graph the parabola, then determine the domain and range. f(x)=x^2x-4x-4
assume you mean
y + 4 = x^2 -4x
y + 8 = x^2 - 4 x + 4
y + 8 = (x-2)^2
vertex at ( 2 , -8)
so range is y >/= - 8
domain is all real x
so do i write the domain as (-infinity,infinity)
not sure how to write the range you showed. It has to be in interval notation
would it be (infinity,-8)?
To graph the parabola, we first need to rewrite the equation in the standard form of a quadratic function. The given equation, f(x) = x^2x - 4x - 4, seems to have a typographical error. It is likely that you meant f(x) = x^2 - 4x - 4.
To graph the parabola, we will follow these steps:
Step 1: Rewrite the equation in standard form: f(x) = ax^2 + bx + c
In our case, a = 1, b = -4, and c = -4. So the equation becomes: f(x) = x^2 - 4x - 4.
Step 2: Determine the vertex of the parabola:
The x-coordinate of the vertex can be found using the formula: x = -b / (2a).
Substituting our values: x = -(-4) / (2*1) = 2
To find the corresponding y-coordinate, substitute the x-coordinate back into the original equation: f(2) = 2^2 - 4(2) - 4 = 4 - 8 - 4 = -8
So the vertex of the parabola is (2, -8).
Step 3: Find the x-intercepts (if any):
Set f(x) = 0 and solve for x: x^2 - 4x - 4 = 0.
Using factoring or the quadratic formula, we find x ≈ -0.77 and x ≈ 5.14.
Step 4: Find the y-intercept:
Set x = 0 and solve for f(x): f(0) = 0^2 - 4(0) - 4 = -4.
So the y-intercept is (0, -4).
Step 5: Plot the vertex, x-intercepts, and y-intercept on a coordinate plane.
Now that we have the necessary points, plot them on a coordinate plane.
Step 6: Draw the parabolic curve passing through the plotted points.
With the vertex, x-intercepts, and y-intercept, you can now sketch the parabolic curve. It should open upward since the coefficient of the x^2 term (a) is positive.
To determine the domain and range of the parabola:
The domain represents all possible x-values that the parabola can take. In this case, since the parabola is a continuous curve, the domain is all real numbers.
The range represents all possible y-values that the parabola can take. For this upward-opening parabola, the range is y ≥ -8, as y has a minimum value of -8 at the vertex.
Note: The graph of the parabola can provide a visual representation for a better understanding, but for the exact values and visualization, it's recommended to use graphing software or a graphing calculator.