This is one of the problems submitted. Reference to which one is deliberately removed to prevent easy copying.
a) Find the vertex.
b) Determine whether there is a maximum or minimum value and find that value.
c) Find the range.
d) Find the intervals on which the function is increasing and the intervals
on which the function is decreasing.
f(x) = -2x2 - 24x - 64
| a) Find the vertex. | |
The vertex is found using the formula for (x,y): -b -b ( ---- , f(----) ) 2a 2a | |
X: -b -(-24) 24 ---- = ------- = ---- = -6 2a 2(-2) -4 |
Y: f(x) = -2x2 - 24x - 64 f(-6) = -2(-6)2 - 24(-6) - 64 f(-6) = -2(36) - (-144) - 64 f(-6) = -72 + 144 - 64 f(-6) = 8 |
The vertex is (-6, 8). |
|
| b) Determine whether there is a maximum or minimum value and find that value. |
| In the parabola equation, ax2 + bx +c, if a is negative, the parabola has a maximum. If it is positive, the parabola has a minimum. (If a=0, the equation is a line, not a parabola.) |
For the equation: f(x) = -2x2 - 24x - 64 a=-2 As a is negative, the parabola has a maximum. The maximum is the Y value from the vertex, found above: 8. |
The maximum y is 8. This happens when x=-6. |
| c) Find the range. |
Given the vertex of (-6,8), the range is: (-∞,8] |
| d) Find the intervals on which the function is increasing and the intervals on which the function is decreasing. |
Given the vertex of (-6,8): The function is increasing on: (-∞,-6) The function is decreasing on: (-6,∞) |