1. **Problem statement:** Given the quadratic function $m(x) = 3x^2 + 9x - 5$, answer the following questions about its graph. 2. **Formula and rules:** A quadratic function $ax^2 + bx + c$ graphs as a parabola. - If $a > 0$, the parabola opens upward. - If $a < 0$, it opens downward. - The vertex is at $x = -\frac{b}{2a}$. - The axis of symmetry is the vertical line through the vertex. - The y-intercept is $m(0) = c$. - The x-intercepts are solutions to $3x^2 + 9x - 5 = 0$. - The minimum or maximum value is the y-value of the vertex. 3. **(a) Direction of opening:** Since $a = 3 > 0$, the parabola opens **upward**. 4. **(b) Vertex:** Calculate $x$-coordinate: $$x = -\frac{b}{2a} = -\frac{9}{2 \times 3} = -\frac{9}{6} = -\frac{3}{2}$$ Calculate $y$-coordinate: $$m\left(-\frac{3}{2}\right) = 3\left(-\frac{3}{2}\right)^2 + 9\left(-\frac{3}{2}\right) - 5 = 3\times \frac{9}{4} - \frac{27}{2} - 5 = \frac{27}{4} - \frac{54}{4} - \frac{20}{4} = \frac{27 - 54 - 20}{4} = -\frac{47}{4}$$ Vertex is at $$\left(-\frac{3}{2}, -\frac{47}{4}\right)$$. 5. **(c) x-intercepts:** Solve $3x^2 + 9x - 5 = 0$ using quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-9 \pm \sqrt{81 - 4 \times 3 \times (-5)}}{2 \times 3} = \frac{-9 \pm \sqrt{81 + 60}}{6} = \frac{-9 \pm \sqrt{141}}{6}$$ So the x-intercepts are: $$x = \frac{-9 + \sqrt{141}}{6} \quad \text{and} \quad x = \frac{-9 - \sqrt{141}}{6}$$ 6. **(d) y-intercept:** Evaluate $m(0)$: $$m(0) = 3 \times 0^2 + 9 \times 0 - 5 = -5$$ So the y-intercept is at $(0, -5)$. 7. **(e) Sketch:** The parabola opens upward with vertex at $\left(-\frac{3}{2}, -\frac{47}{4}\right)$, crosses y-axis at $(0,-5)$, and crosses x-axis at the two points above. 8. **(f) Axis of symmetry:** The axis of symmetry is the vertical line through the vertex: $$x = -\frac{3}{2}$$ 9. **(g) Minimum or maximum:** Since parabola opens upward, vertex is a minimum point. Minimum value is the y-coordinate of vertex: $$m_{min} = -\frac{47}{4}$$ 10. **(h) Domain and range:** - Domain of any quadratic is all real numbers: $$(-\infty, \infty)$$ - Range is all $y$ values greater than or equal to minimum: $$\left[-\frac{47}{4}, \infty\right)$$