1. **State the problem:** We need to analyze the rational function $$y=\frac{-3x^2+1}{2x^2-4x-5}$$. 2. **Recall the formula and rules:** This is a rational function, which is a ratio of two polynomials. Important points include finding domain restrictions (where denominator is zero), intercepts, and behavior. 3. **Find the domain:** Set denominator equal to zero: $$2x^2 - 4x - 5 = 0$$ Use quadratic formula: $$x = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 2 \cdot (-5)}}{2 \cdot 2} = \frac{4 \pm \sqrt{16 + 40}}{4} = \frac{4 \pm \sqrt{56}}{4} = \frac{4 \pm 2\sqrt{14}}{4} = 1 \pm \frac{\sqrt{14}}{2}$$ So domain excludes $$x = 1 + \frac{\sqrt{14}}{2}$$ and $$x = 1 - \frac{\sqrt{14}}{2}$$. 4. **Find x-intercepts:** Set numerator equal to zero: $$-3x^2 + 1 = 0 \implies 3x^2 = 1 \implies x^2 = \frac{1}{3} \implies x = \pm \frac{1}{\sqrt{3}} = \pm \frac{\sqrt{3}}{3}$$ 5. **Find y-intercept:** Set $$x=0$$: $$y = \frac{-3(0)^2 + 1}{2(0)^2 - 4(0) - 5} = \frac{1}{-5} = -\frac{1}{5}$$ 6. **Summary:** - Domain: all real numbers except $$x = 1 \pm \frac{\sqrt{14}}{2}$$ - x-intercepts: $$x = \pm \frac{\sqrt{3}}{3}$$ - y-intercept: $$y = -\frac{1}{5}$$ This completes the analysis of the given rational function.