To graph the function $f(x) = -\frac{3}{2}x$, we first recognize that this is a linear function. The function represents a straight line with the slope $-\frac{3}{2}$ and the y-intercept at 0 (since there is no constant term). Now, let's break this down step by step:

1. Slope and Y-intercept:

• The slope is $-\frac{3}{2}$, which means that for every 2 units you move to the right (in the positive direction along the x-axis), the graph goes down by 3 units.
• The y-intercept is 0, which means the line passes through the origin, point $(0, 0)$.

2. Domain:

The function is linear, so it is defined for all real numbers. Thus, the domain of $f(x)$ is: $\text{Domain: } (-\infty, \infty)$

3. Range:

Since this is a linear function without any restrictions, the y-values can also take any real number. Therefore, the range of $f(x)$ is: $\text{Range: } (-\infty, \infty)$

4. Graph:

The graph of $f(x) = -\frac{3}{2}x$ is a straight line with a downward slope that passes through the origin. Some key points on the graph:

• At $x = 0$, $f(x) = 0$ (the origin).
• At $x = 2$, $f(x) = -3$ (point $(2, -3)$).
• At $x = -2$, $f(x) = 3$ (point $(-2, 3)$).

These points will help you sketch the line.

Would you like me to plot this graph for you or provide more details? Let me know if you have any questions!

5 Related Questions:

1. What happens to the graph if the slope is changed to a positive number?
2. How would the graph change if we added a constant term to the function (e.g., $f(x) = -\frac{3}{2}x + 1$)?
3. What is the interpretation of the slope in the context of real-world problems?
4. How do we find the x-intercept and y-intercept of a linear function?
5. How can you determine if two lines are parallel or perpendicular based on their slopes?

Tip:

When graphing linear functions, always plot at least two points (often the intercepts) and use the slope to sketch the line accurately.