Answer:
1) The roots of the equation Y = -x² - 12·x - 37 are x = -6 - √73 and -6 + √73
2) The factors of the function Y = -x² - 12·x - 37 are (x + (-6 - √73)) and (x - (-6 + √73))
3) The vertex is (-6, -1)
4) The Y-intercept is (0, -37)
5) The axis of symmetry is x = -6
6) The parabola opens downward
7) The maximum point is (-6, -1)
8) The graph of the function created with Microsoft Excel is attached
9) The domain of the function is -∞ < x < ∞
10) The range of the function is -∞ < Y ≤ -1
Step-by-step explanation:
The given function is Y = -x² - 12·x - 37
1) The roots of the equation Y = -x² - 12·x - 37 is given as follows;
0 = -x² - 12·x - 37
∴ -x² - 12·x - 37 = 0
The roots are;
x = (12 ± √((-12)² - 4 × 1 × (-37)))/(-2)
∴ x = -6 - √73 and -6 + √73
2) The factors are (x + (-6 - √73)) and (x - (-6 + √73))
3) The vertex is given by h = -b/(2·a) = 12/(2) = -6
k = c - b²/(4·a) = -37 - 144/(-4) = -1
The vertex = (h, k) = (-6, -1)
4) The intercept with the "Y" axis is given when x = 0, therefore, the y-intercept is (0, -37)
5) The axis of symmetry can be found at the vertex or the line x = -6
6) The sign of the coefficient of x² is negative therefore, the parabola opens downward
7) The maximum point is given as follows;
At the maximum point, dY/dx = d(-x² - 12·x - 37)/dx = -2·x - 12 = 0
x = 12/(-2) = -6, which is the vertex, therefore Y = -(-6)² - 12×(-6) - 37 = -1
The maximum point is the vertex (-6, -1)
8) Please find attached the graph of the function created with Microsoft Excel
9) The domain of the function is -∞ < x < ∞
10) The range of the function is -∞ < Y ≤ -1
