1. **Problem Statement:** Determine the key characteristics of a linear function with a negative slope passing through the origin. 2. **Parent Function:** The linear function is generally given by the formula $$y = mx + b$$ where $m$ is the slope and $b$ is the y-intercept. 3. **Given Information:** The line passes through the origin $(0,0)$, so $b=0$. The slope $m$ is negative, indicating the line descends from top-left to bottom-right. 4. **Domain:** For any linear function, the domain is all real numbers, written as $$(-\infty, \infty)$$. 5. **Range:** Since the line extends infinitely in both vertical directions, the range is also $$(-\infty, \infty)$$. 6. **Increasing/Decreasing Intervals:** Because the slope $m$ is negative, the function is decreasing on the entire domain, so: - Increasing: none - Decreasing: $$(-\infty, \infty)$$ 7. **Maximum and Minimum:** A linear function with a negative slope has no maximum or minimum values because it extends infinitely in both directions. 8. **Intercepts:** - X-intercept: The point where $y=0$. Since the line passes through the origin, the x-intercept is $(0,0)$. - Y-intercept: The point where $x=0$. This is also $(0,0)$. **Final summary:** - Domain: $$(-\infty, \infty)$$ - Range: $$(-\infty, \infty)$$ - Increasing: none - Decreasing: $$(-\infty, \infty)$$ - Maximum: none - Minimum: none - X-intercept(s): $(0,0)$ - Y-intercept: $(0,0)$