For the quadratic function \( f(x)=x^{2}-2 x-8 \), answer parts (a) through (c) (a) Graph the quadratic function by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, \( y \)-intercept, and \( x \)-intercepts, if any.

We are given f(x) = x² – 2x – 8. Let’s find the key features step by step. 1. Opening Direction:  • The coefficient of x² is 1 (which is positive), so the parabola opens upward. 2. Vertex:  • The x-coordinate of the vertex is found using –b/(2a). Here, a = 1 and b = –2:   x = –(–2)/(2·1) = 2/2 = 1.  • Now substitute x = 1 into f(x) to find the y-coordinate:   f(1) = (1)² – 2·1 – 8 = 1 – 2 – 8 = –9.  • So, the vertex is (1, –9). 3. Axis of Symmetry:  • The axis of symmetry is the vertical line through the vertex, which is x = 1. 4. y-intercept:  • The y-intercept is found by evaluating f(0):   f(0) = (0)² – 2·0 – 8 = –8.  • Therefore, the y-intercept is (0, –8). 5. x-intercepts:  • To find the x-intercepts, solve f(x) = 0:   x² – 2x – 8 = 0.  • Compute the discriminant:   D = (–2)² – 4·1·(–8) = 4 + 32 = 36.  • Use the quadratic formula:   x = [–(–2) ± √36] / (2·1) = [2 ± 6] / 2.   This gives two solutions:    x = (2 + 6)/2 = 8/2 = 4 and x = (2 – 6)/2 = –4/2 = –2.  • Thus, the x-intercepts are (4, 0) and (–2, 0). Summary:  – The graph opens upward.  – The vertex is at (1, –9).  – The axis of symmetry is the line x = 1.  – The y-intercept is (0, –8).  – The x-intercepts are (–2, 0) and (4, 0). These features allow you to sketch the parabola accurately.