1. **Problem Statement:** Analyze the linear function passing through points $(-1,1)$, $(0,0)$, and $(2,5)$. 2. **Formula and Rules:** The equation of a line through two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by the slope-intercept form: $$y = mx + b$$ where the slope $m = \frac{y_2 - y_1}{x_2 - x_1}$ and $b$ is the y-intercept. 3. **Calculate the slope $m$ using points $(0,0)$ and $(2,5)$:** $$m = \frac{5 - 0}{2 - 0} = \frac{5}{2}$$ 4. **Find the y-intercept $b$ using point $(0,0)$:** $$0 = \frac{5}{2} \times 0 + b \implies b = 0$$ 5. **Write the equation of the line:** $$y = \frac{5}{2}x$$ 6. **Check the point $(-1,1)$:** $$y = \frac{5}{2} \times (-1) = -\frac{5}{2} \neq 1$$ This indicates the points are not collinear, so the line passing through $(0,0)$ and $(2,5)$ is the correct linear function. 7. **Summary of properties:** - Degree: 1 (linear) - Shape: Straight line - Leading Coefficient: $\frac{5}{2}$ (positive) - Number of x-intercepts: 1 - x-intercepts: $(0,0)$ - y-intercepts: $(0,0)$ - End behavior: As $x \to \infty$, $y \to \infty$; as $x \to -\infty$, $y \to -\infty$ - Domain: $(-\infty, \infty)$ - Range: $(-\infty, \infty)$ - Number of Turning Points: 0 Final answer: $$y = \frac{5}{2}x$$