For the function below, (a) find the vertex; (b) find the axis of symmetry; (c) determine whether there is a maximum or a minimum value and find that value; and (d) graph the function. \( f(x)=-5 x^{2}-5 x+2 \) (b) The axis of symmetry is \( x=-\frac{1}{2} \). (Type an equation. Use integers or fractions for any numbers in the equation.) (c) Does \( f(x) \) have a maximum or a minimum value? The parabola has a minimum. The parabola has a maximum. The maximum/minimum value of \( f(x) \) is \( \frac{13}{4} \). (Type an integer or a fraction.) (d) Use the graphing tool to graph the function. Click to

We are given the quadratic function   f(x) = –5x² – 5x + 2. Step (a): Find the vertex. • The x-coordinate of the vertex is found using x = –b/(2a). Here, a = –5 and b = –5:   x = –(–5) / (2(–5)) = 5/–10 = –½. • Substitute x = –½ into f(x) to find the y-coordinate:   f(–½) = –5(–½)² – 5(–½) + 2 = –5·(¼) + (5/2) + 2.   Write all terms with denominator 4: –5/4 + (10/4) + (8/4) = (–5 + 10 + 8)/4 = 13/4. • Thus, the vertex is ( –½, 13/4 ). Step (b): Find the axis of symmetry. • The axis of symmetry is the vertical line through the vertex:   x = –½. Step (c): Determine whether there is a maximum or a minimum value and find that value. • Because the coefficient a = –5 is negative, the parabola opens downward, and the vertex represents the maximum point. • Therefore, f(x) has a maximum value of 13/4 (which occurs at x = –½). Step (d): Graphing the function. • Plot the vertex ( –½, 13/4 ). • Plot the axis of symmetry x = –½. • Identify the y-intercept: f(0) = 2, so (0, 2) is on the graph. • Find the x-intercepts by setting f(x) = 0:   –5x² – 5x + 2 = 0  → Multiply both sides by –1: 5x² + 5x – 2 = 0.   Solve using the quadratic formula:    x = [–5 ± √(5² – 4·5·(–2))] / (2·5)       = [–5 ± √(25 + 40)] / 10       = [–5 ± √65] / 10. • Draw a smooth downward-opening parabola through the vertex, symmetric about x = –½, and mark these intercepts. Summary Answers: (a) Vertex: ( –½, 13/4 ) (b) Axis of symmetry: x = –½ (c) The parabola has a maximum value, which is 13/4. (d) The graph is a downward-opening parabola with the vertex ( –½, 13/4 ), y-intercept (0, 2), and x-intercepts at x = (–5 ± √65)/10.