Question: Use the quadratic function $f(x) = -4x^2 + 9x + 8$ to answer the following questions. a) Use the vertex formula to determine the vertex. The vertex is __. (Type an ordered pair. Simplify your answer.) b) Does the graph "open up" or "open down"? Up Down c) What is the equation of the axis of symmetry? x = __ (Simplify your answer.) d) Find the y-intercept. Select the correct choice below and, if necessary, fill in the answer box within your choice. A. The y-intercept is __. (Type an integer or a fraction.) B. There is no y-intercept. e) Sketch the graph. Which of the following is the graph of $f(x) = -4x^2 + 9x + 8$? A. B. C. D. Graph shapes: top-left: A downward-opening parabola with its vertex above the x-axis, centered slightly left of the y-axis. top-right: A downward-opening parabola with its vertex above the x-axis, centered near the middle. center-left: A downward-opening parabola with its vertex above the x-axis, centered slightly right of the y-axis. center-right: An upward-opening parabola with its vertex below the x-axis, centered near the middle.

1. **State the problem:** We are given the quadratic function $$f(x) = -4x^2 + 9x + 8$$ and need to find the vertex, direction of opening, axis of symmetry, y-intercept, and identify the correct graph. 2. **Vertex formula:** The vertex of a parabola given by $$f(x) = ax^2 + bx + c$$ is at $$x = -\frac{b}{2a}$$. 3. **Calculate the x-coordinate of the vertex:** $$a = -4, \quad b = 9$$ $$x = -\frac{9}{2 \times (-4)} = -\frac{9}{-8} = \frac{9}{8}$$ 4. **Calculate the y-coordinate of the vertex:** $$f\left(\frac{9}{8}\right) = -4\left(\frac{9}{8}\right)^2 + 9\left(\frac{9}{8}\right) + 8$$ $$= -4 \times \frac{81}{64} + \frac{81}{8} + 8$$ $$= -\frac{324}{64} + \frac{81}{8} + 8$$ Simplify fractions: $$-\frac{324}{64} = -\frac{81}{16}$$ $$\frac{81}{8} = \frac{162}{16}$$ $$8 = \frac{128}{16}$$ Sum: $$-\frac{81}{16} + \frac{162}{16} + \frac{128}{16} = \frac{-81 + 162 + 128}{16} = \frac{209}{16}$$ 5. **Vertex:** $$\left(\frac{9}{8}, \frac{209}{16}\right)$$ 6. **Direction of opening:** Since $$a = -4 < 0$$, the parabola opens **down**. 7. **Axis of symmetry:** The axis of symmetry is the vertical line through the vertex's x-coordinate: $$x = \frac{9}{8}$$ 8. **Y-intercept:** The y-intercept is the value of $$f(0)$$: $$f(0) = -4(0)^2 + 9(0) + 8 = 8$$ So the y-intercept is $$8$$. 9. **Graph choice:** The parabola opens down, vertex is above the x-axis at $$x=\frac{9}{8} \approx 1.125$$, so the vertex is slightly right of the y-axis and above it. The correct graph is the one with a downward-opening parabola with vertex above x-axis centered slightly right of y-axis, which corresponds to **center-left** (option C). **Final answers:** - Vertex: $$\left(\frac{9}{8}, \frac{209}{16}\right)$$ - Opens: Down - Axis of symmetry: $$x = \frac{9}{8}$$ - Y-intercept: $$8$$ - Graph: C