-3x^2+12y^=84
What are the domain and range
My answer
domain: all real numbers
range:
y (less than -) -sqrt7 or sqrt7
-3x^2+12y^=84
3x^2 - 12y^2 = -84
x^2 /28 - y^2 /7 = -1
You have a vertical hyperbola, with vertices
(0,√7) and (0,-√7)
your answer is correct
thank you
Well, the domain is indeed all real numbers, so you got that right! As for the range, I'm afraid your answer went off on a bit of a tangential journey.
To find the range, we can rearrange the equation to solve for y:
12y = 3x^2 + 84
y = (3/12)x^2 + 7
y = (1/4)x^2 + 7
Now, let's analyze the quadratic equation y = (1/4)x^2 + 7. Since the coefficient in front of x^2 is positive (1/4), the parabola will open upwards. This means that the lowest point on the parabola (or vertex) will be the minimum value of y.
Since the coefficient in front of x^2 is (1/4), the minimum value of y will be at the vertex, which occurs at x = 0. Plugging x = 0 into the equation, we find that y = 7.
So, the range of the equation is y ≥ 7, which means y is greater than or equal to 7.
Now, I hope I didn't make your mathematical journey seem too much like a circus act!
To determine the domain and range of the equation -3x^2 + 12y^2 = 84, let's solve the equation and analyze the variables.
First, let's solve the equation for y:
-3x^2 + 12y^2 = 84
Rearranging the equation:
12y^2 = 3x^2 + 84
Divide both sides by 12:
y^2 = (1/4)x^2 + 7
Taking the square root of both sides:
y = ± √((1/4)x^2 + 7)
Now, let's analyze the variables.
Domain:
Since there are no restrictions on the x-values, the domain is all real numbers.
Range:
The y-values will vary as the x-values vary. Looking at the equation y = ± √((1/4)x^2 + 7), we can see that the y-values will be real numbers, both positive and negative.
Therefore, the range is all real numbers.
In conclusion:
Domain: all real numbers
Range: all real numbers
To find the domain and range of the given equation -3x^2 + 12y = 84, we need to understand the definitions of domain and range.
The domain refers to the set of possible values for the independent variable, which in this case is 'x'. Since there are no restrictions or limitations mentioned in the equation, the domain is indeed all real numbers.
The range represents the set of possible values for the dependent variable, which here is 'y'. To determine the range, we need to isolate 'y' in the equation:
-3x^2 + 12y = 84
First, let's simplify the equation by dividing both sides by 12:
-1/4x^2 + y = 7
Next, isolate 'y' by subtracting -1/4x^2 from both sides:
y = -1/4x^2 + 7
From this form, we can observe that the term -1/4x^2 is the squared term, and since a squared term can never be negative, the range of 'y' is all real numbers greater than or equal to 7.
So, in conclusion:
Domain: All real numbers
Range: y ≥ 7