1. **State the problem:** We are given the quadratic function $$f(x) = -2x^2 + 5x + 3$$ and asked to complete the table for the domain $$-2 \leq x \leq 5$$, graph the function including the axis of symmetry, and find the roots of the quadratic equation. 2. **Recall the quadratic function formula:** $$f(x) = ax^2 + bx + c$$ where $$a = -2$$, $$b = 5$$, and $$c = 3$$. 3. **Complete the table:** For each $$x$$ value, calculate $$f(x)$$ by substituting into the formula. - For $$x = -2$$: $$f(-2) = -2(-2)^2 + 5(-2) + 3 = -2(4) - 10 + 3 = -8 - 10 + 3 = -15$$ - For $$x = -1$$: $$f(-1) = -2(1) + 5(-1) + 3 = -2 - 5 + 3 = -4$$ - For $$x = 0$$: $$f(0) = 0 + 0 + 3 = 3$$ - For $$x = 1$$: $$f(1) = -2(1) + 5(1) + 3 = -2 + 5 + 3 = 6$$ - For $$x = 2$$: $$f(2) = -2(4) + 10 + 3 = -8 + 10 + 3 = 5$$ - For $$x = 3$$: $$f(3) = -2(9) + 15 + 3 = -18 + 15 + 3 = 0$$ - For $$x = 4$$: $$f(4) = -2(16) + 20 + 3 = -32 + 20 + 3 = -9$$ - For $$x = 5$$: $$f(5) = -2(25) + 25 + 3 = -50 + 25 + 3 = -22$$ 4. **Axis of symmetry:** The axis of symmetry for a quadratic $$ax^2 + bx + c$$ is given by $$x = -\frac{b}{2a}$$. Calculate: $$x = -\frac{5}{2(-2)} = -\frac{5}{-4} = \frac{5}{4} = 1.25$$ So the axis of symmetry is $$x = 1.25$$. 5. **Find the roots:** Solve $$-2x^2 + 5x + 3 = 0$$ using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Calculate the discriminant: $$\Delta = b^2 - 4ac = 5^2 - 4(-2)(3) = 25 + 24 = 49$$ Calculate roots: $$x = \frac{-5 \pm \sqrt{49}}{2(-2)} = \frac{-5 \pm 7}{-4}$$ - For $$+$$ sign: $$x = \frac{-5 + 7}{-4} = \frac{2}{-4} = -0.5$$ - For $$-$$ sign: $$x = \frac{-5 - 7}{-4} = \frac{-12}{-4} = 3$$ **Final answers:** - Completed table values match the calculations above. - Axis of symmetry: $$x = 1.25$$ - Roots: $$x = -0.5$$ and $$x = 3$$