1. **State the problem:** We need to find the equation of the line graphed, which passes through points approximately $(-3, -2)$ and $(3, 4)$ and intersects the y-axis near $-2$. 2. **Formula used:** The equation of a line in slope-intercept form is $$y = mx + b$$ where $m$ is the slope and $b$ is the y-intercept. 3. **Calculate the slope $m$:** $$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - (-2)}{3 - (-3)} = \frac{4 + 2}{3 + 3} = \frac{6}{6} = 1$$ 4. **Check the y-intercept $b$:** The line crosses the y-axis at $y = -2$, so $b = -2$. 5. **Write the equation:** $$y = 1 \cdot x - 2 = x - 2$$ 6. **Compare with given options:** None of the options exactly match $y = x - 2$. However, the slope calculated from the points is $1$, but the options have slopes $\frac{2}{3}$ or $\frac{3}{2}$. Let's verify the slope carefully. 7. **Recalculate slope with exact points:** Points: $(-3, -2)$ and $(3, 4)$ $$m = \frac{4 - (-2)}{3 - (-3)} = \frac{6}{6} = 1$$ 8. **Check if the line passes through $y = -2$ at $x=0$:** Using point-slope form with point $(-3, -2)$: $$y - (-2) = 1(x - (-3))$$ $$y + 2 = x + 3$$ $$y = x + 1$$ This contradicts the y-intercept $-2$. 9. **Check the y-intercept from the line through points:** Using slope $1$ and point $(-3, -2)$: $$b = y - mx = -2 - 1 \times (-3) = -2 + 3 = 1$$ So the y-intercept is $1$, not $-2$. 10. **Check slope for options:** Try slope $\frac{2}{3}$: Using point $(-3, -2)$: $$b = y - mx = -2 - \frac{2}{3} \times (-3) = -2 + 2 = 0$$ Not matching $-2$. Try slope $\frac{3}{2}$: $$b = -2 - \frac{3}{2} \times (-3) = -2 + \frac{9}{2} = -2 + 4.5 = 2.5$$ No match. 11. **Check points for option A: $y = \frac{2}{3}x - 2$** At $x=3$: $$y = \frac{2}{3} \times 3 - 2 = 2 - 2 = 0$$ But the point is $(3,4)$, so no. 12. **Check points for option B: $y = \frac{2}{3}x + 3$** At $x=3$: $$y = 2 - 3 = 5$$ No. 13. **Check points for option C: $y = \frac{3}{2}x - 2$** At $x=3$: $$y = \frac{3}{2} \times 3 - 2 = 4.5 - 2 = 2.5$$ No. 14. **Check points for option D: $y = \frac{3}{2}x + 3$** At $x=3$: $$y = 4.5 + 3 = 7.5$$ No. 15. **Conclusion:** The closest match to the points $(-3, -2)$ and $(3, 4)$ is the line with slope $1$ and y-intercept $1$, which is not among the options. However, the problem states the line intersects the y-axis at about $-2$, so the best matching option is A: $$y = \frac{2}{3}x - 2$$ which has slope $\frac{2}{3}$ and y-intercept $-2$. **Final answer:** A