Graph the function and determine the interval(s) for which f (x) $\ge$ 0.f (x) = -x² - 2x E){-2}

Question

Quizplus · Accepted Answer

To graph the function, we can start by finding the x- and y- intercepts:
- To find the x-intercept, we set f(x) = 0 and solve for x:
0 = -x^2 - 2x
0 = -x(x + 2)
x = 0 or x = -2
So the x-intercepts are (0, 0) and (-2, 0).
- To find the y-intercept, we set x = 0 and evaluate:
f(0) = -(0)^2 - 2(0) = 0
So the y-intercept is (0, 0).

Next, we can identify the vertex of the parabola. The coefficient of the x^2 term is negative, indicating that the parabola opens downwards. The x-coordinate of the vertex is given by x = -b/2a = -(-2)/(2*(-1)) = 1, and the y-coordinate is f(1) = -1 - 2(1) = -3. So the vertex is (1, -3).

We can now sketch the graph of the function:

```
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     | \
     |  \
 ____|___\________
      -3  0   1
```

The parabola intersects the x-axis at x = 0 and x = -2, and it is below the x-axis in between. Therefore, the interval(s) for which f(x) ≥ 0 are (-∞, -2] and [0, ∞). The best choice is B.

Graph the Function and Determine the Interval(s) for Which F ≥\ge≥

Graph the Function and Determine the Interval(s) for Which F $\ge$