Verified answer

Step-by-step explanation:

c. Direction of the opening of the graph:

- The coefficient of \(x^2\) term is positive (2 in this case), which means the graph opens upwards.

d. Axis of symmetry:

- The axis of symmetry for a quadratic function in the form \(y = ax^2 + bx + c\) is given by \(x = \frac{-b}{2a}\). In this case, \(a = 2\) and \(b = 20\), so the axis of symmetry is \(x = \frac{-20}{2(2)} = -5\).

e. Domain:

- The domain of a quadratic function is all real numbers, since it is defined for all possible values of \(x\).

f. Range:

- Since the graph opens upwards, the minimum value occurs at the vertex. The range will be all real numbers greater than or equal to the y-coordinate of the vertex.

To find the y-coordinate of the vertex, substitute \(x = -5\) into the equation:

\(y = 2(-5)^2 + 20(-5) + 50 = 50 - 100 + 50 = 0\).

So, the range is all real numbers \((-\infty, \infty)\).

h. Sketching the graph:

- I can't draw graphs directly, but I can describe it to you. Since the graph opens upwards and the vertex is at (-5, 0), it will be a parabola that touches the x-axis at \(x = -5\) and extends upwards.