac{3}{4}$ instead of $-\frac{3}{4}$?
3. How do you convert the slope-intercept form to standard form?
4. What are the x- and y-intercepts of this line?
5. How would this graph change if the y-intercept were $-2$ instead of $-6$?

Tip:

When graphing, always use at least two points to ensure accuracy, and confirm by substituting $x$-values into the equation.