graph the parabola, then determine the domain and range. f(x)=x^2x-4x-4

domain: any real x

range: y ≥ the y value of the vertex

To graph the parabola and determine its domain and range, we need to follow a few steps:

Step 1: Rewrite the equation of the parabola in the standard form: f(x) = ax^2 + bx + c.
Given f(x) = x^2x - 4x - 4, it seems there is a mistake in the equation. Please double-check the equation and correct it if needed.

Assuming the correct equation is f(x) = x^2 - 4x - 4, we can move on to the next steps.

Step 2: Determine the vertex of the parabola.
The vertex of a parabola with the equation f(x) = ax^2 + bx + c can be found using the formula x = -b / (2a).
In our case, a = 1, and b = -4. Plugging these values into the formula, we find x = -(-4) / (2 * 1) = 4 / 2 = 2.
So the x-coordinate of the vertex is 2.

To find the y-coordinate of the vertex, we substitute the x-value (2) into the equation:
f(2) = 2^2 - 4(2) - 4
f(2) = 4 - 8 - 4
f(2) = -8

Thus, the vertex of the parabola is (2, -8).

Step 3: Find the axis of symmetry.
The axis of symmetry is a vertical line that passes through the vertex, so its equation is x = 2.

Step 4: Determine other key points.
To find additional points on the parabola, we substitute different x-values into the equation and calculate the corresponding y-values.

For example, let's use x = 0:
f(0) = 0^2 - 4(0) - 4
f(0) = 0 - 0 - 4
f(0) = -4

So we have the point (0, -4) on the parabola.

Step 5: Plot the points and sketch the parabola.
Using the information gathered in steps 2, 4, and 5, we can now plot the points (2, -8) and (0, -4) on a graph and sketch the parabola passing through these points. Make sure to include the axis of symmetry x = 2.

Step 6: Determine the domain.
The domain of a parabola is the set of all possible x-values. In this case, there are no restrictions on x, so the domain is (-∞, ∞), which means all real numbers.

Step 7: Determine the range.
The range of a parabola is the set of all possible y-values. Since the vertex of this parabola is at the lowest point (2, -8), the range starts from -8 and goes up towards positive infinity. Thus, the range of the function f(x) = x^2 - 4x - 4 is [-8, ∞).

By following these steps, you should be able to graph the parabola, determine its domain, and find its range.