\( f(x)=2 x^{2}+16 x-10 \) (a) The function's vertex is at the point \( (-4,-42) \) (Type an ordered pair, using integers or fractions.) (b) Determine whether the parabola has a maximum value or a minimum value and find the value. Select the correct choice below and fill in the answer box to complete your choice. (Type an integer or a fraction) A. The parabola opens downward and has a maximum value of B. The parabola opens upward and has a minimum value of -42 (c) The range of \( f(x) \) is (Type your answer in interval notation. Use integers or fractions for any numbers in the expression )

To analyze the function \( f(x) = 2x^2 + 16x - 10 \), we can answer the questions step by step. ### (a) Vertex The vertex of the parabola is given as \( (-4, -42) \). This is correct as stated. ### (b) Maximum or Minimum Value The coefficient of \( x^2 \) in the function \( f(x) = 2x^2 + 16x - 10 \) is \( 2 \), which is positive. This means the parabola opens upward. Since the parabola opens upward, it has a minimum value at the vertex. The minimum value is the y-coordinate of the vertex, which is \( -42 \). So the correct choice is: **B. The parabola opens upward and has a minimum value of -42.** ### (c) Range of \( f(x) \) Since the parabola opens upward and has a minimum value of \( -42 \), the range of the function starts from \( -42 \) and goes to positive infinity. In interval notation, the range is: \[ [-42, \infty) \] ### Summary of Answers (a) Vertex: \( (-4, -42) \) (b) Minimum value: \( -42 \) (c) Range: \( [-42, \infty) \)