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. Does the graph of \( f \) open up or down? up down

To analyze the quadratic function \( f(x) = x^2 - 2x - 8 \), we will determine whether the graph opens up or down, find the vertex, axis of symmetry, \( y \)-intercept, and \( x \)-intercepts. ### (a) Determine if the graph opens up or down The coefficient of \( x^2 \) in the quadratic function \( f(x) = ax^2 + bx + c \) is \( a = 1 \). Since \( a > 0 \), the graph opens **up**. ### Vertex The vertex of a quadratic function in the form \( f(x) = ax^2 + bx + c \) can be found using the formula for the x-coordinate of the vertex: \[ x = -\frac{b}{2a} \] Here, \( a = 1 \) and \( b = -2 \): \[ x = -\frac{-2}{2 \cdot 1} = \frac{2}{2} = 1 \] Now, substitute \( x = 1 \) back into the function to find the y-coordinate of the vertex: \[ f(1) = (1)^2 - 2(1) - 8 = 1 - 2 - 8 = -9 \] Thus, the vertex is \( (1, -9) \). ### Axis of Symmetry The axis of symmetry is the vertical line that passes through the vertex. It is given by the equation: \[ x = 1 \] ### \( y \)-intercept To find the \( y \)-intercept, we evaluate \( f(0) \): \[ f(0) = (0)^2 - 2(0) - 8 = -8 \] Thus, the \( y \)-intercept is \( (0, -8) \). ### \( x \)-intercepts To find the \( x \)-intercepts, we set \( f(x) = 0 \): \[ x^2 - 2x - 8 = 0 \] We can factor this quadratic: \[ (x - 4)(x + 2) = 0 \] Setting each factor to zero gives: \[ x - 4 = 0 \quad \Rightarrow \quad x = 4 \] \[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \] Thus, the \( x \)-intercepts are \( (4, 0) \) and \( (-2, 0) \). ### Summary of Results - The graph opens **up**. - Vertex: \( (1, -9) \) - Axis of symmetry: \( x = 1 \) - \( y \)-intercept: \( (0, -8) \) - \( x \)-intercepts: \( (4, 0) \) and \( (-2, 0) \) You can now graph the function using this information. The vertex is the lowest point on the graph, and the intercepts will help you plot the curve accurately.